Mathematics Notes

General Mathematics

Existence proof

[2021-09-27 Mon 20:43] One kind of the importance theorem in mathematics is the existence theorem which states something with certain property exists. Basically, there are two kinds of proof of the existence theorem.

• Construction. E.g. existence of smooth but not differentiable function.
• Proof by contradiction. E.g. infiniteness of prime numbers, Brouwer fixed point theorem.

Flavors of mathematics

[2021-10-15 Fri 17:44] I want to quote the word of Charles Fefferman to show my flavors of mathematics.

Mathematics at the highest level has several flavors. On seeing it, one might say

1. What amazing technical power!
2. What a grand synthesis!
3. How could anyone not have seen this before?
4. Where on earth did this come from?

In my opinion, the proof of Fermat’s last theorem by Wiles of kind 1; Langlands conjecutre is of kind 2; the mirror symmetry conjecture or its computation on Calabi-Yau quintics is of kind 3; Tao and Green’s proof of the primes contain arbitrarily long arithmetic progressions of kind 4.

Some advices

[2021-12-28 Tue 17:44] From Vakil’s page on advice for listening to talks.

1. a definition you want to remember (e.g. “a K3 surface is…")
2. a theorem you want to remember (“the moduli space of polarized K3 surfaces is smooth”)
3. a motivating or key example (“a quartic is an example of a K3 surface”)
4. a motivating problem (“why are all moduli spaces of polarized K3 surfaces the same dimension?")
5. a question you want to ask the speaker (“why is that hypothesis in your theorem?")
6. a question you want to ask someone else (a definition, motivation, a question about a connection etc.)
7. anything else of a similar flavor: something specific that made you think. Something vague (“I liked the part where she talked about groups”) does not count as a “thing”.

From Kedlaya’s page, One important point to keep in mind is that unlike at earlier stages in one’s education, in graduate school and beyond one very often learns mathematics “in reverse”. That is, it is often necessary to work through a particular piece of mathematics before one is familiar with its logical prerequisites; the switch from forward to reverse study of mathematics is quite a difficult transition (at least according to my own experience as a student).

Category Theory

TODO A_∞ algebra

TODO A_∞ category

Abelian

[2021-12-19 Sun 20:59] A category is abelian if it is Exact and Additive.

Additive

[2021-12-19 Sun 20:53] A category \(\mathcal{C}\) is called additive if the following conditions hold.

1. For any two object \(\Hom(X,Y)\) has an abelian group structure and compositions are bilinear.
2. There is a zero object, denoted by \(0_{\mathcal{C}}\).
3. For any two objects \(X, Y\), the direct sum \(X \oplus Y\) exists in \(\mathcal{C}\).

A functor is called additive if it induces group homomorphism on homomorphisms.

Adjoint

[2021-12-19 Sun 20:57] Let \(F: \mathcal{C} \to \mathcal{D}\) and \(G: \mathcal{D} \to \mathcal{C}\) two functors. We say that \(G\) is right adjoint to \(F\) and \(F\) is left adjoint to \(G\) if there is an isomorphism \[\Hom_{\mathcal{D}}(F(X), Y) \cong \Hom_{\mathcal{C}}(X, G(Y))\]

Anodyne extension

[2021-12-17 Fri 20:09] Anodyne extensions denote the class of morphisms \(\Lambda_{k}^{n} \subset \Delta^{n}\).

Referenced: Kan fibration.

Associativity constraint

[2021-09-25 Sat 19:10] For \(\mathscr{C}\) category with functor \(\otimes\), an associativity constraint is a functorial isomorphism \[\phi: X \otimes (Y \otimes Z) \to (X \otimes Y) \otimes Z\] satisfying pentagon identity.

Referenced: Monoidal category, Tensor category, Rigid, Tensor functor.

TODO Bialgebra

Category

[2021-12-19 Sun 18:49] A category \(\mathcal{C}\) consists of the following data

1. A collection of objects.
2. For every pair of objects \(X, Y \in \mathcal{C}\), a set \(\Hom_{\mathcal{C}}(X,Y)\) of morphisms from \(X\) to \(Y\).
3. For every object \(X \in \mathcal{C}\), an identity morphism \(\id_{X} \in \Hom_{\mathcal{C}}(X, X)\).
4. For every triple of objects, a composition map \[\Hom_{\mathcal{C}}(X, Y) \times \Hom_{\mathcal{C}}(Y,Z) \to \Hom_{\mathcal{C}}(X, Z)\]
5. We have for every morphism \(f: X \to Y\), \(\id_{Y} \circ f = f = f \circ \id_{X}\).
6. The composition satisfies the associativity condition.

I will use symbol like \(\cat{Cat}\) to denote categories.

Classifying space

[2021-12-12 Sun 21:54] For \(\mathscr{C}\) a small category, the classifying space (or nerve) \(B\mathscr{C}\) isthe Simplicial set with \(B \mathscr{C} = \hom_{\cat{Cat}}(\bm{n}, \mathscr{C})\).

A standard example is the \(BG\), where \(G\) is a group and denoted by the same notion corresponding Groupoid, with one object and morphism \(\Hom(*, *) = G\) and \(\abs{BG}\) is the Eilenberg-Mac Lane space \(K(G,1)\).

Cocomplete

[2021-10-07 Thu 16:47] A category is called cocomplete if all Colimits exist.

Referenced: Category CGWH.

TODO Cofiber

Colimit

[2021-10-07 Thu 16:40] Let \(C\) be an ∞-category and let \(p: K \to C\) be an arbitrary map of simplicial sets. A colimit for \(p\) is an initial object of \(C_{p/}\). A limit for \(p\) is a final object of \(C_{/p}\).

Referenced: Cocomplete, Compact.

Commutativity constraint

[2021-09-30 Thu 13:58] The commutativity constraint is \(\psi_{X,Y}: X \otimes Y \to Y \otimes X\) for all \(X, Y\) such that \[\psi_{Y,X} \circ \psi_{X,Y} = \id_{X \otimes Y}\] with some compatibility condition.

Referenced: Tensor category, Rigid, Tensor functor.

Comodule

[2021-10-08 Fri 14:05] Let \(A\) be a Hopf algebra. Then \(V\) with \(k\)-linear map \(\rho: V \to V \otimes A\) satisfying \(V \stackrel{\rho}{\to} V \otimes A \stackrel{\id \otimes \epsilon}{\to} V \otimes k \cong V\) is identity and \(V \stackrel{\rho}{\to} V \otimes A \overset{\rho \otimes \id}{\underset{\id \otimes \Delta}{\rightrightarrows}} V \otimes A \otimes A\) are same.

Compact

[2021-11-29 Mon 18:45] An object \(A\) in category \(\mathscr{C}\) is called compact if the functor \(\Hom(A, -): \mathscr{C} \to \cat{Set}\) preserves the Colimit.

Complete

[2021-10-07 Thu 16:47] A category is called complete if all Limits exist.

Referenced: Category CGWH.

Complex

[2021-12-19 Sun 21:00] A complex of Abelian category \(\mathcal{A}\) is a chain \[A^{\bullet}: \cdots \to A^{-1} \xrightarrow{d^{-1}} A^{0} \xrightarrow{d^{0}} A^{1} \to \cdots\] where the differential satisfies \(d^{k+1} \circ d^{k} = 0\). A morphism between \(A^{\bullet}\) and \(B^{\bullet}\) is a set of morphisms between \(A^{k}\) and \(B^{k}\) commutes with differential map. The category of complexes is denoted by \(C(\mathcal{A})\).

The cohomology of the complex id \(H^{k}(A^{\bullet}) = \frac{\ker d^{k}}{\im d^{k - 1}}\).

We introduce the following associated truncated complexes by \[\tau_{\leq m}A^{\bullet}: \to A^{m -1} \xrightarrow{d^{m - 1}} \ker d^{m} \to 0 \to \] \[\tau^{\leq m}A^{\bullet}: \to A^{m -1} \xrightarrow{d^{m - 1}} A^{m} \xrightarrow{d^{m}} \im d^{m} \to 0 \to\] We also define \[\tau_{\geq m}A^{\bullet} = A^{\bullet}/ \tau_{m -1} A^{\bullet}\] \[\tau^{\geq m}A^{\bullet} = A^{\bullet}/ \tau^{m -1} A^{\bullet}\]

We have shift functor \(X[1]^{i} = X^{i + 1}\).

Acyclic

[2021-12-19 Sun 21:03] A complex \(A^{\bullet}\) is called acyclic if \(H^{k}(A^{\bullet}) = 0\) for all \(k \in \ZZ\).

Mapping cone

[2021-12-19 Sun 21:12] Let \(u: X^{\bullet} \to Y^{\bullet}\) be a morphism. The mapping cone is the complex given by \(C_{u}^{\bullet} = Y^{\bullet} \oplus (X^{\bullet}[1])\), where \(d_{u}(y, x) = (\dif y + u(x), - \dif x)\). It also denoted by \({\rm Cone}(u)\).

Quasi-isomorphism

[2021-12-19 Sun 21:04] A morphism \(u: X^{\bullet} \to Y^{\bullet}\) is a quasi-isomorphism if it induces isomorphism on cohomology groups.

Homotopy

[2021-12-19 Sun 21:05] Two complex morphism \(u,v: X^{\bullet} \to Y^{\bullet}\) is called homotopic if there exists a homotopy \(h \in \Hom^{-1}(X^{\bullet}, Y^{\bullet})\) such that \(u - v = \dif_{Y} h + h \dif_{X}\).

Two complexes are homotopy equivalent if there exists \(f: X \to Y\) and \(g: Y \to X\) such that \(v \circ u \sim \id_{X}\) and \(u \circ v \sim \id_{Y}\).

Cone

[2021-12-25 Sat 21:31] The left cone \(K^{\vartriangleleft}\) is defined to be \(\Delta^{0} \star K\). And dually, the right cone \(K^{\vartriangleright}\). Either cone contains a distinguished vertex which we will refer to as the cone point.

Cylinder object

[2021-12-18 Sat 17:34] A cylinder boject for an object \(A\) in a Closed model category is the commutative triangle

where \(\nabla\) is the canonical fold map identity on each component.

Referenced: Left homotopy, Path object.

Derived category

[2021-10-15 Fri 20:20] The derived category of the Abelian category \(\mathcal{A}\) is the Triangulated category \(D(\mathcal{A})\) obtained from the Homotopical category \(K(\mathcal{A})\) by localization with respect to the multiplicative system formed by all the quasi-isomorphisms in \(K(\mathcal{A})\).

We have category \(D^{b}(\mathcal{A})\) (resp. \(D^{+}(\mathcal{A}), D^{-}(\mathcal{A})\)) equivalent to the full triangulated subcategory of \(D(\mathcal{A})\) consisting of objects \(X\) such that \(H^{n}(X) = 0\) for all \(\abs{n} \gg 0\) (resp. \(n \ll 0, n \gg 0\)).

Derived functor

[2021-10-15 Fri 20:20] If \(F\) is left exact, we define the right derived functor to be \(RF\) making the diagram commutes and satisfying the universality property.

Similarly, we have left derived functor.

Drinfeld double

[2021-11-14 Sun 14:15] Let \((H, H^{\vee}, \langle , \rangle)\) Dual pair of Hopf algebra, and let \(H^{0}\) be he Hopf algebra \(H^{\vee}\) with the opposite comultiplication. Then \(D(H) = H \otimes H^{0}\) has an algebra structure uniquely determined by \[\Delta_{D}(a)( R) = R \Delta_{D}^{op}(a)\] for any \(a \in D(H)\) and \(R = \sum_{i} (e_{i} \otimes 1_{H^{0}}) \otimes (1_{H} \otimes e^{i})\), where \(e_{i}\) is basis in \(H\) and \(e^{i}\) basis in \(H^{\vee}\). Such construction is called Drinfeld double.

Dual

[2021-09-30 Thu 14:46] We define the dual to be \(X^{\vee} = \ihom(X, U)\), where \(U\) is the Identity object.

Cf. Tensor category.

Referenced: Reflexive.

Essentially surjective

[2021-12-26 Sun 15:20] For Simplicial set, a functor is essentially surjective if the induced functor on Homotopy categories is essentially surjective. Similarly statement holds for fully faithful.

Exact

[2021-12-19 Sun 20:52] A category is exact if there are Zero object, all the morphisms have kernels and cokernels, and \({\rm Coim}(f) \to {\rm Im}(f)\) is an isomorphism.

F-acyclic

[2021-12-19 Sun 21:42] Let \(F: \mathcal{A} \to \mathcal{B}\) be a left Derived functor. An object \(X\) in \(\mathcal{A}\) is F-acyclic if \(R^{i}F = 0\) for all \(i > 0\).

Fiber

Fibre functor

[2021-09-25 Sat 19:09] For Tensor category \((\mathscr{C}, \otimes)\), a fibre functor \(F\) is a functor \(\mathscr{C} \to \cat{Vec}\) satisfies that \(F(V_{1} \otimes V_{2}) \cong F(V_{1}) \otimes F(V_{2})\).

Final

[2021-12-26 Sun 15:23] An object in Simplicial set is called final if it is final in the Homotopy category. Similar result holds for initial.

Fully faithful

[2021-12-26 Sun 15:22] See Essentially surjective.

Function complex

[2021-12-17 Fri 20:13] The function complex \(\bm{Hom}(X,Y)\) is the simplicial set defined by \[\bm{Hom}(X,Y)_{n} = \hom_{\bm{S}}(X \times \Delta^{n}, Y)\]

Functor

Referenced: Continuous, Simplicial set.

Continuous

[2021-11-07 Sun 10:21] Let \(C\) and \(C'\) be Sites with associated Topoi \(T\) and \(T'\), then a Functor \(C' \to C\) is called continuous if for every \(X \in C'\) and \(\{X_{i} \to X\} \in \cat{Cov}(X)\) the family \(\{f(X_{i}) \to f(X)\}\) is in \(\cat{Cov}(f(X))\) and if \(f\) commutes with fiber prooducts when they exists in \(C'\).

Given a functor \(f: C' \to C\), we have a functor of presheaves: \(f_{* }: \hat{C} \to \hat{C}'\). That is \((f_{* }F)(T) := F(f(X))\). When functor is continuous, then \(f_{* }\) sends sheaves to sheaves. A functor \(F\) between two categories \(\mathcal{C}\) and \(\mathcal{D}\) is a collection of data, for \(X \in \mathcal{C}\), \(X \to F(X)\) and for \(f \in \Hom_{\mathcal{C}}(X,Y) \to F(f) \in \Hom_{\mathcal{D}}(F(X), F(Y))\) satisfying the associativity and identity condition.

Referenced: Sheaf.

Fully faithful

[2021-12-19 Sun 20:56] A functor \(F\) is called fully faithful if \(\Hom(X,Y) \to \Hom(F(X), F(Y))\) is a bijection.

Functorial factorization

[2021-12-18 Sat 17:53] A functorial factorization is an ordered pair \((\alpha, \beta)\) of functors \(\cat{Map}(\mathcal{C}) \to \cat{Map}(\mathcal{C})\) such that \(f = \beta(f) \circ \alpha(f)\) for all \(f \in \cat{Map}(\mathcal{C})\).

Referenced: Model category, Cofibrant replacement.

Group object

Groupoid

[2021-10-07 Thu 14:59] A groupoid is a category whose morphisms are all isomorphisms.

Referenced: Classifying space, Seifert-Van Kampen theorem.

Heart

[2021-10-19 Tue 18:33] Let \((D^{\leq 0}, D^{\geq 0})\) be t-structure for category \(D\), then \(C = D^{\leq 0} \cap D^{\geq 0}\) is called heart of t-structure.

Referenced: Perverse sheaf.

Homotopic

[2021-12-18 Sat 18:19] Maps \(f,g\) are a homotopic which is both Left homotopic and right homotopic, denoted by \(f \sim g\).

Homotopical category

[2021-12-19 Sun 21:22] We can define the homotopical category of complexes \(K(\mathcal{A})\) with the same object as \(C(\mathcal{A})\) but \(\Hom_{K(\mathcal{A})}(X, Y) = \Hom(X^{\bullet}, Y^{\bullet})/ \sim = H^{0}(\Hom^{\bullet}(X,Y))\).

The family of triangles in \(K(\mathcal{A})\) which are isomorphic to a standard triangle (Mapping cone) is called distinguished triangle.

Homotopy category

[2021-12-18 Sat 18:11] Suppose \(\mathcal{C}\) is a category with a subcategory of weak equivalence \(\mathcal{W}\). Define the homotopy category \(\cat{Ho}(\mathcal{C})\) as follows. Form the free category \(F(\mathcal{C}, \mathcal{W}^{-1})\) on the arrows of \(\mathcal{C}\) and reversals of the arrows of \(W\). \(\cat{Ho}(\mathcal{C})\) is the quotient category of \(F(\mathcal{C}, \mathcal{W}^{-1})\) by the relations \(1_{A} = (1_{A})\) for all objects \(A\), \((f, g) = (g \circ f)\) for all composable arrows \(f,g\) and \(1_{\text{dom } w} = (w, w^{-1})\) and \(1_{\text{codom } w} = (w^{-1}, w)\).

Let \(\mathcal{C}_{cf}\) denote the full subcategory of Cofibrant and Fibrant objects of \(\mathcal{C}\). Then we have

Cf. Model category, Homotopy equivalence.

Cf. Cofibrant replacement, Fibrant replacement.

The homotopy category can also be defined for Topological category \(\mathcal{C}\). It is then defined to have the same object \(\mathcal{C}\) and morphism is defined to be \(\Hom_{h \mathcal{C}}(X,Y) = \pi_{0} {\rm Map}_{\mathcal{C}}(X,Y)\), or \({\rm Map}_{h \mathcal{C}}(X,Y) = [{\rm Map}_{\mathcal{C}}(X,Y)]\). The latter denote the equivalence class in Category CGWH by inverting all Weak homotopy equivalence, category \(\mathcal{H}\).

If \(S\) is a simplicial set, then the homotopy category \(hS\) is defined to be the homotopy category \(h \mathfrak{C}[S]\) of Simplicial category \(\mathfrak{C}[S]\). Here \(\mathfrak{C}[S]\) follows the definition of Higher Topos p. 21-22. You can also construct homotopy category by functor \(\pi\), p. 29 Higher Topos.

Homotopy equivalence

[2021-12-18 Sat 18:22] A map \(f\) is a homotopy equivalence if there exists a map \(f\) such that \(hf \sim 1\) and \(fh \sim 1\).

Referenced: Homotopy category.

Hopf algebra

[2021-10-08 Fri 14:05] A bialgebra \((A, \Delta, \epsilon)\) equipped with an antiautomorphism \(S: A \to A\), that is \(S(ab) = S(b)S(a)\) and \(\Delta(S(a)) = (S \otimes S)(\Delta^{op}(a))\) is called a Hopf algebra if the following diagram is commutative

Referenced: Comodule, Drinfeld double, Dual pair, Quasitriangular, Ribbon.

Dual pair

[2021-11-04 Thu 17:51] \((H_{1}, H_{2}, \langle \rangle: H_{1} \otimes H_{2} \to k)\) is a dual pair of Hopf algebra if \(\langle, \rangle\) is nondegerate and \[\langle lm, a \rangle = \langle l \otimes m , \Delta_{H_{2}}(a) \rangle\] \[\langle e, ab \rangle = \langle \Delta_{H_{1}}(e), a \otimes b \rangle\] \[\langle S_{H_{1}}(e), a \rangle = \langle e, S_{H_{2}}(a) \rangle\] \[\langle 1_{H_{1}}, a \rangle = \epsilon_{H_{2}}(a)\] \[\langle e, 1_{H_{2}} \rangle = \epsilon_{H_{1}}(e)\]

However, the dual pair is not unique.

Referenced: Drinfeld double.

Quasitriangular

[2021-11-14 Sun 14:13] A Hopf algebra \((A, R \in A \hat{\otimes} A)\) is called quasitriangular if \[\Delta^{op}(a) = R \Delta(a)R^{-1}\] \[(\Delta \otimes \id)( R) = R_{13}R_{23} \in A^{\otimes 3}\] \[(\id \otimes \Delta)( R) = R_{13}R_{12} \in A^{\otimes 3}\]

Ribbon

[2021-11-14 Sun 14:21] A Hopf algebra is called ribbon if \(v \in Z(A)\) such that \[\Delta_{v} = (v \otimes v)(R_{21}R)^{-1} = (R_{21}R)^{-1}(v \otimes v)\]

Cf. Ribbon category.

Horn

[2021-12-17 Fri 20:16] The horn \(\Lambda^{n}_{k}\) is defined by

as coequalizer in \(\bm{S}\).

Identity object

[2021-09-30 Thu 14:06] See Tensor category.

Referenced: Dual.

Initial

[2021-12-26 Sun 15:25] See Final.

Injective

[2021-12-19 Sun 21:32] An object \(I\) in an abelian category \(\mathcal{A}\) is injective if the functor \(\Hom(-, I)\) is exact. A complex \(I^{\bullet}\) is injective if all the term \(I^{k}\) are injective.

An abelian category has enough injective objects if any object \(X\) in \(\mathcal{C}\) is a suboject of an injective object \(I\) in \(\mathcal{C}\).

∞-category

[2021-10-19 Tue 18:32] We follow Jacob Lurie’s terminology. Joyal call then quasi-categories.

One of the possible candidate for \(\infty\)-category is Topological category, but it is quite hard to work with.

Cf. Horn.

It is also called weak Kan complexes. (Cf. Kan complex)

The weak Kan complexes approach is equivalent to topological category approach.

The functor (Cf. Functor) between two \(\infty\)-categories is a functor between simplicial sets.

∞-category of spaces

[2021-12-26 Sun 15:41] Let \(\cat{Kan}\) denote the full subcategory of \(\cat{Set}_{\Delta}\) spanned by the collection of Kan complex. Then \(S = N(\cat{Kan})\) the Nerve of \(\cat{Kan}\) is called \(\infty\)-category of spaces.

(\infinity,n)-category

[2021-12-19 Sun 19:01] A \(\infty,n\)-category is a \(\infty\)-category with \(k\)-morphisms are invertible for \(k > n\).

Internal adjunction

[2021-09-30 Thu 14:45] With the notation of Internal Hom, we have internal adjunction \(\ihom(Z, \ihom(X,Y)) \ito \ihom(Z \otimes X, Y)\).

Cf. Tensor category.

Internal Hom

[2021-09-30 Thu 14:07] If \(\mathscr{C}^{op} \to \cat{Set}\), \(T \to \Hom(T \otimes X, Y)\) is representable, then write the representing object \(\ihom(X,Y)\) called internal Hom. \(\ihom(T \otimes X, Y) \cong \Hom(T, \ihom(X,Y))\). So we have \({\rm ev}_{X,Y}: \ihom(X, Y) \otimes X \to Y\) isomorphic to \({\rm id}_{\ihom(X,Y)}\).

Cf. Tensor category.

Referenced: Internal adjunction.

Invertible

[2021-09-30 Thu 14:06] See Tensor category.

Join

[2021-12-25 Sat 21:07] Let \(S\) and \(S'\) be simplicial sets, then the simplicial set \(S \star S'\) is defined as \[(S \start S')(J) = \prod_{J = I \cup I'} S(I) \times S'(I')\] That is \[(S \star S')_{n} = S_{n} \cup S'_{n} \cup \cup_{i + j = n - 1} S_{i} \times S'_{j}\]

Cf. ∞-category.

Left homotopy

[2021-12-18 Sat 17:36] A left homotopy of maps \(f,g: A \to B\) is a commutative diagram where \((f, g)\) is the map on the disjoint union. \(\tilde{A}\) is some choice of Cylinder object.

A similar definition of right homotopy for Path object.

Referenced: Homotopic.

Left lifting property

[2021-12-17 Fri 20:08] See Right lifting property.

Left Quillen functor

[2021-12-18 Sat 18:32] A functor is a left Quillen functor if \(F\) is a left adjoint and preserves cofibrations and trivial cofibrations. A functor is a right Quillen functor if \(U\) is a right adjoint and preserves fibrations and trivial fibrations. We call \((F, U, \phi)\) Quillen adjunction if \(F\) is a left Quillen functor and \(\phi\) makes \(U\) a right adjoint of \(F\).

Referenced: Quillen adjunction, Right Quillen functor, Total left derived functor.

Limit

[2021-10-07 Thu 16:40] See Colimit.

Referenced: Complete.

Map

[2021-12-18 Sat 17:51] For a category \(\mathcal{C}\), we can form a category \(\cat{Map}(\mathcal{C})\), whose objects are morphisms of \(\mathcal{C}\) and whose morphisms are commutative squares.

For \(S\) simplicial subset containing vertices \(x\) and \(y\). We define \({\rm Map}_{S}(x,y) = {\rm Map}_{hS}(x,y) \in \mathcal{H}\) to be the object of \(\mathcal{H}\) representing the space of maps from \(x\) to \(y\) in \(S\).

Model category

[2021-12-17 Fri 20:38] A model structure on a category is three classes of morphisms called cofibrations, fibrations, and weak equivalences, satisfying

1. The category is closed under all finite limits and colimits.

2. Suppose that the following diagram commutes,

   if any two \(f, g\) and \(h\) are weak equivalence, then so is the third.

3. If \(f\) is a Retract of \(g\) and \(g\) is a weak ewquivalence, fibration or cofibration, then so if \(f\).

4. Suppose that we are given a commutative solid arrow diagram

   where \(i\) is a cofibration and \(p\) is a fibration. Then the dotted arrow exists, making the diagram commute, if either \(i\) or \(p\) is also a weak equivalence.

5. Any map \(f: X \to Y\) may be factored \(f = p \circ i\), where \(p\) is fibration and \(i\) is trivial cofibration and \(f = q \circ j\) where \(q\) is a trivial fibration and \(j\) a cofibration. (We can further require Functorial factorization)

A model category is a category with all small limits and colimits together with a model structure.

If \(\mathcal{C}\) is a model category, then it has an initial object, the colimit of the empty diagram, and a terminal object, the limit of the empty diagram.

Referenced: Cylinder object, Homotopy category, Cofibrant, Path object.

Cofibrant

[2021-12-18 Sat 18:00] An object of Model category is called cofibrant if the map from the initial object \(0\) to it is a cofibration.

An object is called fibrant if the map from it to the terminal object is a fibration.

Referenced: Homotopy category, Fibrant, Cofibrant replacement, Quillen equivalence, Total left derived functor.

Fibrant

[2021-12-18 Sat 18:04] See Cofibrant.

Referenced: Homotopy category, Quillen equivalence.

Fibrant replacement

[2021-12-18 Sat 18:10] See Cofibrant replacement.

Referenced: Homotopy category.

Cofibrant replacement

[2021-12-18 Sat 18:08] By applying factorization to \(0 \to X\), we get a cofibrant replacement functor \(X \to QX\) (Functorial factorization is required here) such that \(QX\) is Cofibrant and \(QX \to X\) is a trivial fibration.

Similar, we have fibrant replacement functor \(X \to RX\) by applying factorization to \(X \to 1\).

Referenced: Homotopy category, Fibrant replacement.

Monoid

[2021-10-08 Fri 14:17] A monoid is a group without inverse. In other words, a group is a monoid with inverse.

Monoidal category

[2021-10-19 Tue 18:07] A monoidal category is a category \(C\) with \(\otimes\) which has Associativity constraint and there exists unit \(1 \in \mathop{Ob}( C)\) such that \(1 \otimes V \cong V \otimes 1 \cong V\).

A monoidal functor is a functor respect the monoid structure.

Cf. Tensor category. In fact, I don’t know their differences.

Referenced: Strict, Pivotal, Braided, Rigid braided monoidal category.

Strict

[2021-10-19 Tue 18:10] A Monoidal category is called strict if \((A \otimes B) \otimes C = A \otimes (B \otimes C)\) and the same is for morphisms.

Pivotal

[2021-11-04 Thu 17:51] A Rigid Monoidal category is called pivotal if there exists a system of functorial isomorphism \[\mu = \{\mu_{X}: X \to X^{**}\}\] such that \[\mu_{X \otimes Y} = \mu_{X} \otimes \mu_{Y}\]

Referenced: Spherical.

Trace

[2021-11-14 Sun 14:01] We introduce the categorical trace by \[\tr_{f}^{L} = 1 \xrightarrow{i} V \otimes V^{*} \xrightarrow{f \otimes \id} V \otimes V^{*} \xrightarrow{\mu_{V} \otimes \id} V^{**} \otimes V^{*} \xrightarrow{e_{V^{*}}} 1\] Similarly, we can define the right trace \(\tr_{f}^{R}\).

We have following properties similar to classical trace \(\tr^{L,R}(fg) = \tr^{L,R}(gf)\) and \(\tr^{L,R}(f \otimes g) = \tr^{L,R}(f) \otimes \tr^{L,R}(g)\).

Referenced: Spherical, Dimension.

Spherical

[2021-11-14 Sun 14:07] A Pivotal category is spherical if for any \(V\) and \(f: V \to V\), we have \(\tr^{L}(f) = \tr^{R}(f)\).

Cf. Trace.

Referenced: Dimension.

Dimension

[2021-11-14 Sun 14:07] In Spherical category we can define the categorical dimension by \(\dim V = \tr(\id_{V})\) since the two Traces are the same.

Braided

[2021-11-14 Sun 14:09] A Monoidal category is braided if there exists \(c = \{c_{V,W}\}\), \(c_{V,W}: V \otimes W \to W \otimes V\) with some compatibility conditions.

Referenced: Rigid braided monoidal category.

Natural transformation

[2021-12-19 Sun 18:56] A natural transformation \(\alpha\) between two Functors \(F,G: \mathcal{C} \to \mathcal{D}\) is a collection of morphisms \(\{\alpha_{C}: F( C) \to G( C)\}_{C \in \mathcal{C}}\) satisfying the commutative diagram

Nerve

[2021-12-19 Sun 19:35] To any category \(\mathcal{C}\), we can associate a Simplicial set \(N(\mathcal{C})\) called nerve of \(\mathcal{C}\). For each \(n \geq 0\), we let \(N(\mathcal{C})_{n} = {\rm Map}_{\cat{Set}_{\Delta}}(\Delta^{n}, N(\mathcal{C}))\) denote the set of all functors \([n] \to \mathcal{C}\).

The objects of \(\mathcal{C}\) are simply the vertices of \(N(\mathcal{C})\).

There exists a useful characterisation of the nerve.

Cf. Homotopy category.

Ordinal number category

[2021-12-12 Sun 21:29] See Simplicial set.

Overcategory

[2021-12-25 Sat 21:40] See the following diagram of overcategory

For any category \(C'\), we have bijection \[\Hom(C', C_{/X}) \con \Hom_{x}(C' \star [0], C)\] where \(x: [0] \to [C]\) is the canonical functor.

Cf. Join.

One defines \((S_{/p})_{n}\) to be \(\Hom_{p}(\Delta^{n} \star K, S)\).

Path object

[2021-12-17 Fri 20:29] Let \(\bm{S}_{f}\) be the full subcategory of Simplicial set category (or Closed model category) whose objects are the Kan complex. A path object for \(X \in \bm{S}_{f}\) is a commutative diagram

where \(s\) is a Weak equivalence and \(d_{0}, d_{1}\) are Kan fibrations (in fact necessarily Trivial fibrations).

Cf. Cylinder object.

Referenced: Left homotopy.

Pointed

[2021-12-18 Sat 18:05] An category is called pointed if the map from the initial object to the termninal object is an isomorphism.

Quillen adjunction

[2021-12-18 Sat 18:35] Cf. Left Quillen functor.

Referenced: Quillen equivalence.

Quillen equivalence

[2021-12-18 Sat 18:41] A Quillen adjunction \((F, U, \phi): \mathcal{C} \to \mathcal{D}\) is called a Quillen equivalence if and only if, for all Cofibrant \(X\) in \(\mathcal{C}\) and Fibrant \(Y\) in \(\mathcal{D}\), a map \(f: FX \to Y\) is a weak equivalence in \(\mathcal{D}\) if and only if \(\phi(f): X \to UY\) is a weak equivalence in \(\mathcal{C}\).

Reflexive

[2021-09-30 Thu 14:47] An object \(X\) is called reflexive if \(i_{X}: X \ito X^{\vee \vee}\).

Cf. Dual, Tensor category.

Referenced: Rigid.

Representable

[2021-12-26 Sun 16:45] We say that a functor \(F: \cat{Sch} \to \cat{Sets}\) is representable by a scheme if there exists a scheme \(X\) and an isomorphism of functors \(F \cong h_{X}\). A morphism \(F \to G\) of functors is representable by schemes if for any map \(S \to G\) from a scheme \(S\), the fiber product \(F \times_{G} S\) is representable by a scheme.

A morphism \(F \to G\) is open immersion if for any morphism \(S \to G\) from a scheme \(S\), \(F \times_{G} S\) is representable by an open subscheme of \(S\).

We say that a set of open subfunctors \(\{F_{i}\}\) is a Zariski-open cover of \(F\) if for any morphism \(S \to F\) from a scheme \(S\), \(\{F_{i} \times_{F} S\}\) is a Zariski-open cover of \(S\).

If we have\(F \cong h_{X}\), then the universal family of \(F\) is the object \(U \in F(X)\) corresponding to the identity morphism \(\id_{X}\)

A morphism \(X \to Y\) of Prestacks over \(\cat{Sch}\) is representable if for every morphism \(V \to Y\) form a scheme \(V\), the fiber product \(X \times_{Y} V\) is an Algebraic space.

.

Retract

[2021-12-18 Sat 17:44] A map in \(\mathcal{C}\) is a retract of a map \(g \in \mathcal{C}\) if and only if there is a commutative diagram of the form

where horizontal composites are identities.

Referenced: Model category.

Ribbon category

[2021-11-14 Sun 14:19] A ribbon category is a Rigid braided monoidal category with \(v = \{v_{X}\}\) such that \[v_{X \otimes Y} = (v_{X} \otimes v_{Y}) c_{XY}^{-1}c_{YX}^{-1}\]

Referenced: Ribbon.

Right lifting property

[2021-12-17 Fri 20:04] A map \(p: X \to Y\) is said to have the right lifting property (RLP) with repsect to a class \(M\) is in every solid arrow diagram

with \(i \in M\) the dotted arrow exists making the diagram commute.

The left lifting property is defined by same diagram with \(i\) and \(p\) in definition changed.

Referenced: Left lifting property, Kan fibration.

Right morphism

[2021-12-25 Sat 20:50] For \(S\) ∞-category, we define right morphism from \(x\) to \(y\) to be a new simplicial set \(\Hom_{S}^{R}(x,y)\), by letting \(\Hom_{\cat{Set}_{\Delta}}(\Delta^{n}, \Hom_{S}^{R}(x,y))\) denote the set of all \(z: \Delta^{n + 1} \to S\) such that \(z| \Delta^{n + 1} = y\) and \(z| \Delta^{0, \cdots, n}\) is a constant simplex at the vertex \(x\).

Cf. Kan complex.

Right Quillen functor

[2021-12-18 Sat 18:34] See Left Quillen functor.

Rigid braided monoidal category

[2021-11-14 Sun 14:10] A rigid braided monoidal category is a Rigid Braided Monoidal category such that \[c_{A^{*}B^{*}} = (c_{B,A})^{*}\]

Referenced: Ribbon category.

Saturated

[2021-12-17 Fri 20:02] A class \(M\) of monomorphisms of \(\cat{S}\) is said to be saturated if the following conditions are satisfied:

1. All isomorphisms are in \(M\).
2. \(M\) is closed under pushout.
3. Each retract of an element of \(M\) is in \(M\).
4. \(M\) is closed under countable compositions and arbitrary direct sums.

Simplicial category

[2021-12-25 Sat 20:40] A simplicial category is a category which is enriched over the category \(\cat{Set}_{\Delta}\) of simplical sets. The category of simplicial categories will be denoted by \(\cat{Cat}_{\Delta}\).

A functor \(\mathcal{C} \to \mathcal{C}'\) between simplicial categories is an equivalence if the induced functor \(h \mathcal{C} \to h \mathcal{C}'\) is an equivalence of \(\mathcal{H}\)-enriched categories.

Simplicial group

[2021-12-17 Fri 20:00] A simplicial group is a simplical object in the category of groups, that is a contravariant functor from \(\bm{\Delta}\) to the category \(\cat{Group}\).

Simplicial homotopy

[2021-12-17 Fri 20:17] Let \(f, g: K \to X\) be simplicial maps. We say that there is a simplicial homotopy \(f \cong g\) from \(f\) to \(g\) if there is a commutative diagram

The map \(h\) is called homotopy.

This is the analog of topological Homotopy.

Simplicial homotopy group

[2021-12-17 Fri 20:20] Let \(X\) be a Fibrant Simplicial set and \(v \in X_{0}\) be a vertex of \(X\). Define the simplicial homotopy group \(\pi_{n}(X, v), n \geq 1\) to be the set of homotopy classes of maps \(\alpha: \Delta^{n} \to X (\text{rel } \partial \Delta^{n})\) fit into the diagram

\(\pi_{0}(X)\) is usually called path components of \(X\). The simplicial set \(X\) is said to be connected if \(\pi_{0}(X)\) is trivial.

The group law is given by \([\alpha] * [\beta]\) to be \(\dif_{n}(\omega)\) of the extension

where \((v_{0}, \cdots, v_{n - 1}, , v_{n+1}) = (v, v, \cdots, v, \alpha, , \beta)\).

Cf. Homotopy group.

Simplicial set

[2021-11-28 Sun 14:02] Let \(\bm{\Delta}\) be the category of finite ordianl numbers with order preserving maps between them. The object of \(\bm{\Delta}\) is denoted by \(\bm{n}\) which is a category denoted by \(0 \to 1 \to 2 \to \cdots \to n\), where identity and composition morphism is omitted. \(\bm{\Delta}\) is called ordinal number category. A simplicial set is a contravariant Functor \(X: \bm{\Delta^{op}} \to \cat{Sets}\).

One of the important example is standard covarinat functor \(\bm{\Delta} \to \cat{Top}\) mapping \(\bm{n}\) to standard n-simplex \(\abs{\Delta^{n}}\) with \[\abs{\Delta^{n}} = \{(t_{0}, \cdots, t_{n}) \in \RR^{n + 1} \mid \sum_{i = 0}^{n}t_{i} = 1, t_{i} \geq 0\}\]

In this language, the Singular homology can be describe by singular set \(S(T)\) \[\bm{n} \to \hom(\abs{\Delta^{n}, T})\]

We have morphism in \(\bm{\Delta}\) \(d_{i}: \bm{n - 1} \to \bm{n}\) (cofaces) and \(s^{j}: \bm{n + 1} \to \bm{n}\) (codegeneracies), defined as \[d^{i}(0 \to 1 \to \cdots \to n - 1) = (0 \to 1 \to \cdots \to i - 1 \to i + 1 \to \cdots \to n)\] \[s^{j}(0 \to 1 \to \cdots \to n + 1) = (0 \to 1 \to \cdots \to j \to j \to \cdots \to n)\]

We have cosimplicial identities

\begin{align*} \begin{cases} d^{j}d^{i} = d^{i}d^{j - 1} & \text{if } i < j \\ s^{j}d^{i} = d^{i}s^{j - 1} & \text{if } i < j \\ s^{j}d^{j} =1 = s^{j}d^{j+1} &\\ s^{j}d^{i} = d^{i - 1} s^{j} & \text{if } i > j + 1 \\ s^{j}d^{i} = s^{i}s^{j+1} & \text{if } i \leq j \\ \end{cases} \end{align*}

We have similar simplicial identities for \(Y_{n}\), where \(Y\) is a simplicial set.

Referenced: Classifying space, Ordinal number category, Path object, Simplicial homotopy group, Realization, Fibrant, Weak equivalence.

Realization

[2021-12-17 Fri 19:52] A realization of a Simplicial set \(X\), \(\abs{X}\), is defined to be \[\abs{X} = \varinjlim_{\Delta^{n} \to X} \abs{\Delta^{n}}\] It is a functor from simplicial set category to topology space category (in fact \(\cat{CGWH}\), Category CGWH). In fact, we have adjoint property \[\hom_{\cat{Top}}(\abs{X}, Y) \cong \hom_{\cat{S}} (X, SY)\]

Kan fibration

[2021-12-17 Fri 19:56] A map \(p: X \to Y\) of simplicial set is called Kan fibration if it satisfies the extension condition that for every commutative diagram

there exists \(\theta: \Delta \to X\) making the diagram commute.

Kan fibration is the categorical analog of Fibration.

Cf. Anodyne extension， Right lifting property.

Referenced: Path object, Fibrant, Trivial fibration.

Fibrant

[2021-12-17 Fri 19:58] A Simplicial set is called fibrant or Kan complex if the canonical map \(Y \to *\) is a Kan fibration.

Referenced: Simplicial homotopy group, Kan complex, Weak equivalence.

Kan complex

[2021-12-17 Fri 20:30] See Fibrant.

Referenced: Path object.

Weak equivalence

[2021-12-17 Fri 20:31] The map \(f: X \to Y\) between Fibrant Simplicial sets is called weak equivalence if

1. For each vertex \(x\) of \(X\) the induced map \(f_{*}: \pi_{i}(X, x) \to \pi_{i}(Y, f(x))\) is an isomorphism for \(i \geq 1\).
2. The map \(f_{*}: \pi_{0}(X) \to \pi_{0}(Y)\) is a bijection.

Cf. Weak homotopy equivalence.

Referenced: Path object, Trivial fibration.

Trivial fibration

[2021-12-17 Fri 20:33] A map \(p: X \to Y\) in \(\bm{S}_{f}\) is called trivial fibration if it is a Kan fibration and a Weak equivalence.

Referenced: Path object.

TODO Stable ∞-category

Strongly final

[2021-12-26 Sun 15:25] Let \(\mathcal{C}\) be a Simplicial set, then a vertex \(X\) is strongly final if the projection \(\mathcal{C}_{/X} \to \mathcal{C}\) is a trivial fibration of simplicial sets. That is for any map \(\f_{0}: \partial \Delta^{n} \to \mathcal{C}\) such that \(f_{0}(n) = X\) can be extended to a map \(f: \Delta^{n} \to \mathcal{C}\).

Cf. ∞-category.

Cf. Final.

The discussion can be generalized to Initial.

Tensor category

[2021-09-15 Wed 19:04] A tensor category is data \((\mathscr{C}, \otimes, \phi, \psi)\) where \(\phi\) is the Associativity constraint and \(\psi\) the Commutativity constraint.

The identity object in tensor cateogry is a pair \((U, u)\) where \(U \in \mathscr{C}\) with \(u: U \ito U \otimes U\) such that \(U \otimes -\) is an equivalence.

The object \(L\) is called invertible if \(L \otimes -\) is an equivalence. If \(L\) is invertible, then \(L\) has inverse with \(\delta: L \otimes L^{-1} \ito U\).

Referenced: Dual, Fibre functor, Identity object, Internal adjunction, Internal Hom, Invertible, Monoidal category, Reflexive, Rigid, Tensor functor, Criterion for category being representation category.

Rigid

[2021-09-30 Thu 14:48] A Tensor category is called rigid if we have isomorphism \[\ihom(X_{1}, Y_{1}) \otimes \ihom(X_{2}, Y_{2}) \to \ihom(X_{1} \otimes X_{2}, Y_{1} \otimes Y_{2})\] for all \(X_{i}, Y_{i}\) and all objects are Reflexive.

In rigid category, we have trace \(ihom(X, X) \to X^{\vee} \otimes X \to U\) called trace. And we define the rank to be \(\tr_{X}({\rm id}_{X})\).

The following is a criterion for rigidity.

Cf. Associativity constraint, Commutativity constraint, Tensor category.

Referenced: Pivotal, Rigid braided monoidal category, Criterion for category being representation category.

Tensor functor

[2021-09-30 Thu 14:51] A tensor functor between Tensor category is a functor with morphism \[c:FX \otimes FY \ito F(X \otimes Y)\] functorial in \(X, Y\) with compatibility with Commutativity constraint and Associativity constraint.

t-exact

[2021-10-30 Sat 16:17] A functor is called t-exact if it is compatible with the t-structure.

Topological category

[2021-12-19 Sun 19:19] A topological category is a category which is enriched over \(\cat{CGWH}\). The category of topological category will be denoted by \(\cat{Cat}_{top}\).

Cf. Category CGWH.

Torsion pair

[2021-10-19 Tue 18:33] A torsion pair \((\mathcal{P}, \mathcal{Q})\) is a pair of full subcategory of \(\mathcal{C}\) such that

1. \(\hom(P, Q) = 0\) for every \(P \in \mathcal{P}, Q \in \mathcal{Q}\).
2. \(\hom(\mathcal{P}, X) = 0\), then \(X \in \mathcal{Q}\).
3. \(\hom(X, \mathcal{Q}) = 0\), then \(X \in \mathcal{P}\).
4. For every \(C\) in \(\mathcal{C}\), there exists \(P \in \mathcal{P}, Q \in \mathcal{Q}\) such that \[0 \to P \to C \to Q \to 0\] is exact.

Total left derived functor

[2021-12-18 Sat 18:36] If \(F: \mathcal{C} \to \mathcal{D}\) is a Left Quillen functor, define the total left derived functor \(LF: \cat{Ho}(\mathcal{C}) \to \cat{Ho}(\mathcal{D})\) to be the composite \[\cat{Ho}(\mathcal{C}) \xrightarrow{\cat{Ho}(Q)} \cat{Ho}(\mathcal{C}_{c}) \xrightarrow{\cat{Ho}(F)} \cat{Ho}(\mathcal{D})\] where \(\mathcal{C}_{c}\) means the full subcategory with Cofibrant objects. For natural transformation, we have total derived natural transformation.

Similarly, we have total right derived functor.

Referenced: Total right derived functor.

Total right derived functor

[2021-12-18 Sat 18:39] See Total left derived functor.

Triangulated category

[2021-10-15 Fri 20:20] An Additive category \(\mathcal{C}\) endowed with a shift self-equivalence \(T\) and a family of distinguished triangles \(\mathcal{T}\) is a triangulated category if these data satisfying the following axioms, with \(X[1] = TX\).

1. Any triangle isomprhic to a distinguished triangle is distinguished.

2. \(X \to X \to 0 \to \) is distinguished.

3. Any \(u: X \to Y\) is part of distinguished triangle \(X \to Y \to Z \to \).

4. A triangle \(X \to Y \to Z \to \) is distinguished if and only if the triangle \(Y \to Z \to X[1] \to\) is distinguished.

5. Any diagram

   can be completed to commutative diagram.

6. For any pair of morphism \(u: X \to Y\) and \(v: Y \to Z\) and triple \(X \to Y \to A \to \) \(Y \to Z \to B \to \) and \(X \to Z \to C \to X\) there are isomorphisms \(a: A \to C\), \(b: C \to B\) and the triangle \(A \to C \to B \to A\) is distinguished.

A full additive subcategory \(\mathcal{D} \subset \mathcal{C}\) is called a triangulated subcategory if \(T(\mathcal{D}) \subset \mathcal{D}\) and if two vertices in a distinguished triangle in \(\mathcal{T}\) are in \(\mathcal{D}\), then so is the third.

Referenced: t-structure.

t-structure

[2021-10-19 Tue 18:33] Let \(D\) be a Triangulated category and \(D^{\leq 0}, D^{\geq 0}\) full subcategories, then \((D^{\leq 0}, D^{\geq 0})\) is called t-structure if the followings are satisfied

1. \(D^{\leq 1} \subset D^{\leq 0}\), \(D^{\geq 1} \subset D^{\geq 0}\).
2. \(\Hom_{D}(X, Y) = 0\) for \(X \in D^{\leq 0}, Y \in D^{\geq 1}\).
3. For \(X \in D\), there exists distinguished triangle \[X_{0} \in D^{\leq 0} \to X \to X_{1} \in D^{\geq 1} \to X_{0}[1]\]

Referenced: Heart, t-exact, Bounded, Perverse sheaf.

Bounded

[2021-10-30 Sat 16:14] A t-structure is called bounded if \(D^{\leq n} = D = D^{\geq m}\) for \(m \ll 0\) and \(n \gg 0\).

Undercategory

[2021-12-25 Sat 21:42] Cf. Overcategory.

Weak equivalence

[2021-12-19 Sun 20:20] Let \(F: \mathcal{C} \to \mathcal{D}\) be functor between Topological categories, we say that \(F\) is a weak equivalence (or equivalence) if the induced functor \(h \mathcal{C} \to h \mathcal{D}\) is an equivalence of \(\mathcal{H}\)-enriched categories.

Cf. Homotopy category.

TODO Zero object

Commutative Algebra

TODO Commutative Frobenius algebra

Ext group

[2021-10-13 Wed 21:11] Let \(M, N\) be two \(R\)-modules. Let \(P_{\bullet} \to M\) be a free \(R\)-module resolution of \(M\) \[\cdots \to P_{n} \to \cdots \to P_{1} \to P_{0} \to M \to 0\] is an exact sequence of \(R\)-modules and \(P_{i}\)’s are free. Then we define the Ext group by \[\Ext^{k}_{R}(M, N) = H^{k}(\Hom(P_{\bullet}, N))\] and the Tor group \[\Tor^{R}_{k}(M, N) = H_{k}(P_{\bullet} \otimes_{R} N)\]

Referenced: Tor group.

TODO Finite length

TODO K-theory

Module

TODO Injective

TODO Resolution

Semisimple

[2021-10-30 Sat 16:19] An object in a category is called semisimple if it is the direct sum of the Simple object.

TODO Simple

Referenced: Semisimple.

Tor group

[2021-10-13 Wed 21:11] See Ext group.

Sheaf Theory

Constructible sheaf

[2021-10-23 Sat 14:20] A sheaf is called constructible if \(F|_{X_{\alpha}}\) is a Local system for some Stratification \((X_{\alpha})\). And we define the category \(D_{c}^{b}(X) \subset D^{b}(X)\) full subcategory with constructible cohomology.

Dual complex

[2021-10-23 Sat 14:19] We define dual complex to be \(D_{X} = w_{X}[n]\), where \(n = \dim X\). and \(w_{X}(U) = \Hom(H_{c}^{n}(U), k)\).

Referenced: Verdier duality.

TODO Equivariant sheaf

TODO G-resolution

[2021-11-04 Thu 18:08] Equivariant pullback

TODO Intersection cohomology

TODO Local system

[2021-10-15 Fri 19:00] The local system also has its dual.

Referenced: Constructible sheaf, Verdier duality.

Perverse sheaf

[2021-10-15 Fri 19:01] For \(D = D^{b}_{c}(X)\), we can take t-structure as \[D^{\leq 0} = \{F | \dim \supp \mathscr{H}^{j}(F) \leq -j\}\] \[D^{\geq 0} = \{F | \dim \supp \mathscr{H}^{j}(\mathbb{D}_{X}F) \leq -j\}\] And we define the perverse sheaf to be the Heart of the t-structure. That is \[\cat{Perv}(X) = D^{\leq 0} \cap D^{\geq 0}\]

The operator \(\mathbb{D}_{X}\) preverse the category \(\cat{Perv}(X)\).

For perverse sheaf category, we have functor \(^{p}F\).

TODO Six functors

[2021-10-15 Fri 19:01] \(f^{-1}\) is exact, and \(f_{* }\) and \(f_{!}\) only left exact.

We have following important properties for operators.

• Adjunction pairs: \((f^{*}, Rf_{* })\), \((Rf_{!}, f^{!})\), \((- \otimes \mathcal{F}, \shom(\mathcal{F}, -)\).

• The morphism between functors \(f_{!} \to f_{* }\), which is isomorphism if \(f\) is proper. Also, we have \(j^{!} = j^{* }\) for \(j\) an open inclusion.

• For \(X = U \coprod Z\) where \(U\) is open and \(Z\) is closed, denote the includion \(i: U \to X\), \(j: Z \to X\), then we have distinguished triangles \[i_{!}i^{!} \to \id \to j_{* } j^{* }\] \[j_{!}j^{!} \to \id \to i_{* } i^{* }\]

• See Verdier duality.

• For \(f: X \to Y\) smooth of relative dimension \(d\) then one has an isomorphism \[f^{!}k = k[2d]\]

• \(H^{n}(X) = H^{n}(f_{* }k_{X})\), \(H_{n}^{!}(X) = H^{n}(f_{* } \omega_{X})\), \(H_{!}^{n} = H^{n}(f_{!}k_{X})\), \(H_{n}(X) = H^{-n}(f_{!} \omega_{X})\)

• For pullback diagram

  We have isomorphism of functors \[f^{* } \circ g_{!} \cong (g')_{!} \circ (f')^{!}\]

Stratification

[2021-10-23 Sat 14:20] A stratification of \(X\) is a decomposition \[X = \coprod_{\alpha \in A} X_{\alpha}\] which is locally finite, \(X_{\alpha}\) locally closed and smooth and satisfies \[\bar{X}_{\alpha} = \coprod_{\beta \in B} X_{\beta}\]

Referenced: Constructible sheaf.

Verdier duality

[2021-10-23 Sat 14:29] Let \(D_{X} \in D^{b}(X)\) be the Dual complex, we define \(\mathbb{D}_{X}(E) = \mathbb{R} \shom_{X}(E, D_{X})\) gives functor \(\mathbb{D}_{X}: D^{b}(X) \to D^{b}(X)\).

We have \(\mathbb{D}^{2} = \id\).

The Verdier duality is the eqauility \[\mathbb{D}_{Y} \circ Rf_{!} = Rf_{*} \circ \mathbb{D}_{X}\] Also, we have \[f^{!} \circ \mathbb{D}_{Y} = \mathbb{D}_{X} \circ f^{-1}\]

As a speical case, when \(\mathcal{L}\) is Local system and \(X\) is smooth, we have \[\mathbb{D} \mathcal{L} \cong \mathcal{L}^{\vee}[2 d_{X}]\]

Referenced: Six functors.

TODO Whitney stratification

Representation theory

The reference of representation of finite groups includes [steinberg2012] Chapter 1-6. The basics of the quivers are from [kirillov2016].

Burnside Theorem

TODO Characters

\(L(G)\); character; irreducible character; class function; multiplicity; regular representation; character table

The irreducible characters of \(G\) form an orthonormal set of class functions. and \({\rm dim} Z(L(G)) = |Cl(G)|\). Therefore the number of irreducible representations is smaller than the conjugacy groups. In fact, two numbers are equal.

Cf. Orthogonal relations.

TODO Class function

Completely decomposition

Criterion for category being representation category

Cf. Tensor category, Rigid, Group scheme.

Dimension theorem

这里值得注意的是，证明中使用了一些代数数论的结论，主要是关于 algebraic number 和 algebraic integer。

Fourier analysis of finite groups

fourier transform; fourier inversion; convolution; dual group;

nonabelian Fourier transform (Wedderburn)

Orthogonal relations

Group algebra

Referenced: Characters and class functions.

Point group

Quivers

quiver; subquiver; oriented cycle; loop; morphism; direct sum; subrepresentation; irreducible representation; indecomposable representation; quivers of finite type; path; product of path; path algebra; one-way path

Reconstruction theorem

[2021-10-08 Fri 14:18] Let \(\cat{Rep}_{k}(G)\) be Representation category of affine Group scheme \(G\). And let \({\rm For}: \cat{Rep}_{k}(G) \to \cat{Vec}_{k}\) be forgetful functor. Then we have \(\cat{Aut}^{\otimes}({\rm For})( R) \ni \lambda\) is data of \(\lambda_{X}: X \otimes R \to X \otimes R\) for every \(X \in \cat{Rep}_{k}(G)\) with compatibilities

• \(\lambda_{X_{1} \otimes X_{2}} = \lambda_{X_{1}} \otimes \lambda_{X_{2}}\)
• \(\lambda_{\bold 1} = \id_{k \otimes R = R}\)

And for all \(\alpha: X \to Y\) in \(\cat{Rep}_{k}(G)\), it commutes with \(\lambda_{X}\). We observe that \(g \in G( R)\) gives an element of \(\cat{Aut}^{\otimes}({\rm For})( R)\). In fact, we have

Representation category

[2021-09-25 Sat 19:03] For a group \(G\), we may define the category \(\cat{Rep}(G)\) for finite dimensional representation of \(G\) with morphisms of representations.

Cf. Algebraic group.

Referenced: Reconstruction theorem.

Representation of finite groups

equivalence; G-invariant subspace; irreducible representation; completely reducible representation; indecomposable representation; unitary representation

Schur lemma

Tensor product of irreducible representations

TODO Dykin diagram

Linear Algebra

Simultaneously diagonalizable

[2021-09-17 Fri 21:45] For two linear endomorphism \(A, B\), if \([A,B] = 0\), then \(A, B\) can be simultaneously diagonalized.

Lie Algebra

Base

[2021-09-25 Sat 18:56] A base of a Root system \(\Delta \in \phi\) is a basis for \(V\) such that each \(\alpha \in \phi\) is a positive or negative integer linear combination from \(\Delta\).

For a base \(\Delta\) we have \(\NN \Delta \cap \phi\) system of Positive roots. And positive roots can give a bases by simple roots.

Referenced: Fundamental weight.

Dominant

[2021-09-25 Sat 19:00] For \(\{\lambda_{i}\}\) Fundamental weights, \(\lambda \in P(\phi) = \oplus_{i} \ZZ \lambda_{i}\) is called dominant if \(\langle \lambda, \alpha_{j}^{\vee} \rangle \geq 0\) for all \(j\).

Cf. Root system.

Fundamental weight

[2021-09-25 Sat 18:58] For \(\Delta = \{\alpha_{1}, \cdots, \alpha_{n}\}\) Base for \(\phi\), we define the fundamental weights \(\{\lambda_{1}, \cdots, \lambda_{n}\}\) basis for \(V\) dual to \(\Delta^{\vee}\) so that \(\langle \lambda_{i}, \alpha_{j}^{\vee}\rangle = \delta_{ij}\).

Cf. Root system.

Referenced: Dominant.

TODO Poinccaré-Birkhoff-Witt theorem

Positive root

[2021-09-25 Sat 18:53] Let \((G, T)\) split Reductive, there exists Borel \(B \supset T\), and therefore \(\mathfrak{b} \supset \mathfrak{t}\). Let \(\phi\) be the Root system of \(G\), we define the positive root to by \(\phi^{+ }(B) = \{\alpha \in \phi \mid \mathfrak{g}_{\alpha} \subset \mathfrak{b}\}\).

Referenced: Base.

TODO Root system

Referenced: Base, Dominant, Fundamental weight, Positive root.

TODO simple root

Referenced: Base.

Differential Geometry

Cartan’s magic formula

Collar

[2021-08-26 Thu 19:56] A collar of a smooth boundary Manifold \(M\) is a diffeomorphism \(\kappa: \partial M \times \rinterval{0}{1} \to M\) where the image is an open neighborhood of \(\partial M\) in \(M\) and \(\kappa(x, 0) = x\).

For collar, we have following proposition

The proposition is similar to the existence of the tubular neighborhood for Normal bundle.

Complete integral distribution

[2021-09-09 Thu 20:35] Given a rank \(k\) Distribution \(D \subset TM\), let us say that a smooth coordinate chart \((U, \phi)\) on \(M\) is flat for \(D\) if \(\phi(U)\) is a cube in \(\RR^{n}\) and at points of \(U\), \(D\) is spanned by the first \(k\) coordinate vector fields \(\partial/ \partial x^{1}, \cdots, \partial/ \partial x^{k}\).

A distribution \(D \subset TM\) is completely integrable if there exists a flat chart for \(D\) in a neighborhood of each point of \(M\).

Referenced: Frobenius theorem.

Connected sum

[2021-08-31 Tue 17:46] Given two \(n\)-manifolds \(M_{1}\), and \(M_{2}\), we define the connected sum of \(M_{1}\) and \(M_{2}\) by remove two \(D^{n}\) in \(M_{1}\) and \(M_{2}\) and glue two \(S^{n - 1}\) together, denoted by \(M_{1} \shar M_{2}\).

Connection

[2021-09-06 Mon 19:32] A connection on a Vector bundle \(E \to M\) is a linear map \(\nabla^{E} : C^{\infty}(E) \to C^{\infty}(E \otimes T^{ *}M)\) satisfying the condition \[\nabla^{E}(\alpha e) = \alpha \nabla^{E} e + e \otimes \dif \alpha\] whenever \(e \in C^{\infty}\) is a smooth section of \(E\) and \(\alpha\) is a smooth function on \(M\).

A connection on a Principal bundle is a linear map \(A: TP \to \ker(\pi_{*})\) that equals the identity on the kernal of \(\pi_{*}\) and is equivariant with respect to the action of \(G\) on \(P\). Here \(\pi: P \to M\) is the projection map, \(\ker(\pi_{*})\) is often called vertical subbundle. Equivariant means that if \(g \in G\) and \(v \in TP\), then \((m_{g})_{*} (Av) = A((m_{g})_{*} v)\). \(\ker(A)\) is often called horizontal subbundle.

By isomorphism of \(\psi: P \times \mathfrak{lie}(G) \to \ker(\pi_{*})\) we can view a connection as a map \[A: TP \to \mathfrak{lie}(G)\] (or in other words \(A: \mathfrak{lie}(G) \to \mathfrak{lie}(G) \times T^{*}M\)) with \(A(\psi(p,m)) = m\) and \(A(m_{g*}v) = gA(v)g^{-1}\)

Referenced: Flat, Covariant derivative, Curvature, Holonomy group, Horizontal subbundle, Vertical subbunel, Chern class, Chern-Simons functional.

Flat

[2021-12-12 Sun 21:20] A Connection is called flat if the Curvature of the connection is identically zero.

Referenced: Chern-Simons functional.

Covariant derivative

[2021-12-12 Sun 20:51] The covariant derivative for Vector bundle is same as Connection. (Not sure)

Given a connection on a Principal bundle \(P\), we defines a covariant derivative \(\nabla_{A}\) on \(P \times_{\rho} V\) where \(\rho\) is a representation of \(G\) on \(V\). The Horizontal subbundle \(H_{A} \subset TP\) is canonically isomorphic to \(\pi^{*}TM\). For section \(s\) of \(E\), we can view it as \(G\)-equivariant map \(s^{P}:P \to V\). Then the restriction to \(H_{A}\) of the homomorphism \((s^{P})_{*}: TP \to V\) defines a covariant derivative of \(s\). In other word, let \(x \in M\) and \(v \in TM|_{x}\). Pick \(p \in P|_{X}\), then there exists a unique \(v_{A} \in H_{A}|_{p}\) such that \(\pi_{*} v_{A} = v\). Then \(\nabla_{A}s\) defined to send \(v\) to the equivalence class in \(E|_{x} = (P|_{x} \times_{\rho} V)\) of the pair \((p, (s^{p})_{*}v_{A})\).

Referenced: Exterior covarint derivative.

Curvature

[2021-09-06 Mon 19:41] For a given Connection and therefore Exterior covarint derivative \(\nabla^{E}\), there exists a unique smooth section \(R(\nabla^{E}) \in C^{\infty}(\End(E) \otimes \wedge^{2} T^{* }M)\) called the curvature of \(\nabla^{E}\), that satisfies the equation \[R(\nabla^{E}) \cdot (e \otimes v \wedge w) = \nabla_{v}^{E} \nabla_{w}^{E} e - \nabla_{w}^{E} \nabla_{v}^{E} e - \nabla^{E}_{ [v,w]}e\] for all \(v,w \in C^{\infty}( TM)\) and \(e \in C^{\infty}(E)\).

Or we can define curvature \(F_{\nabla}\) by \(\dif_{\nabla}^{2}m = F_{\nabla} \wedge m\).

The famous Bianchi identity is \[\dif_{\nabla} F_{\nabla} = 0\]

For Principal bundle, we define the \(F_{A}\) to be the section of the bundle \((P \times_{ad} \mathfrak{lie}(G)) \otimes \wedge^{2} T^{*}M\).

Referenced: Flat, Chern class, Chern-Simons functional, Yang-Mills equation.

Degree

[2021-09-09 Thu 20:25] Suppose \(M\) and \(N\) are compact, connected, Oriented, smooth manifolds of dimension \(n\), and \(F: M \to N\) is a smooth map. There exists a unique integer \(k\) called the degree of \(F\), that satisfies

1. For every smooth \(n\)-form \(\omega\) on \(N\) \[\int_{M} F^{* } \omega = k \int_{N} \omega\]
2. If \(q\) is a Regular value of \(F\), then \[k = \sum_{x \in F^{-1}(q)} \sgn(x)\] where \(\sgn(x) = 1\) if \(\dif F_{x}\) is orientation-preserving and \(-1\) if orientation-reversing.

de Rham cohomology

[2021-09-09 Thu 20:15] We define the de Rham cohomology group in degree \(p\) to be the quotient vector space \[H_{dR}^{p}(M) = \frac{Z^{p}(M)}{B^{p}(M)}\] where \(Z^{p}(M)\) is the closed \(p\)-forms on \(M\). then \(B^{p}(M)\) is the exact \(p\)-forms.

Referenced: de Rham theorem, Cohomology.

de Rham theorem

Cf. Singular cohomology and de Rham cohomology.

Differential form

[2021-09-09 Thu 19:42] A section of \(\Lambda^{k}T^{* }M\) is called a differential \(k\)-form. The integer \(k\) is called the degree of the form.

Cf. Tangent bundle.

Referenced: Exterior derivative, Invariant formula for the exterior derivative, Stokes’s theorem.

Distribution

[2021-09-09 Thu 20:28] Cf. Tangent bundle.

A distribution on \(M\) of rank \(k\) is a rank \(k\) subbundle of \(TM\). It is called a smooth distribution if it is a smooth subbundle.

Referenced: Complete integral distribution, Integral distribution, Involutive distribution.

Exterior derivative

[2021-09-09 Thu 20:02] We define the exterior differential of Differential form \[\dif (\sum_{J} \omega_{J} \dif x^{J}) = \sum_{J} \dif \omega_{J} \wedge \dif x^{J}\]

Referenced: Exterior covarint derivative, Invariant formula for the exterior derivative, Stokes’s theorem.

Exterior covarint derivative

[2021-12-12 Sun 21:07] Cf. Exterior derivative.

Given a Covariant derivative \(\nabla\), we can define the exterior covariant derivative to be \(\dif_{\nabla}\) such that if \(\omega\) is a \(p\) form and \(s\) is a section of \(E\), then \(\dif_{\nabla}(s \omega) = \nabla(s) \wedge \omega + s \dif \omega\)

Referenced: Curvature.

Frobenius theorem

[2021-09-09 Thu 20:39] By definition, we have Complete integral distribution implies Integral distribution and integral distribution implies Involutive distribution. The Frobenius theorem suggests the reverse.

Holonomy group

[2021-09-06 Mon 19:44] Let \(M\) be a Manifold, \(E\) a Vector bundle over \(M\), and \(\nabla^{E}\) a Connection on \(E\). Suppose \(\gamma: [0,1] \to M\) is smooth, with \(\gamma(0) = x\) and \(\gamma(1) = y\), where \(x,y \in M\). Then for each \(e \in E_{x}\) there exists a unique smooth section \(s\) of \(\gamma^{* }(E)\) satisfying \(\nabla_{\dot{\gamma}(t)}^{E} s(t) = 0\) for \(t \in [0,1]\) with \(s(0) = e\). Define \(P_{\gamma}(e) = s(1)\). Then \(P_{\gamma}: E_{x} \to E_{y}\) is a well-defined linear map called the parallel transport map.

Fix a point \(x \in M\), we define the holonomy group \(\hol_{x}(\nabla^{E}) = \{P_{\gamma}: \gamma \text{ is a loop based at } x\} \subset \GL(E_{x})\).

Referenced: Kähler manifold with holomorphic symplectic structure.

Berger classification theorem

Horizontal subbundle

[2021-12-12 Sun 20:45] See Connection.

Referenced: Covariant derivative.

Immersion

[2021-08-26 Thu 19:17] A smooth map \(f\) is an immersion if each \(T_{x}f\) is injective and submersion if each \(T_{x}f\) is surjective. Cf. Tangent space.

Referenced: Normal bundle, Submersion, Knot, Link.

Integral distribution

[2021-09-09 Thu 20:31] A nonempty immersed submanifold \(N \subset M\) is called an integral manifold of \(D\) if \(T_{p}N = D_{p}\) at each point \(p \in N\). And a Distribution is called integral if every point of \(M\) is contained in an integral manifold.

Referenced: Frobenius theorem.

Invariant formula for the exterior derivative

[2021-09-09 Thu 19:44] Let \(M\) be a smooth Manifold, and \(\omega \in \Omega^{k}(M)\). For any smooth vector fields \(X_{1}, \cdots, X_{k + 1}\) on \(M\).

\begin{align}\dif \omega (X_{1}, \cdots, X_{k + 1}) &= \\ & \sum_{1 \leq i \leq k + 1}(-1)^{i - 1}X_{i}(\omega(X_{1}, \cdots, \widehat{X_{i}}, \cdots, X_{k + 1})) \\ & + \sum_{1 \leq i < j \leq k + 1} (-1)^{i + j} \omega([X_{i}, X_{j}], X_{1}, \cdots, \widehat{X}_{i}, \cdots, \widehat{X_{j}}, \cdots, X_{k + 1}) \end{align}

Cf. Differential form, Exterior derivative.

Involutive distribution

[2021-09-09 Thu 20:32] The Distribution \(D\) is involutive if given a pair of smooth local sections of \(D\), their Lie bracket is also local section of \(D\).

Referenced: Frobenius theorem.

Lie derivative

[2021-09-02 Thu 10:53] Given a smooth covariant tensor field \(A\) on \(M\), we define the Lie derivative of \(A\) with respect to \(V\), denoted by \(\mathscr{L}_{V}A\) by \[(\mathscr{L}_{V}A)_{p} = \frac{\dif}{\dif t} \mid_{t = 0} (\theta_{t}^{* }A)_{p}\] Here \(V\) is a smooth vector field on \(M\), and \(\theta\) is its flow. \(\theta_{t}\) is locally an isomorphism.

Referenced: Liouville vector field.

Lie group

Cf. Manifold.

Referenced: Principal bundle.

TODO E8

Referenced: Intersection forms.

Manifold

The definition has three requirements, where the countable basis is hardly used.

We call a compact manifold without boundary a closed manifold.

Referenced: Collar, Connected sum, Holonomy group, Invariant formula for the exterior derivative, Lie group, Orientation, Riemannian metric, Tangent space, Whitney embedding theorem, Fundamental class, Local homology group, Poincaré duality, Symplectic manifold, Freedman’s classification of topological 4-manifold, Rokhlin Theorem, Spin structure.

Normal bundle

[2021-08-26 Thu 19:48] Let \(f: M \to N\) be an Immersion. We define the quotient bundle of the bundle morphism \(i: TM \to f^{* }TN|_{M}\) to be the normal bundle. There exists a neighborhood of the zero section of normal bundle diffeomorphic to the neighborhood of \(N\) in \(M\). The diffeomorphic map is called tubular map and the target is called tubular neighborhood.

Cf. Vector bundle, Tangent bundle.

Referenced: Collar.

Orientable

[2021-10-08 Fri 13:36] See Orientation.

Referenced: Heegaard diagram, Trisection.

Orientation

[2021-08-26 Thu 19:38] An atlas is called orienting if any two charts are positively related, that is every two charts has the same orientation as a subset of \(\RR^{n}\). If a Manifold \(M\) has an orienting atlas, we call \(M\) orientable. Notice that if a manifold is orientable, it naturally has two orientation.

Given Local homology group, a generator of the homology group \(H_{n}(M, M \backslash x; R) \cong R\) is called a local \(R\)-orientation. Let \(M\) be a \(n\)-manifold and \(A \subset M\). An \(R\)-orientation of \(M\) along \(A\) is a section \(s \subset \Gamma(A; R)\) of \(\omega: H_{n}(M, M \backslash \bullet; R) \to M\) such that \(s(a) \in H_{n}(M, M \backslash a; R) \cong R\) is a generator for each \(a \in A\). For \(A = M\), we called it \(R\)-orientation and for \(R = \ZZ\), it is called orientation.

By characteristic class theory, a manifold is orientable if the first Stiefel-Whitney class vanishes that is \(w_{1}(E) = 0\).

Referenced: Degree, Orientable, Oriented, Euler class, Fundamental class, Poincaré duality, Lickorish-Wallace theorem.

Oriented

[2021-10-08 Fri 13:36] See Orientation.

Referenced: Linking number, Seiberg-Witten moduli space, Seifert surface, Trisection.

Poincaré lemma

Principal bundle

[2021-12-12 Sun 20:34] A principal bundle is a fiber bundle with fiber Lie group \(G\) and free \(G\) actions fiberwise.

Referenced: Connection, Covariant derivative, Curvature.

Regular point

[2021-08-26 Thu 19:20] For a smooth map \(f: M \to N\), a point \(x \in M\) is called a regular point if \(T_{x}f\) is surjective and a point \(y \in N\) is called a regular value if each \(x \in f^{-1}(y)\) is a regular point, otherwise is called a singular value. Cf. Tangent space.

Referenced: Degree, Sard theorem, Critical point.

Riemannian metric

[2021-09-09 Thu 19:29] A Riemannian metric on \(M\) is a smooth symmetric covariant \(2\)-tensor field on \(M\) that is positive definite at each point. A Riemannian manifold is a pair \((M, g)\), where \(M\) is a smooth manifold and \(g\) is a Riemannian metric on \(M\).

Cf. Manifold.

Referenced: Spin^C connection.

Sard theorem

Cf. Regular point.

Referenced: Whitney embedding theorem, Seiberg-Witten moduli space.

Stein manifold

[2021-12-12 Sun 10:07] A stein manifold is a complex manifold which is a holomoprhically convex and holomorphically separable subset of \(\CC^{n}\). This is equivalent to \(\Sigma\) admits a proper holomorphic immersion \(\Sigma \hookrightarrow \CC^{n}\). In particular, analytification of any affine variety over \(\CC\) is a Stein space, but the converse is not true.

Stokes’s theorem

Cf. Differential form, Exterior derivative.

Submersion

[2021-09-13 Mon 18:14] See Immersion.

Tangent bundle

[2021-08-26 Thu 19:45] The Tangent space at each point constitutes a Vector bundle called the tangent bundle.

Referenced: Differential form, Distribution, Normal bundle.

Tangent space

[2021-08-26 Thu 19:17] We denote the tangent space at point \(p\) by \(T_{p}(M)\), and to each map \(f: M \to N\) associated a linear map \(T_{p}(f): T_{p}(M) \to T_{f(p)}N\) the differential of \(f\) at \(p\), such that \[T_{p}(gf) = T_{f(p)} g \circ T_{p}f\] The elements of \(T_{p}(M)\) are the tangent vectors of \(M\) at \(p\). Cf. Manifold.

Referenced: Immersion, Regular point, Tangent bundle, Predeformation functor.

TODO Uhlenbeck compactification

Vertical subbunel

[2021-12-12 Sun 20:45] See Connection.

Wedge product

[2021-09-09 Thu 19:39] For \(w \in \Lambda^{k}(V^{* })\) and \(\eta \in \Lambda^{l}(V^{* })\), we define the wedge product or exterior product to be \[\omega \wedge \eta = \frac{(k+l)!}{k!l!} {\rm Alt}(\omega \otimes \eta)\]

Whitney embedding theorem

Cf. Sard theorem, Manifold.

Algebraic Topology

Alexander-Whitney map

[2021-09-26 Sun 21:25] See Eilenberg-Zilber theorem.

Referenced: Cup product.

Base

[2021-10-07 Thu 15:12] See Fibre bundle.

Borsuk-Ulam theorem

Cf. Degree.

Bott periodicity

[2021-08-24 Tue 10:55] We consider the Grothendieck ring of vector bundles. We have that

Cf. Complex line bundle over CP^1.

The Bott periodicity is the following deep result

Brouwer fixed point theorem

[2021-09-27 Mon 20:40] The Brouwer fixed point theorem is an example of non construction existence proof.

The Brouwer fixed point theorem can give a proof of the fundamental theorem of algebra.

Referenced: Existence proof.

Category CGWH

[2021-10-08 Fri 15:51] denote the full subcategory of Compactly generated Weak Hausdorff space by \(\cat{CGWH}\).

Cf. Complete, Cocomplete.

Also, we have for \(X, Y \in \cat{CGWH}\), \[{\rm Map}(X, Y) \in \cat{CGWH}\] and the exponential law holds.

Referenced: Realization, Loop space.

Cellular

[2021-10-11 Mon 20:52] Let \((X, Y)\) be CW complexes. A map \(f: X \to Y\) is called cellular if \(f(X^{n}) \subset Y^{n}\) for any \(n\). We define the category \((cat{CW})\) whose objects are CW complexes and morphisms are cellular maps.

Referenced: Cellular approxiamtion theorem, Cellular homotopy.

Cellular approxiamtion theorem

Cf. Cellular homotopy, Cellular and Relative CW complex.

Cellular homotopy

[2021-10-11 Mon 20:54] A cellular homotopy between two Cellular maps \(X \to Y\) is a Homotopy \(X \times I \to Y\) that itself a cellular map. The quotient category is called \(\cat{hCW}\).

Referenced: Cellular approxiamtion theorem.

Chern class

[2021-09-22 Wed 20:40] Consider complex Vector bundle \(E \to X\), we wan to find sections linear independent everywhere. Then we have primary obstruction Chern class \(c_{n - k + 1} \in H^{2 (n - k + 1)}(X; \ZZ)\). Axioms of Chern class include \(c_{0}(E) = 1\), naturality and product structure and normalization \(c(T \CC \PP^{n}) = (1 + w)^{n + 1}\) for \(w = \pd [\CC \PP^{n - 1}]\).

All characteristic class (classes satisfies the naturality condition) for complex vector bundles are combination of Chern classes.

For complex manifod, with Connection \(\nabla_{A}\), we can calculate the Chern class analytically by Chern-Weil formula

Cf. Curvature.

This can be generalized to principal bundles.

Referenced: Pontryagin class, Chern-Weil theory.

Classifying Space

[2021-08-23 Mon 19:50] We consider the functor \(B(-, G)\) from the category of topological space to sets, where \(B(B,G)\) denote the set of isomorphism classes of Numerable \(G\) principal bundles over \(B\). We have that functor representable by classifying space \(BG\), with a numerable \(EG \to BG\). The total space \(EG\) is called universal if each numerable free \(G\) space \(E\) has up to \(G\)-map \(E \to EG\).

We can construct \(EG, BG\) by join. We consider a family of topological space \(X_{j} \in J\). Then we have \[X = \star_{j \in J} X_{j} = \{(t_{i}X_{i})_{i \in J}| \sum_{i \in J}t_{i} = 1\}/ \sim\] where the summation has finite nonzero summand \((t_{i}X_{i}) \sim (s_{i}Y_{i})\) if \(t_{i} = s_{i}\) and \(X_{i} = Y_{i}\) when \(t_{i} = s_{i} \neq 0\). The topology of the space \(X\) is the coarsest topology where the function \(t_{i}: X \to \mathbb{R}, (t_{i}X_{i}) \to t_{i}\) and function \(t_{i}:t_{i}^{-1}(\linterval{0}{1}) \to X_{i}, (t_{i}X_{i}) \to X_{i}\) is continuous.

Then we construct the Milnor space \[EG = G \star G \star G \star \cdots\] and \(BG = EG/G\). The action of \(G\) is obvious.

For a discrete group, \(BG\) is an Eilenberg-Mac Lane space of type \(K(G,1)\).

Cobordism hypothesis

Cofibered

[2021-10-09 Sat 15:12] We say a pair \((X, A)\) is cofibered if the inclusion \(A \subset X\) is a Cofibration.

Cofiber exact sequence

[2021-10-11 Mon 20:14] Cf. Exact Puppe sequence. Let \(f: X \to Y\) in \(\cat{CGWH*}\) between well-pointed spaces. The following sequence is co-exact in \(\cat{hCGWH*}\) \[X \to Y \to C_{* f} \to \Sigma X \to \Sigma Y \to \Sigma C_{* f} \to \Sigma^{2}X \to \cdots\] Cf. Cone.

Cofibration

[2021-10-09 Sat 15:05] A map \(i: A \to X\) is called a cofibration if it has Homotopy extension property for any spaces.

An equivalent saying is the follow one.

Referenced: Cofibered, Well-pointed.

Cohomology

There are various cohomology theories. The can be defined by common axioms.

Cf. Excision theorem, Mayer-Vietoris sequence.

Given a cohomology theory, we called \(h^{n}(X,A)\) the \(n\)-th cohomology group. The \(\delta^{n}\) are called the coboundary operator. And we write \(h^{n}(X) := h^{n}(X, \emptyset)\).

As in homology theory, we called \(h^{n}(pt)\) the coefficient groups of the theory and a cohomology theory may have following additional axioms

• dimension axiom. \(h^{n}(pt) = 0\) for \(n \neq 0\).
• additive axiom. \[h^{n}(\coprod X_{j}, \coprod A_{j}) \to \coprod_{j} h^{n}(X_{j}, A_{j})\] is always an isomorphism.

Also, we have triple exact sequence \[ \cdots \to h^{n - 1}(A, B) \stackrel{\delta}{\to} h^{n}(X, A) \to h^{n}(X, B) \to h^{n}(A, B) \stackrel{\delta}{\to} \cdots\]

Cf. de Rham cohomology.

Referenced: Cap product, Eilenberg-Mac Lane space, Excision theorem, Mayer-Vietoris sequence, Reduced cohomology, Category of modules.

Singular cohomology

[2021-09-09 Thu 20:23] Cf. Singular homology.

We consider the cochain complex \(S^{n}(X; G) = \Hom(S_{n}(X), G)\) and define the i-th singular cohomology by the i-th cohomology group of the cochain complex.

Referenced: de Rham theorem.

Relative singular cohomology

[2021-10-13 Wed 21:00] We define the relative singular cochain complex by \(S^{\bullet}(X, A; G) = \Hom(S_{\bullet}(X)/S_{\bullet}(A), G)\). Its cohomology is called the relative singular cohomology.

Compactly supported cohomology

[2021-09-27 Mon 20:24] Let \(\mathscr{K}\) denote the set of compact subspaces of \(X\). We define compactly supported cohomology of \(X\) by \[H_{c}^{k}(X) :=\colim_{K \in \mathscr{K}} H^{k}(X, X - K)\] In particular if \(X\) is compact, we have \(H_{c}^{k}(X) = H^{k}(X)\).

The compactly supported cohomology also satisfies Mayer-Vietoris sequence.

Referenced: Poincaré duality.

Cup product

[2021-09-03 Fri 20:36] One of the most difference between homology theory and cohomology theory is the multiplicative structure making cohomology group a cohomology rings. \[h^{m}(X, A) \otimes_{R} h^{n}(X, B) \to h^{m + n}(X, A \cap B)\] We call \(x \cap y\) the cup product of \(x, y\). The cup product is defined if \(A\) or \(B\) is empty or \(A = B\). We have axioms of cup product

• Naturality: For triads \(f:(X, A, B) \to (X'; A', B')\) we have \[f^{ *}(x \cup y) = f^{ *}(x) \cup f^{ *}(y)\]
• Stability: TODO
• Unit element: There is a unit \(1 \in h^{0}(pt)\) such that \(1_{X} = p^{*}(1)\) is the unit of multiplication in cohomology ring.
• Associativity: \((x \cup y) \cup z = x \cup (y \cup z)\).
• Commutativity: \(x \cup y = (-1)^{\abs{x} \abs{y}} y \cup x\). Here \(\abs{x}\) is the grade in the cohomology ring.

For the most importanct case \(A = B = \emptyset\), the cup product can be given by Alexander-Whitney map, that is \[(\alpha \cup \beta)(\sigma) = \alpha(_{p}\sigma) \cdot \beta(\sigma_{q})\]

The morphism \(H^{* }(-; R)\) is natural, that is it is a functor from the category of topological spaces to the category of graded commutative rings.

Cap product

[2021-09-26 Sun 21:34] We have cap product between Cohomology and Homology. That is we first consider the pairing \[\langle -, - \rangle: S^{\bullet}(X; R) \times_{R} S_{\bullet}(X;R) \to R\] for \(\alpha \in S^{p}(X)\), \(\sigma \in S_{p}(X)\), \(r \in R\), defined as \(\langle \alpha, (\sigma \otimes r) \rangle = \alpha(\sigma) \cdot r\).

The we define the cap product by composing the above pairing by \(1 \otimes \Delta\), so we have \[S^{\bullet}(X;R) \otimes S_{\bullet}(X;R) \to S^{\bullet}(X;R) \otimes S_{\bullet}(X \times X; R) \to S^{\bullet}(X;R) \otimes S_{\bullet}(X;R) \otimes S_{\bullet}(X;R) \to S_{\bullet}(X;R)\] which induced the map \[\cap : H^{p}(X; R) \otimes H_{p + q}(X; R) \to H_{q}(X; R)\]

The cap product makes \(H_{\bullet}(X; R)\) an \(H^{\bullet}(X;R)\)-module.

External product

[2021-09-07 Tue 21:42] For details see, section 17.3 in [dieck2008].

The external product is a family of \(R\)-linear maps \[h^{m}(X, A) \otimes_{R} h^{n}(Y, B) \to h^{m + n}((X, A) \times (Y, B)), x \otimes y \to x \times y\] The external products satisfying some properties.

Compactly generated

[2021-10-08 Fri 15:43] A subset \(Y \subset X\) is called compactly closed or \(k\)-closed if \(f^{-1}(Y)\) is closed in \(K\) for every continuous map \(f: K \to X\) with \(K\) compact Hausdorff. We define a new topology on \(X\) denoted by \(X\), where close subsets of \(kX\) are compactly closed subsets of \(X\). The map \(kX \to X\) is a continuous map. \(X\) is called compactly generated if \(kX = X\). The full subcategory of \(\cat{Top}\) consisting of compactly generated space is denoted by \(\cat{CG}\).

We have exponential law in \(\cat{CG}\).

Referenced: Category CGWH.

Complex line bundle over CP^1

[2021-08-24 Tue 10:18] We have that \(H(k) = (\mathbb{C}^{2} \backslash 0) \times_{\mathbb{C}^{* }} \mathbb{C}\) with equivalence relation \(((z_{0}, z_{1}), u) \sim ((\lambda z_{0}, \lambda z_{1}), \lambda^{k} u) \). Then we have that \(H(k)\) represent the isomorphism classes of complex line bundles.

Cf. Vector bundle

Referenced: Bott periodicity.

Cone

[2021-10-11 Mon 20:18] Let \((X, x_{0}) \in \cat{CGWH*}\). We define its cone by \[C_{* }X = X \wedge I = X \times I / (X \times \{0\} \cap \{x_{0}\} \times I)\] Cf. Smash product.

Referenced: Cofiber exact sequence, Homotopy cofiber.

TODO Contractible

Referenced: Integer homology 3-sphere.

Covering

[2021-10-07 Thu 15:10] A covering (space) is a Locally trivial map \(p: E \to B\) with discrete Fibre \(F\). A covering map which is a trivial Fibre bundle is also called a trivial covering.

Cf. Path connected, Semi-locally simply connected.

Referenced: Deck transformation, Fibration, Fundamental group, G-principal covering, Lifting, Universal covering.

CW approximation

[2021-10-11 Mon 20:58] A CW approximation of a topological space \(Y\) is a CW complex \(X\) with a Weak homotopy equivalence \(f: X \to Y\).

The following theorem making the study of topological space reduces to the study of CW complex up to weak homotopy equivalence.

TODO CW complex

Referenced: Cellular, CW approximation, Eilenberg-Mac Lane space, Whitehead theorem.

Deck transformation

[2021-10-07 Thu 16:29] Let \(B\) be a Path connected and \(p: E \to B\) be a connected Covering. A deck transformation is a homeomorphism \(f: E \to E\) such that \(p \circ f = p\). We denote the group of deck transformations by \( \aut(p)\).

Deformation retract

[2021-10-07 Thu 16:19] See Retraction.

Referenced: Neighborhood deformation retract.

Degree

For degree of isogeny, see Isogeny.

For degree of a map \(f: S^{n} \to S^{n}\), the degree its image in the Homotopy group \(\pi_{n}(S^{n}) \cong \ZZ\).

Referenced: Borsuk-Ulam theorem.

Eilenberg-Mac Lane space

Eilenberg-Mac Lane space is the representation space for Cohomology, that is we have the following

Cf. CW complex.

Referenced: Classifying space, Classifying Space.

Eilenberg-Zilber theorem

The proof of the theorem relies on Spectral sequence and is rather abstract. However, for Singular homology, there exists a explicit form of the Eilenberg-Zilber map, called Alexander-Whitney map. First for \(\sigma: \Delta^{n} \to X\), we introduce the map \( _{p} \sigma: \Delta^{p} \to X, _{p} \sigma(t_{0}, \cdots, t_{p}) = \sigma(t_{0}, \cdots, t_{p}, 0 ,\cdots, 0)\) and \(\sigma_{q}(t_{0}, \cdots, t_{q}) = \sigma(0, \cdots, 0, t_{0}, \cdots, t_{q})\), and then we have Alexander-Whitney map \[AW: S_{\bullet}(X \times Y) \to S_{\bullet}(X) \otimes S_{\bullet}(Y)\] given by \[AW(\sigma) = \sum_{p + q = n} {}_{p}(\pi_{X} \circ \sigma) \otimes (\pi_{Y} \circ \Sigma)_{q}\] By Eilenberg-Zilber theorem, the above map is a chain homotopy equivalence.

Cf. Homology.

Referenced: Alexander-Whitney map.

TODO Equivariant cohomology

[2021-08-29 Sun 15:25] Compared with classical cohomology theory, the equivariant cohomology is more general and simpler.

Referenced: Geometric representation theory.

TODO Equivariant K-theory

Referenced: Geometric representation theory.

Euler class

[2021-09-22 Wed 20:34] Let \(p: E \to X\) be a real Oriented Vector bundle of dimension \(n\), we want to find a nowhere vanishing section \(s: X \to E\), the primary obstruction is defined as the Euler class \(e(E) \in H^{n}(X; \ZZ)\). A geometric definition is taht we consider a generic section \(s: X \to E\) which is transverse to zero section, then \(s^{-1}(0)\) is a submanifold of \(X\) of codimension \(n\) and \(e(E)\) can be taken as Poincaré dual of \([s^{-1}(0)]\).

Referenced: Poincaré-Hopf thereom.

Exact Puppe sequence

[2021-10-09 Sat 14:26] A sequence in \(\cat{hCGWH}\) is exact if for any \(Y\), the sequence \(\cdots \to [Y, X_{n + 1}]_{0} \to [Y, X]_{0} \to [Y, X_{n - 1}]_{0} \to \cdots\) is exact. Let Homotopy fiber \(F_{f}\) be the pullback of \(p_{1}:P_{y_{0}}Y \to Y\) and \(f:X \to Y\) in category \(\cat{CGWH*}\). Then we have

We also have that \(\Omega Y = F_{F_{f} \to X}\). So we have exact sequence

Also by considering the adjunction formula \([\sigma Y, X]_{0} = [Y, \Omega X]_{0}\), we have that

Combining two lemmas above, we get the exact Puppe sequence

Cf. Fibration, Homotopy group.

Referenced: Cofiber exact sequence.

Excision theorem

[2021-10-12 Tue 18:07] The Mayer-Vietoris sequence has a equivalent description called excision theorem.

Cf. Homology and Cohomology.

Referenced: Cohomology.

Fibration

[2021-10-07 Thu 15:42] A map \(p: E \to B\) is a fibration if \(p\) has Homotopy lifting property.

Cf. Covering.

By the corollary in Homotopy fiber, we see that fibration is similar to Fibre bundle. In the converse, we have following criterion for fibration

Referenced: Kan fibration, Exact Puppe sequence, Homotopy fiber.

Fibre

[2021-10-07 Thu 15:12] See Fibre bundle.

Referenced: Covering.

Fibre bundle

[2021-10-07 Thu 15:07] Let \(p: E \to B\) be in \(\cat{Top}\). A trivialization of \(p\) over an open set \(U \subset B\) is a homeomorphism \(\phi:p^{-1}(U) \to U \times F\) over \(U\), such that the following diagram commutes

\(p\) is called locally trivial if there exists an open cover \(\mathscr{U}\) of \(B\) such that \(p\) has trivialization over each open \(U \in \mathscr{U}\). Such \(p\) is called a fibre bundle, \(F\) is called the fiber and \(B\) is called the base. And we denoted by \[F \to E \to B\]

Cf. Vector bundle.

Referenced: Base, Covering, Fibration, Fibre, Homotopy fiber, Local trivial, Trvialization.

Fibre homotopy

[2021-10-08 Fri 21:32] For two fibrations \(p_{1}:E_{1} \to B\) and \(p_{2}: E_{2} \to B\) and two fiber maps \(f_{0}, f_{1}:p_{1} \to p_{2}\) are said to be fiber homotopic if there exists a homootpy \(F: E_{1} \times I \to E_{2}\) such that \(F(-, t)\) is a fiber map for each \(t \in I\). \(f:p_{1} \to p_{2}\) is a fiber homotopic equivalence if there exists an inverse.

Fundamental class

[2021-08-31 Tue 18:12] Theorem 16.4.1 in [dieck2008].

Cf. Orientation, Local homology group.

If \(M\) is orientable, then a generator will be called fundamental class. If \(M\) is a manifold with boundary, then we call \(z \in H_{n}(M, \partial M)\) a fundamental class if for each \(x \in M \backslash \partial M\) the restriction of \(z\) is a generator in \(H_{n}(M, M \backslash x)\), cf. Local homology group.

Theorem 16.5.1 (ibid.)

Referenced: Poincaré duality, Virtual fundamental class.

TODO Fundamental group

Cf. Covering.

Cf. G-principal covering and above theorem.

Referenced: Homotopy group, Seifert-Van Kampen theorem, Semi-locally simply connected, Braid group.

TODO Fundamental groupoid

Referenced: Seifert-Van Kampen theorem.

G-principal covering

[2021-10-07 Thu 15:54] A left \(G\)-principal covering is a Covering \(p: E \to B\) with a left Properly discontinuous \(G\)-action on \(E\) over \(B\) commutes with the covering map.

Referenced: Fundamental group.

Grothendieck ring

[2021-08-24 Tue 10:55] Given a commutative monoid \(M\) which is a set together with an associative and commutative composition law \(+\) with zero element. We have a group \(K(M) = M \times M / D(M)\). That is elements of \(K(M)\) are of the form \((a,b) \in M \times M\), with equivalence relation \((a, b) \sim (c, d)\) if \((a + n, b + n) = (c + m, d + m)\) for some \(m,n \in M\). Such group is called Grothendieck group. If the monoid is given with a multiplicative structure, which can by inherited by Grothendieck group, the group becomes Grothendieck ring.

Referenced: Bott periodicity, Vector bundle.

TODO Homology

Referenced: Cap product, Eilenberg-Zilber theorem, Excision theorem, Hurewicz theorem, Whitehead theorem.

Singular homology

[2021-09-26 Sun 19:55] A singular n-simplex is a continuous map \(\sigma: \Delta^{n} \to X\), where \(\Delta^{n}\) is the Standard n-simplex. We define \(S_{n}(X)\) to be the free abelian group by all singular \(n\)-simplexes \(S_{n}(X) = \oplus_{\sigma \in \Hom(\Delta^{n}, X)} \ZZ \sigma\). An element of \(S_{n}\) is called a singular n-chain in \(X\). Given \(\sigma: \Delta^{n} \to X\), we have \(\partial^{i} \sigma: \Delta^{n - 1} \to X\) for \(0 \leq i \leq n\), by restricting the \(\sigma\) to the \(i\)-th face of the \(\Delta^{n}\) whose vertices are given by \(\{v_{0}, v_{1}, \cdots, \hat{v}_{i}, \cdots, v_{n}\}\). And we define the boundary map \(\partial: S_{n}(X) \to S_{n - 1}(X)\) to be \(\partial \sigma = \sum_{i = 0}^{n} (-1)^{i} \partial^{(i)} \sigma\). Then \((S_{\bullet}(X), \partial)\) defines a chain complex and we define the \(n\)-th singular homology group of \(X\) by \[H_{n}(X) = H_{n}(S_{\bullet}(X), \partial)\]

The singular homology groups are homotopy invariants.

The following theorem is sometimes called dimension axiom.

Referenced: Simplicial set, Singular cohomology, Eilenberg-Zilber theorem, Künneth formula, Local homology group, Poincaré duality.

Relative singular homology

[2021-10-12 Tue 17:45] Let \(A \subset X\) be a subspace. It induces a natural injective chain map \(S_{\bullet}(A) \to S_{\bullet}(X)\). We define the singular chain complex of \(X\) relative to \(A\) to be \[S_{n}(X, A) = S_{n}(X)/S_{n}(A)\] with the induced differential. Its homology \(H_{n}(X, A) = H_{n}(S_{\bullet}(X, A))\) is called the \(n\)-th relative homology.

TODO Reduced homology

Cf. Suspension.

TODO Cellular homology

TODO Homotopy

Referenced: Simplicial homotopy, Cellular homotopy, Hurewicz theorem.

Homotopy cofiber

[2021-10-11 Mon 20:25] Cf. Homotopy fiber.

Given \(f: X \to Y\) in \(\cat{CGWH*}\), we define homotopy cofiber \(C_{ * f}\) by the push-out of \(X \to C_{*}X\) and \(f: X \to Y\). Cf. Cone.

Homotopy excision theorem

[2021-10-12 Tue 08:17] Let \((A, C)\), \((B, C)\) be Relative CW complex. Let \(X\) be the push-out of \(C \to B\) and \(C \to A\). If \((A, C)\) is m-connected and \((B, C)\) is n-connected, then \[\pi_{i}(A, C) \to \pi_{i}(X, B)\] is an isomorphism for \(i < m + n\) and a surjection for \(i = m + n\).

Homotopy extension property

[2021-10-09 Sat 15:02] A map \(i: A \to X\) is said to have the homotopy extension property (HEP) with respect to \(Y\) if for any map \(f:X \to Y\) and any homotopy \(F:A \times I \to Y\), we have following commutative diagram.

Referenced: Cofibration.

Homotopy fiber

[2021-10-09 Sat 14:52] Let \(f: X \to Y\), we define its homotopy fiber over \(y \in Y\) to be the fiber of \(P_{f} \to Y\) over \(y\). Then we have

Cf. Path connected.

Therefore, Fibration is similar to Fibre bundle.

Referenced: Exact Puppe sequence, Fibration, Homotopy cofiber.

TODO Homotopy group

Unlike the Fundamental group, the higher homotopy group is abelian.

Referenced: Simplicial homotopy group, Degree, Exact Puppe sequence, Smash product.

Homotopy lifting property

[2021-10-07 Thu 15:43] A map \(p: E \to B\) is said to have the homotopy lifting property with respect to \(X\) if for every \(\tilde{f}\) and \(F\) we have \(\tilde{F}\) making the following diagram commutes.

Referenced: Fibration.

Hurewicz theorem

[2021-09-26 Sun 15:54] The Hurewicz theorem relates the Homotopy theory and Homology theory. We first fix generator satisfying the compatibility conditions \(i_{n} \in \tilde{H}_{n}(S^{n}) = H_{n}(D^{n}, S^{n - 1}) = \tilde{H}_{n - 1}(S^{n - 1})\). Then we can define Hurewicz map \[\rho: \pi_{n}(X) \to H_{n}(X)\] by sending \([f: S^{n} \to X] \to f_{* }(i_{n})\). Then we have

The Hurewicz theorem states that in certian circumstances the above homomorphism can be isomorphism.

Notice that for \(n \geq 2\), the homotopy group \(\pi_{n}(X)\) is already abelian.

The theorem can also be written in relative form.

Referenced: Whitehead theorem.

Künneth formula

[2021-09-26 Sun 21:14] The algebraic version of Künneth formula is

Apply the algebraic version to the Singular homology, we have

Lefschetz fixed point theorem

[2021-09-27 Mon 20:33] The Poincaré duality gives the intersection pairing \[\langle -, - \rangle : H_{i}(X) \times H_{n - i}(X) \to H_{0}(X) \cong \mathbb{Z}\] or equivalently \[\langle -, - \rangle : H^{i}(X) \times H^{n - i}(X) \to H^{n}(X) \cong \mathbb{Z}\]

Cf. Intersection forms.

By consider the intersection of the diagonal \(\Delta\) and the graph of the map \(f\), we get the fixed points of \(f\). We define the Lefschetz number of \(f\) by \[L(f) := \sum_{p}(-1)^{p} \tr(f_{* }: H_{p}(X; \QQ) \to H_{p}(X; \QQ))\] Then we have Lefschetz fixed point theorem

TODO Lifting

The following theorem states when a map can be lifted to its Covering space.

Cf. Path connected.

Local homology group

[2021-08-31 Tue 17:57] Let \(h_{* }(-)\) be a homology theory and \(M\) an \(n\)-dimensional Manifold. Then groups of the form \(h_{k}(M, M \backslash x)\) are called local homology group.

By excision theorem, we have that \(h_{k}(M, M \backslash x)\) is \(h_{k}(D^{n}, D^{n} \backslash x)\). For Singular homology theory, we have \(H_{n}(M, M \backslash x; G) \cong G\) and other homology groups are all \(0\).

For \(K \subset L \subset M\), we have homomorphism \(r_{K}^{L}: h_{k}(M, M \backslash L) \to h_{k}(M, M \backslash K)\).

Referenced: Orientation, Fundamental class.

Local trivial

[2021-10-07 Thu 15:13] See Fibre bundle.

Referenced: Covering.

Loop space

[2021-10-08 Fri 21:17] Given \((X, x_{0}) \in \cat{CGWH* }\), we define the based loop space \(\Omega_{x_{0}}X\) or simply \(\Omega X\) by \[\Omega X = {\rm Map}_{* }(S^{1}, X)\] And we define the free loop space \[\mathscr{L}X = {\rm Map}(S^{1}, X)\]

Cf. Category CGWH.

Referenced: Arc space.

Mapping cylinder

[2021-10-09 Sat 15:06] For \(f: A \to X\), we define the mapping cyclinder \(M_{f}\) by the pushout of \(A \times \{0\} \to A \times I\) and \(A \times \{0\} \to X \times \{0\}\).

For based space, let \(f:(X, x_{0}) \to (Y, y_{0}) \in \cat{CGWH*}\). We define its mapping cylinder by \[M_{*f} = M_{f}/ \{x_{0} \times I\}\]

Mayer-Vietoris sequence

There are Mayer-Vietoris sequence for homology and Cohomology theory. For \((A; A_{0}, A_{1}) \subset (X; X_{0}, X_{1})\) be excisive triads that is \(A = A_{0}^{\circ} \cap A_{1}^{\circ}\) and \(X = X_{0}^{\circ} \cap X_{1}^{\circ}\). Set \(X_{01} = X_{0} \cap X_{1}\) and \(A_{01} = A_{0} \cap A_{1}\). Then we have exact sequence \[\cdots \to h^{n - 1}(X_{01}, A_{01}) \to h^{n}(X, A) \to h^{n}(X_{0}, A_{0}) \oplus h^{n}(X_{1}, A_{1}) \to h^{n}(X_{01}, A_{01}) \to \cdots\]

Referenced: Cohomology, Compactly supported cohomology, Excision theorem.

n-connected

[2021-10-11 Mon 20:36] A pair \((X, A)\) is called n-connected if \(\pi_{0}(A) \to \pi_{0}(X)\) is surjective and \[\pi_{k}(X, A; x_{0}) = 0 \; \forall 1 \leq k \leq n, x_{0} \in A\]

Let \(X\) be obtained from \(A\) by attaching \(n\)-cells \((n \geq 1)\), then \((X, A)\) is \((n - 1)\)-connected.

Referenced: Homotopy excision theorem.

Neighborhood deformation retract

[2021-10-11 Mon 20:02] Let \(A\) be a subspace of \(X\). \(A\) is called a neighborhood deformation retract if there exists a continuous map \(u: X \to I\) with \(A = u^{-1}(0)\) and a homotopy \(H: X \times I \to X\) such that

\begin{equation} \begin{cases} H(x, 0) = x & \forall x \in X \\ H(a, t) = a & \text{if } (a, t) \in A \times I \\ H(x, 1) \in A & \text{if } u(x) < 1 \end{cases} \end{equation}

If \(A\) is a NDR of \(X\), then \(A\) is a strong Deformation retract of the open subset \(u^{-1}(\rinterval{0}{1})\).

n-equivalence

[2021-10-11 Mon 20:38] A map \(f: X \to Y\) is called an n-equivalence if for any \(x_{0} \in X\) we have \(f_{* }: \pi_{r}(X,x_{0}) \to \pi_{r}(Y, f(x_{0}))\) bijective for \(r < n\) and \(f_{* }: \pi_{n}(X, x_{0}) \to \pi_{n}(Y, f(x_{0}))\) is surjective.

Referenced: Weak homotopy equivalence.

Numerable

Partition of unity is one of the important properties of the covering which deserves its own definition.

Cf. Vector bundle

Referenced: Riemannian metric, Classifying Space, Vector bundle.

TODO Path connected

Referenced: Covering, Deck transformation, Homotopy fiber, Lifting, Seifert-Van Kampen theorem, Universal covering.

Path space

[2021-10-08 Fri 21:25] Given \(X \in \cat{CGWH}\) and \(x \in X\), we define the free path space \(PX = {\rm Map}(I, X)\) and based path space \(P_{x}X = {\rm Map}((I, 0), (X, x))\).

For \(f: X \to Y\), and \(p_{1}:PY \to Y\), we define the mapping path space \(P_{f}\) by the pull-back \(PY \times_{Y} X\).

Poincaré duality

[2021-09-27 Mon 20:21] The Poincaré duality is the following famous theorem.

Cf. Orientation, Manifold, Compactly supported cohomology, Singular homology, Fundamental class.

Referenced: Lefschetz fixed point theorem.

Poincaré-Hopf thereom

Cf. Euler class, Vector bundle.

Pontryagin class

[2021-09-22 Wed 20:46] The Pontryagin class is defined as \(P_{i}(E) = (-1)^{i}c_{2i}(E \otimes_{\RR} \CC)\), where \(c_{i}\) is the Chern class.

Referenced: Characteristic class for 4-manifold.

TODO Postnikov tower

TODO Properly discontinuous

Referenced: G-principal covering.

Reduced cohomology

The reduced cohomology groups are defined to be \(\tilde{h}^{n}(X) \to \coker(h^{n}(pt) \to h^{n}(X))\) where the map is induced by \(p: X \to pt\).

Cf. Cohomology.

TODO Relative CW complex

Referenced: Cellular approxiamtion theorem, Homotopy excision theorem.

TODO Relative homotopy group

Retraction

[2021-10-07 Thu 16:01] Let \(i: A \subset X\) be an inclusion. A continuous map \(r: X \to A\) is called a retraction if \(r \circ i = 1_{A}\). \(r\) is called a deformation retraction if furthermore we have a homotopy \(i \circ r \cong 1_{X} \text{ rel } A\).

Referenced: Deformation retract.

Seifert-Van Kampen theorem

Theorem

Let \(X = U \cap V\) where \(U, V \subset X\) are open. Then the following diagram

is a pushout in the category \(\cat{Groupoid}\).

Cf. Groupoid, Fundamental groupoid.

Cf. Path connected, Fundamental group.

Semi-locally simply connected

[2021-10-07 Thu 16:21] A space is semi-locally simply connected if for any \(x_{0} \in X\), there is a neighborhood \(U_{0}\) such that the image of the map \(i_{* }: \pi_{1}(U_{0}, x_{0}) \to \pi_{1}(X, x_{0})\) is trivial.

Cf. Simply connected, Fundamental group.

Referenced: Covering, Universal covering.

TODO Simply connected

Referenced: Semi-locally simply connected, Universal covering.

Smash product

[2021-10-07 Thu 16:43] We define the smash product \(\wedge\) by \(X \wedge Y = X \times Y / X \vee Y\).

Cf. Wedge product.

The above example will be used in the computation of Homotopy group of \(S^{n}\).

Referenced: Cone, Suspension.

Standard n-simplex

[2021-10-12 Tue 16:03] We denote the standard n-simplex by \(\Delta^{n} = \{(t_{0}, \cdots, t_{n}) \in \RR^{n + 1} \mid \sum_{i = 0}^{n} t_{i} = 1, t_{i} \geq 0\}\).

Referenced: Singular homology.

Stiefel-Whitney class

[2021-09-22 Wed 20:44] For \(p: E \to X\) real Vector bundle, we want to find sections linear independent. And we define the Stiefel-Whitney class \(w_{* }(E)\) as the obstruction in \(H^{* }(X; \ZZ/2)\). It is universal for characteristic class in \(\ZZ/2\) coeffecients.

Referenced: Orientation, Characteristic class for 4-manifold, Spin^C structure.

Suspension

[2021-10-08 Fri 21:20] We define the suspension of the space \(X\) by \(\Sigma X = X \wedge S^{1}\).

Cf. Smash product.

Referenced: Reduced homology.

TODO Transport functor

Trvialization

[2021-10-07 Thu 15:12] See Fibre bundle.

Universal coefficient theorem

[2021-10-13 Wed 21:45] The universal coefficient theorem for cohomology gives cohomology from homology.

The universal coefficient theorem for homology gives homology with other coefficients from coefficient \(\ZZ\).

Universal covering

[2021-10-07 Thu 16:19] A universal covering is a Covering which is Simply connected.

Cf. Path connected, Semi-locally simply connected.

Vector bundle

Cf. Numerable

The real vector bundles and complex vector bundles have monoid structure and can be transformed into Grothendieck ring denoted by \(KO(X)\) and \(K(X) = KU(X)\).

Referenced: Connection, Covariant derivative, Holonomy group, Normal bundle, Tangent bundle, Bott periodicity, Chern class, Complex line bundle over CP^1, Euler class, Fibre bundle, Numerable, Poincaré-Hopf thereom, Stiefel-Whitney class, Elliptic fibration.

Weak Hausdorff

[2021-10-08 Fri 15:48] A space \(X\) is weak Hausdorff if for every compact Hausdorff \(K\) and every continuous map \(f:K \to X\), the image \(f(K)\) is closed in \(X\). We denote the full subcategory of weak Hausdorff space by \(\cat{WH}\).

Referenced: Category CGWH.

Weak homotopy equivalence

[2021-10-11 Mon 20:41] \(f\) is called weak homotopy equivalence if \(f\) is n-equivalence for any \(n \geq 0\).

Referenced: Weak equivalence, CW approximation, Whitehead theorem.

Wedge product

[2021-10-07 Thu 16:42] We define the wedge product \(\vee\) by \(X \vee Y = X \coprod Y / \sim\), where \(\sim\) identifies \(x_{0} \in X\) and \(y_{0} \in Y\).

Referenced: Smash product.

Well-pointed

[2021-10-11 Mon 20:15] A based space \((X,x_{0})\) is called well-pointed, if the inclusion of the base point \(x_{0} \in X\) is a Cofibration in the unbased sense.

Whitehead theorem

[2021-09-26 Sun 16:14] The Whitehead theorem means that Weak homotopy equivalence is equivalent to homotopy equivalence for CW complexes.

By Hurewicz theorem, the Whitehead theorem also has an Homology version.

TODO Whitehead tower

Algebraic Geometry

[2021-08-23 Mon 19:50] There are two tastes of algebraic geometry. The algebraic geometers view it as a specialized geometry, which is closed related to complex geometry (by GAGA), birational geometry (MMP for example), symplectic geometry (Gromov-Witten invariants) etc. Algebraic geometers are mainly interested in case where the base field is \(\CC\). Experts in other field may view algebraic geometry as useful languages, where vast properties are worked out by algebraic geometers. Therefore, Grothendieck’s style of algebraic geometry provides many insights into number theory, representation theory etc. In both case, the characteristic \(p\) case is more interesting. Examples includes Steinberg variety, p-adic Hodge theory.

The following are the themes in Algebraic Geometry Salt Lake 2015 seminar https://sites.google.com/site/2015summerinstitute/home/schedule : analytic methods, birational geometry and classification, commutative algebra and computational algebraic geometry, Hodge theory, singularities and characteristic p methods, derived algebraic geometry, derived categories, geometric representation theory, Gromov-Witten and Donaldson-Thomas theories, mirror symmetry, tropical geometry, algebraic cycles, cohomology theories, p-adic Hodge theory, rational points and Diophantine problems, topology of algebraic varieties

2-Yoneda lemma

[2021-12-27 Mon 20:15] The 2-Yoneda lemma is Yoneda lemma for 2-morphisms.

Cf. Prestack.

TODO Affine derived scheme

Affine grassmannian

[2021-11-14 Sun 14:31] Let \(G\) affine linear Algebraic group. We define the functor \[{\rm Gr}_{G}( R) = G(R((t)))/G(R[|t|])\] called affine grassmannian. We write \(\mathcal{O} = k[|t|]\) and \(K = k((t))\).

Cf. Loop space, Arc space.

We have Cartan decomposition for \(G(K)\).

The above theorem gives a stratification of \({\rm Gr}_{G}\)

And we have following inclusion properties

Affine toric variety

There are several ways of constructing toric varieties. First, consider a finite subset \(\mathscr{A} = \{\chi^{1}, \cdots, \chi^{l}\} \subset \ZZ^{n}\) a lattice equivalent to the character of a torus \((\CC^{* })^{n}\). Then we may consider a map \(\Phi_{\mathscr{A}}(t) = (\chi^{1}(t), \cdots, \chi^{l}(t)), T_{N} \to \CC^{s}\). The Zariski closure of the image is a toric variety denoted by \(Y_{\mathscr{A}}\). The ideal of the affine toric variety is a Toric ideal. Detailed construction can see [CLS2011] p.14-15.

Another construction concerns Affine semigroup. given a affine semigroup \(S\), we have \(\CC[S] = \{\sum_{m \in S} c_{m} \chi^{m} \mid c_{m} \in \CC \text{ and } c_{m} = 0 \text{ for all but finitely many } m\}\), where \(\chi^{m}\) is dependent on the embedding into the lattice \(M\). The multiplication is induced by \(\chi^{m} \cdot \chi^{m'} = \chi^{m + m'}\). We have

The above construction are equivalent in the sense

We consider the relation between sublattice variety and original variety. ([CLS2011] Proposition 1.3.18) For \(N'\) have finite index in \(N\) a sublattice and quotient \(G = N/N'\) and \(\sigma \subset N_{\RR}' = N_{\RR}\) be a Strongly convex Rational Polyhedral cone. Then we have

1. There are natural isomorphisms \[G \cong \Hom_{\ZZ}(M'/M, \CC^{* }) = \ker(T_{N'} \to T_{N})\]
2. \(G\) acts on \(\CC[\sigma^{\vee} \cap M']\) and \[\CC[\sigma^{\vee} \cap M']^{G} = \CC[\sigma^{\vee} \cap M]\]
3. We have bijection \[U_{\sigma, N'}/G \cong U_{\sigma, N}\]

That is the original variety is Geometric quotient of the sublattice varieties.

Referenced: Toric, Face, Pointed, Saturated, Toric variety.

Algebraic group

[2021-09-07 Tue 20:45] An algebraic group \(G\) over a field \(k\) is a Group scheme \(G/k\) which is separated and smooth.

Referenced: Representation category, Affine grassmannian, Radical, Unipotent, Unipotent radical, Semisimple, Reductive, Parabolic, Isogeny, Langlands dual pair.

Radical

[2021-09-15 Wed 08:52] [milne2017] p.135

Let \(G\) be a connected Algebraic group over \(k\). \(G\) contains a largest connected solvable normal subgroup called the radical, \(R(G)\) of \(G\).

Referenced: Unipotent radical, Semisimple.

Unipotent

[2021-09-15 Wed 09:01] [milne2017] p.135

An Algebraic group \(G\) is said to be unipotent if every nonzero representation of \(G\) has a nonzero fixed point. Or equivalently speaking, every finite-dimensional representation \(r: G \to \GL_{V}\) there is a basis such that \(r(G) \subset \U_{n}\).

Unipotent radical

[2021-09-15 Wed 09:01] [milne2017] p.135

The unipotent radical of \(G\) is the largest connected normal unipotent sub-Algebraic group of \(G\), denoted by \(R_{u}(G)\).

Cf. Radical.

Referenced: Reductive.

Semisimple

[2021-09-15 Wed 08:50] [milne2017] p.135

An Algebraic group is called semisimple if \(R(G_{k^{a}})\) is trivial, where \(k^{a}\) is the algebraic closure of \(k\).

Cf. Radical.

Reductive

[2021-09-15 Wed 08:49] [milne2017] p.135

An Algebraic group is called reductive if \(R_{u}(G_{k^{a}})\) is trivial, where \(k^{a}\) is the algebraic closure of \(k\).

Cf. Unipotent radical.

Referenced: Positive root.

Parabolic

[2021-09-25 Sat 18:50] A smooth subgroup \(P \subset G\) is parabolic if \(G/P\) complete as algebraic variety.

Cf. Algebraic group.

Referenced: Borel.

Borel

[2021-09-25 Sat 18:51] A algebraic subgroup \(B\) is called Borel if it is Parabolic, smooth, connected and solvable.

Referenced: Positive root.

Algebraic space

[2021-12-26 Sun 20:35] An algebraic space is a Sheaf \(X\) on \(\cat{Sch}_{Ét}\) such that there exist a scheme \(U\) and a surjective Étale morphism \(U \to X\) representable by schemes. The morphism \(U \to X\) is called an étale presentation.

Proposition

Let \(P\) be one of the following property of morphisms of algebraic spaces: open immersion, closed immersion, locally closed immersion, affine, or quasi-affine. Consider the cartesian diagram

such that \(Y' \to Y\) is surjective, flat and locally of finite presentation. Then \(X \to Y\) has \(P\) if and only if \(X' \to Y'\) has \(P\).

For more properties, see Algebraic stack.

Algebraic stack

[2021-12-26 Sun 20:38] An algebraic stack is a Stack \(X\) over \(\cat{Sch}_{Ét}\) such that there exist a scheme \(U\) and a surjective, smooth and Representable morphism \(U \to X\). The morphism is called smooth presentation.

A typical example of algebraic stack is the quotient stack, denoted by \([X/G]\), whose object over \(S\) are diagrams

where \(P \to S\) is a \(G\)-Torsor and \(f: P \to X\) \(G\)-equivaraint. And morphism between objects the following diagram

The map \(X \to [X/G]\) is a \(G\)-torsor even when the action is not free.

We define the topological space of the algebraic stack \(\abs{X}\) as set consisting of field-value moprhisms \(x: \spec K \to X\). Two morphism \(x_{1}: \spec X_{1} \to X\) and \(x_{2}: \spec K_{2} \to X\) if there exists field extension \(K_{3}/K_{1}\) and \(K_{3}/K_{2}\) such that \(x_{1}|_{\spec K_{3}}\) and \(x_{2}|_{\spec K_{3}}\) are isomorphic in \(X(K_{3})\). A subset \(V\) is open if there exists open immersion \(U \ito X\) such that \(V\) is the image of \(\abs{U} \to \abs{X}\).

We say that an algebraic stack is quasi-compact, connected, or irreducible if \(\abs{X}\) is.

A point \(x \in \abs{X}\) is of finite type if there exists a representative \(\spec K \to X\) of finite type.

Let \(R \to U\) be a smooth Groupoid of schemes. Define \([U/R]^{pre}\) to be the prestack whose objects are morphism \(T \to U\) from a scheme \(T\). A morphism \((S \xrightarrow{a} U) \to (T \xrightarrow{b} U)\) is the data of a morphism of schemes \(f: S \to T\) and an element \(r \in R(S)\) such that \(s( r) = a\) and \(t( r) = f \circ b\). Define \([U/R]\) to be the stackification of \([U/R]^{pre}\).

Let \(X\) be a noetherian algebraic space and \(x \in \abs{X}\). We define the dimension of \(X\) at \(x\) to be \[\dim_{x} X = \dim_{u} U \in \mathbb{Z}_{\geq 0} \cup \infty\] where \(U \to X\) is any étale presentation and \(u \in U\) is an preimage.

Let \(X\) be an algebraic stack with smooth presentation \(U \to X\) and corresponding smooth groupoid \(s, t: R \to U\), and let \(u \in U\) be a preimage of \(x \in \abs{X}\). We define the dimension of \(X\) at \(x\) to be \[\dim_{x} X = \dim_{u} U - \dim_{e(u)} R_{u} \in \ZZ \cup \infty\] where \(R_{u}\) is the fiber of \(s: R \to U\) over \(u\) and \(e: U \to R\) denotes the identity morphism in the groupoid.

If \(X\) is a noetherian algebraic space or stack, we define the dimension of \(X\) to be \[\dim X = \sup \dim_{x} X \in \ZZ \cup \infty\]

If \(X\) is an algebraic stack and \(x: \spec k \to X\), we define the Zariski tangent space or simply the tangent space of \(X\) at \(x\) as the set \[T_{X,x} = \{f:\spec k[\epsilon] \to X, \alpha: x \cong f|_{spec k}\}/ \sim\] where two pairs are equivalent if there is an isomorphism between them.

Arc space

[2021-11-14 Sun 14:26] We define the arc space to be functor \(Y[|t|]\) such that \(Y[|t|]( R) = Y(R[|t|])\).

Cf. Loop space.

Referenced: Affine grassmannian, Loop space.

Auslander-Buchsbaum formula

[2021-09-08 Wed 20:32] Let \(A\) be a regular ring, then we have \[\dimh(M) + \depth(M) = \dim (A)\] Cf. Homological dimension, Depth.

Calabi-Yau variety

Referenced: Donaldson-Thomas theory, Kähler manifold with holomorphic symplectic structure.

Castelnuovo theorem

[2021-09-25 Sat 18:32] The Castelnuovo theorem classified the mimimal surfaces.

Categorical quotient

[2021-09-07 Tue 20:59] Given an action \(\sigma\) of \(G/S\) on \(X/S\), a pair \((Y, \phi: X \to Y)\) is called a categorical quotient if we have diagram

commutes and universal property that for any \((Z, \psi)\) satisfying the above diagram, there exists a unique morphism \(\chi: Y \to Z\) such that \(\psi = \chi \circ \phi\).

A categorical quotient is called universal if it is stable under base change, and is called uniform if it is stable under flat base change.

Referenced: Geometric quotient.

Category of modules

[2021-11-07 Sun 15:21] Let \(T\) be a Topos and let \(\Lambda\) be a ring in \(T\). Denote by \(\cat{Mod}_{\Lambda}\) the category of \(\Lambda\)-modules in \(T\).

Then we have functor \[\Gamma(T, -): \cat{Mod}_{\Lambda} \to \cat{Ab}\] obtained by \[F \to \Hom_{\cat{Mod}_{\Lambda}}(\Lambda, F)\] This is a left exact functor, and we denote the resulting right derived functor by \(H^{i}(T, -)\), called cohomology functor.

Cf. Cohomology.

Character (of a torus)

[2021-09-08 Wed 19:50] A character of a Torus is a group homomorphism \(\chi: T \to \CC^{* }\).

TODO Coarse moduli space

TODO Cohen strcuture theorem

Deformation functor

[2021-09-19 Sun 20:24] A Predeformation functor is called a deformation functor if it satisfies (1-2) in Schlessinger’s theorem.

Deligne-Mumford stack

[2021-12-26 Sun 20:33] A Deligne-Mumford stack is a stack \(X\) over \(\cat{Sch}_{Ét}\) such that there exist a scheme \(U\) and a surjective, Étale and Representable morphism \(U \to X\). The morphism is also called étale presentation.

For properties, see Algebraic stack.

Cf. Algebraic space.

Depth

[2021-09-08 Wed 20:34] The maximal length of an \(M\)-Regular sequence is called the depth of \(M\).

Referenced: Auslander-Buchsbaum formula.

TODO Derived stack

[2021-11-04 Thu 18:06] Examples: Betti stack, de Rham stack

Determinant bundle

[2021-09-08 Wed 20:50] If \(E\) is locally free of rank \(s\), then the determinant bundle \(\det(E)\) is by definition the line bundle \(\wedge^{s}(E)\). Let \(E\) be arbitrary coherent sheaf. Then we consider a finite locally free resolution \[0 \to E_{n} \to \cdots \to E_{0} \to E \to 0\] And define \(\det(E) = \otimes \det(E_{i})^{(-1)^{i}}\).

If \(\dim(E) \leq \dim(X) - 2\), then we have that \(\det(E) \cong \OO_{X}\).

Different characters may give same toric variety

[2021-09-08 Wed 20:26] Exercise 1.1.6 in [CLS2011]. We consider two map \[\Phi_{1}(s,t) = (s^{2}, st, st^{3}), \quad \Phi_{2}(s,t) = (s^{3}, st, t^{3})\] We can show that \(\Phi_{1}\) and \(\Phi_{2}\) gives same affine toric variety. However \(\Phi_{2}\) is surjective and \(\Phi_{1}\) is not.

Donaldson-Thomas theory

[2021-08-29 Sun 16:00] Cf. Enumerative geometry.

We consider the case when \({\rm dim}_{\CC}(X) = 3\). We fix a class \(\beta \in H_{2}(X, \ZZ)\) and fix an integer \(\chi \in \ZZ\). The Donaldson-Thomas theory describes the intersection theory on Hilbert scheme of \(X\). Due to existence of nonvanishing higher \({\rm Ext}\) group, the deformation-obstruction theory and Virtual fundamental class of virtual fundamental class is not well-defined. To remedy the issue, we consider \(I_{\chi}(X, \beta)\) parametrizing pairs \((E, \phi)\) such that

• \(E\) is a torsion free sheaf on \(X\) of rank \(1\).
• \(E\) is equipped with a trivialization \(\phi: \det E \cong \mathbb{O}_{X}\).
• The numerical invariants of \(E\) is given by \(c_{2}(E) = - \beta\), \(\xi(E) = \xi(\mathbb{O}_{X}) - \chi\).

The natural map \[{\rm Hilb}_{\chi}(X, \beta) \to I_{\chi}(X, \beta)\] \[Z \to J_{Z}\] is an isomorphism on the level of closed points.

The expected dimension is given by \(\dim(\Ext_{0}^{1}(E, E)) - \dim(\Ext_{0}^{2}(E,E))\), where \(\Ext_{0}\) denote the trace free subspace. So we have that \(\virdim I_{\chi}(X, \beta) = \chi(E, E) - \chi(\OO) = -K_{X} \cdot \beta\). In Gromov-Witten theory, we also have the fundamental class of dimension equal to \(-K_{X} \cdot \beta\). So if \(X\) is Calabi-Yau variety, we have \(\virdim I_{\chi}(X, \beta) = 0\).

We don’t have evaluation map in Donaldson-Thomas theory, so we consider \[\tilde{J} \to \hilb(X) \times X\] for the purpose of integration. Given \(\alpha \in H^{*}(X)\), we can define an operator \[\sigma_{0}(\alpha): \pi_{\hilb *}(- \ch_{2}(\tilde{J}) \cup \pi_{X}^{*}(\alpha) \cap \pi_{\hilb}^{*}(-)), H_{*}(\hilb(X)) \to H_{*}(\hilb(X))\] Then the Donaldson-Thomas invariants are defined by

\begin{align}\langle \alpha_{1}, \cdots, \alpha_{n} \rangle^{DT}_{\beta, m} &= \deg \prod_{k} \sigma_{0}(\alpha_{k})[I_{\chi}(X, \beta)]^{vir} \\ &= \int_{ [I_{\chi}(X, \beta)]^{vir}} \prod_{k} \sigma_{0}(\alpha_{k}) \\ &= \int_{ [I_{\chi}(X, \beta)]^{vir}} \prod_{k} \pi_{\hilb *}(- \ch_{2}(\tilde{J}) \cup \pi_{X}^{*}(\alpha_{k}) \cap \pi^{*} \beta)\end{align}

Here note \(-\ch_{2}(I_{Z}) = \beta\) is represented by cycle class \(Z \subset X\).

There may be some mistakes in definition, because I suppose that the second line is not well-defined and the \(\beta\) in the last line is unnatural. However, I don’t know the correct algebraic definition of Donaldson-Thomas theory yet. So I still follow the definition in GW/DT Theory Lecture 11.

Dual sheaf

[2021-09-08 Wed 20:39] Let \(E\) be a coherent sheaf of codimension \(c\). The dual sheaf is defined as \(E^{D} = \sext^{c}_{X} (E, \omega_{X})\).

Referenced: Reflexive.

Enumerative geometry

[2021-08-31 Tue 08:52] There are several aspects of enumerative geometry, beyond the numbers.

• counting curves and sheaves (Gromov-Witten theory, Donaldson-Thomas and related theories)
• gauge theory enumerative geometry (3d gauge theories and Coulomb branches, mirror symmetry, 4d gauge theories, and Vafa-Witten invariants, etc)
• applications of enumerative geometry to categorification and low-dimensional topology
• Hall algebras and their refined versions (cohomological, K-theoretic, derived categories)
• Enumerative geometry of Nakajima quiver variety and the representation of quantum groups (symplectic resolution, etc)

Referenced: Donaldson-Thomas theory, Gromov-Witten theory.

Étale site

[2021-11-06 Sat 21:30] See Site.

:ID: e75a7770-d111-4f5e-9fb0-72a34c894b6e

[2021-11-06 Sat 21:32] See Site.

TODO Exact sequence about extensions

[2021-09-15 Wed 21:29] See [sernesi2006] Proposition 1.1.5, Proposition 1.1.7, and Corollary 1.1.8.

Referenced: Extension.

Ex_A(R,I)

[2021-09-15 Wed 21:15] See [sernesi2006] p.12.

We now consider all isomorphism class of \(A\)-Extensions of \(R\) by \(I\), called \(\ex_{A}(R,I)\). We want to show that \(\ex_{A}(R, I)\) is an \(R\)-module. We first construct the pullback and pushout for extensions. We define the pullback, denoted by \(f^{* }(R', \phi)\) by following diagram:

And define the pushout by the other, denoted by \(\lambda_{*}(R', \phi)\):

Where \[R' \coprod_{I} J = \frac{R' \tilde{\oplus} J}{\{(- \alpha(i), \lambda(i)), i \in I\}}\]

Now we are ready to define the \(R\)-module structure on \(\ex_{A}(R, I)\). If \(r \in R\), we define \(r[R', \phi] = [r_{* }(R', \phi)]\) where \(r: I \to I\) is multiplication by \(r\). And for some, we consider \(R' \times_{R} R''\) as the extension of \(R\) by \(I \oplus I\). And consider the pushout of the function \(\delta: I \oplus I \to I\), such that \(\delta(i \oplus j) = i + j\).

By above construction, \(\ex_{A}(R, -)\) is a covariant functor from \(R\)-modules to \(R\)-modules.

Referenced: Extension, First cotangent module.

Extension

[2021-09-15 Wed 20:28] See [sernesi2006] p.9.

Let \(A \to R\) be a ring homomorphism. An \(A\)-extension of \(R\) by \(I\) is an exact sequence \[(R', \phi) : 0 \to I \to R' \stackrel{\phi}{\to} R \to 0\] where \(R'\) is an \(A\)-algebra and \(\phi\) is a homomorphism of \(A\)-algebras whose kernel \(I\) is an ideal of \(R'\) satisfying \(I^{2} = (0)\) which makes \(I\) a \(R\)-module.

Two extensions are isomorphic if the following diagram commutes:

Here \(\chi\) is an isomorphism of \(A\)-algebras. Similarly, we can define the homomorphism of extensions.

The \(A\)-extension \((R', \phi)\) is called trivial if it has a section, that is we have ring homomorphism \(\psi :R \to R'\) such that \(\phi \circ \phi = 1_{R}\). In that case, we say \((R', \phi)\) splits and \(\psi\) is called a splitting.

Examples: dual number, small extension. See [sernesi2006] p.11.

The extension of algebras can be extended to extension of schemes. An extension of \(X/S\) is a closed immersion \(X \subset X'\), where \(X'\) is an \(S\)-scheme, defined by a sheaf of ideals \(\mathscr{I} \subset \mathscr{O}_{X'}\) such that \(\mathscr{I}^{2} = 0\). The isomorphism, homomorphism, triviality, and \(\ex(X/S, \mathscr{I})\) is similar to the algebra case. For details, see [sernesi2006] p.15. (Cf. Ex_A(R,I) and Exact sequence about extensions)

In particular, we have following theorem relating the deformation to the \(\Ext\) groups.

Referenced: Ex_A(R,I), Versal extension.

Fiber category

[2021-12-26 Sun 20:09] Given a Prestack \(\mathfrak{X}\), the fiber category \(\mathfrak{X}(S)\) is the category of objects over \(S\) whose morphisms are over \(\id_{S}\).

TODO Fine moduli space

First cotangent module

[2021-09-15 Wed 21:33] See [sernesi2006] p.14.

The \(R\)-module \(\ex_{A}(R,R)\) is called the first cotangent module of \(R\) over \(A\), and is denoted by \(T_{R/A}^{1}\). We omit \(A\) if \(A = k\). The first cotagent sheaf is obtained by gluing the first cotangent modules.

Cf. Ex_A(R,I).

Flat

[2021-09-19 Sun 20:23] A \(R\)-module \(M\) is called flat if tensoring with \(M\) is an exact functor.

Fppf site

[2021-11-06 Sat 21:32] See Site.

Framed moduli space

[2021-09-19 Sun 19:28] See [nakajima1999] p.17.

We define \(\mathscr{M}(r,n)\), the framed moduli space of torsion free Sheaves on \(\PP^{2}\) with rank (r) and \(c_{2} = n\). That is the isomorphism class of pairs \((E, \Phi)\) on \(\PP^{2}\) with \(\rk(E) = n\) \(c_{2}(E) = n\) which is locally free in a neighborhood of \(l_{\infty}\) such that \(\Phi: E|_{l_{\infty}} \ito \mathscr{O}_{l_{\infty}}^{\oplus r}\). The latter isomorphism is called framing at infinity.

We have following simple characterization of framed moduli space.

Cf. Hilb(A^2,n).

There is a interesting lemma with basic techniques in linear algebra.

See [nakajima1999] p.26. We have \(S^{n}(\CC^{2}) \cong \{(B_{1}, B_{2}, i, j) \mid [B_{1}, B_{2}] + i j = 0\} // \GL_{n}(\CC)\), where \(//\) denote the algebro-geometric quotient.

Cf. Geometric quotient.

Generalized Serre’s condition

[2021-09-08 Wed 20:37] We define the generalized Serre’s condition \(S_{k,c}\) as \[\depth(E_{x} \geq \min \{k, \dim(\OO_{X,x}) - c\}) \text{ for all } x \in \supp(E)\]

Geometric quotient

Given an action \(\sigma\) of \(G/S\) on \(X/S\), a pair \((Y, \phi: X \to Y)\) is called a categorical quotient if we have

• diagram

  commutes.

• \(\phi\) is surjective and image of \(\Psi = \phi_{\id_{X}}\) is \(X \times_{S} X\) (equivalently the geometric fiber of \(\phi\) are precisely the orbits of the geometric points of \(X\) for geometric points over an algebraic closed field of sufficiently high transcendence degree).

• \(\phi\) is submersion that is \(U \subset Y\) is open if and only if \(\phi^{-1}(U)\) is open in \(X\).

• The fundamental sheaf \(\OO_{Y}\) is the subsheaf of \(\phi_{* }(\OO_{X})\) consisting of invariant functions, i.e., if \(f \in \Gamma(U, \phi_{* }(\OO_{X})) = \Gamma(\phi^{-1}(U), \OO_{X})\), then \(f \in \Gamma(U, \OO_{Y})\) if and only if

  

commutes, where \(F\) is the function induced by \(f\).

A geometric quotient is called universal if it is stable under base change, and is called uniform if it is stable under flat base change.

The geometric quotient is related to Categorical quotient by the following proposition.

Referenced: Affine toric variety, Framed moduli space.

Gromov-Witten theory

[2021-08-25 Wed 19:53] Cf. Enumerative geometry.

We have natural maps for Moduli space of Stable Maps

And we have Gromov-Witten class \[I_{g,n,\beta}(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}) = \pi_{2!} (\pi_{1}^{* }(\alpha_{1} \otimes \cdots \otimes \alpha_{n}))\] Also we have Gromov-Witten invariant \[\langle I_{g,n,\beta}(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}) \rangle = \int_{\overline{M}_{g,n}} I_{g,n,\beta}(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}) \]

The Gromov-Witten invariants has natural enumerative geometric interpretation. \(\langle I_{g,n,\beta}(\alpha_{1}, \cdots, \alpha_{n}) \rangle\) equals the number of stable curves in \((C, p_{1}, \cdots, p_{n}) \in \overline{M}_{g,n}\) for which we can find \(f\) such that \(f(p_{i}) \in Z_{i} = \alpha_{i}^{\vee}\) and \(f_{*}( C) = \beta\).

By above interpretation we have natural axioms about Gromov-Witten invariants.

Referenced: Donaldson-Thomas theory, Linear axiom, Effectivity axiom, Degree axiom, Equivariant axiom, Fundamental class axiom, Divisor axiom, Point mapping axiom, Splitting axiom, Reduction axiom, Deformation axiom, Kontsevich recursion formula.

Linear axiom

[2021-08-25 Wed 20:35] Cf. Gromov-Witten theory. \(I_{g,n,\beta}\) is linear in each variable.

Effectivity axiom

[2021-08-25 Wed 20:35] Cf. Gromov-Witten theory. For \(X\) projective, \(I_{g,n,\beta} = 0\) if \(\beta\) is not an effective curve class.

Degree axiom

[2021-08-25 Wed 20:36] Cf. Gromov-Witten theory. For \(\alpha_{1}, \cdots, \alpha_{n} \in H^{* }(X, \mathbb{Q})^{\otimes n}\), \(I_{g,n, \beta}(\alpha_{1}, \cdots, \alpha_{n}) \in H^{* }(\bar{M}_{g,n}, \mathbb{Q})\) has degree \(2 (g - 1) {\rm dim} X + 2 \int_{\beta} \omega_{X} + \sum_{i} {\rm deg} \alpha_{i}\)

Equivariant axiom

[2021-08-25 Wed 20:39] Cf. Gromov-Witten theory. The map \(I_{g,n, \beta}\) is \(S_{n}\) equivariant where \(S^{n}\) acts on \(H^{* }(X, \mathbb{Q})^{\otimes n}\) naturally and acts on \(H^{ * }(\overline{M}_{g,n}), \mathbb{Q}\) by permuting \(p_{i}\)’s of the pointed curves.

Fundamental class axiom

[2021-08-25 Wed 20:41] Cf. Gromov-Witten theory. If \(n + 2g \geq 4\) then there exists a natural forgetting map \[\pi_{n}: \bar{M}_{g,n} \to \bar{M}_{g,n - 1}\] if \([X] \in H^{0}(X, \mathbb{Q})\) fundamental cycle, then we have that \[I_{g,n, \beta}(\alpha_{1}, \cdots, \alpha_{n - 1}, [X]) = \pi_{n}^{* }I_{g,n-1, \beta}(\alpha_{1}, \cdots, \alpha_{n - 1})\]

Divisor axiom

[2021-08-25 Wed 20:43] Cf. Gromov-Witten theory. If \(n + 2g \geq 4\), then \(\pi_{n}: \bar{M}_{g,n} \to \bar{M}_{g,n - 1}\) as above, if \(\alpha_{n} \in H^{2}(X, \mathbb{Q})\) then we have \[\pi_{n * } I_{g,n, \beta}(\alpha_{1}, \cdots, \alpha_{n}) = (\int_{\beta} \alpha_{n}) \times I_{g,n -1, \beta}(\alpha_{1}, \cdots, \alpha_{n-1})\]

Point mapping axiom

[2021-08-25 Wed 20:45] Cf. Gromov-Witten theory. When \(\beta = 0\), if \(\alpha_{i}\) are homogeneous cohomology classes, then \[I_{0,n,0}(\alpha_{1}, \cdots, \alpha_{n}) = \int_{X}(\alpha_{1} \cup \cdots \cup \alpha_{n})[\bar{M}_{g,n}]\] if \(\sum_{i} {\rm deg} \alpha_{i} = 2 {\rm dim} X\) and is zero otherwise.

Splitting axiom

[2021-08-25 Wed 20:48] Cf. Gromov-Witten theory. By gluing \(2\) points, we have natural map \[C: \bar{M}_{g_{1}, n_{1} + 1} \times \bar{M}_{g_{2}, n_{2} +1} \to \bar{M}_{g,n}\] where \(g = g_{1} + g_{2}\) and \(n = n_{1} + n_{2}\). Then we have \[\phi^{* }I_{g,n, \beta} (\alpha_{1}, \cdots, \alpha_{n}) = \sum_{\beta = \beta_{1} + \beta_{2}} \sum_{i,j} g^{ij}I_{g_{1}, n_{1} + 1, \beta}(\alpha_{1}, \cdots, \alpha_{n_{1}}, T_{i}) \otimes I_{g_{2}, n_{2} +1, \beta_{2}}(T_{j}, \alpha_{n_{1}+ 1}, \cdots, \alpha_{n})\] where \(\phi\) is the diagonal map and \(g^{ij}\) is the inverse matrix of \(g_{ij} = \int_{X} T_{i} \cup T_{j}\).

Reduction axiom

[2021-08-25 Wed 20:52] Cf. Gromov-Witten theory. Gluing two point together, we have natural map \(\psi: \bar{M}_{g - 1, n + 2} \to \bar{M}_{g, n}\). Then we have \[\psi^{* } I_{g,n, \beta}(\alpha_{1}, \cdots, \alpha_{n}) = \sum_{i,j}g^{ij}I_{g - 1, n +2, \beta}(\alpha_{1}, \cdots, \alpha_{n}, T_{i}, T_{j})\]

Deformation axiom

[2021-08-25 Wed 20:54] Cf. Gromov-Witten theory. Let \(X \to T\) be a smooth proper amp with connected base \(T\). And set \(X_{t}\) is the fiber. Then for each \(t \in T\) and \(\beta_{t} \in H_{2}(X_{t}, \mathbb{Z})\), we obtain a map \(I_{g,n, \beta_{t}}: H^{* }(X_{t}, \mathbb{Q}) \to H^{* }(M_{g,n}, \mathbb{Q})\). Then if \(\beta_{t}\) is locally constant section of \(H_{2}(X_{t}, \mathbb{Q})\) and \(\alpha_{1}, \cdots, \alpha_{n}\) are locally constant, then we have \(I_{g,n, \beta_{t}}(\alpha_{1}, \cdots, \alpha_{n})\) is constant.

Kontsevich recursion formula

[2021-08-26 Thu 15:23] Cf. Gromov-Witten theory. For \(X = \mathbb{P}^{2}\), we denote that \(N_{d} = \langle I_{0, 3d - 1, d}([pt], \cdots, [pt]) \rangle \) where \([pt]\) occur \(3d - 1\) times for \(d \geq 1\). Then we have following Kontsevich recursion formula \[N_{d} = \sum_{d = d_{1} + d_{2}, d_{1}, d_{2} > 0} N_{d_{1}}N_{d_{2}} (d_{1}^{2}d_{2}^{2} \binom{3d - 4}{3d_{1} - 2} - d_{1}^{3}d_{2} \binom{3d - 4}{3d_{1} - 1})\]

Grothendieck topology

[2021-11-06 Sat 21:20] Let \(C\) be a category. A Grothendieck topology on \(C\) consists of a set \(\cat{Cov}(X)\) of collections of moprhisms \(\{X_{i} \to X\}_{i \in I}\) for every object \(X \in C\) such that the following hold:

1. If \(V \to X\) is an isomorphism, then \(\{V \to X\} \in \cat{Cov}(X)\).
2. If \(\{X_{i} \to X\}_{i \in I} \in \cat{Cov}(X)\) and \(Y \to X\) is any arrow in \(C\), then the fiber products \(X_{i} \times_{X} Y\) exist in \(C\) and the collection \[\{X_{i} \times_{X} Y \to Y\}_{i \in I}\] is in \(\cat{Cov}(Y)\).
3. If \(\{X_{i} \to X\}_{i \in I} \in \cat{Cov}(X)\) and if for every \(i \in I\) we are given \(\{V_{ij} \to X_{i}\}_{j \in J_{i}} \in \cat{Cov}(X)\), then the collection of compositions \[\{V_{ij} \to X_{i} \to X\}_{i \in I, j \in J_{i}}\] is in \(\cat{Cov}(X)\).

The Grothendieck topology is also called a pre-topology.

Referenced: Site.

Group scheme

[2021-09-07 Tue 20:42] A group scheme \(G/S\) is a morphism \(\pi: G \to S\) of schemes with \(S\)-morphism multiplication \(\mu: G \times_{S} G \to G\), inverse \(\beta: G \to G\) identity \(e: S \to G\) satisfying associativity, law of inverse and law of identity.

A group scheme \(G\) action on a scheme \(X/S\) is given by a morphism \(\sigma: G \times_{S} X \to X\) with associativity property and identity.

Let \(f: T \to X\) be a \(T\)-valued point. We have morphism \(\phi_{f} = \sigma \circ (1_{G} \times f): G \times_{S} T \to X \times_{S} T\). The image of \(\phi_{f}\) is called the orbit of \(f\) and denoted by \(O(f)\). The fibre product \(S(f)\) is called stabilizer of \(f\). The orbit and stabilizer is the algebraic version of orbit and stabilizer in action on topological spaces.

Referenced: Criterion for category being representation category, Reconstruction theorem, Algebraic group.

Groupoid of schemes

[2021-12-28 Tue 20:49] See Introduction to Stacks and Moduli p.79.

Hilb(A^2,n)

[2021-09-16 Thu 20:35] See [nakajima1999], Theorem 1.9.

We have a specific characterization of \(\hilb(A^{2}, n)\). Let \[\tilde{H} = \{(B_{1}, B_{2}, i) \mid [B_{1}, B_{2}] = 0 \text{ and stable}\}\] The triple \((B_{1}, B_{2}, i)\) is called stable if there exists no proper subspace \(S \subsetneq k^{n}\) such that \(B_{\alpha}(S) \subset S\), \(\alpha = 0,1\) and \(\im i \subset S\). \(B_{\alpha} \in \End(k^{n}) \) and \(i \in \Hom(k, k^{n})\). We have a natural \(\GL_{n}\) action on \(\tilde{H}\) defined by \((B_{1}, B_{2}, i) \to (gB_{1}g^{-1}, gB_{2}g^{-1}, gi)\). And by quotient, we define \(H = \tilde{H}/ \GL_{n}\). And we have that \(H \cong \hilb(\mathbb{A}^{2},n)\).

Cf. Hilbert scheme.

The manifold \(\hilb(\mathbb{A}^{2}, n)\) admits a hyper-Kähler metric.

Referenced: Framed moduli space.

Hilbert-Chow morphism

[2021-09-16 Thu 20:25] For Hilbert scheme of points, we have a well define map \[\pi:X^{ [n]}_{red} \to S^{n}X\] by \(\pi(Z) = \sum_{x \in X}l(Z_{x})[x]\), here \(l\) denote the length.

Referenced: Smoothness of Hilbert scheme.

Hilbert scheme

[2021-08-29 Sun 15:24] We fix a class \(\beta \in H_{2}(X, \ZZ)\) and fix an integer \(\chi \in \ZZ\).

In [nakajima1999], there is another notation of Hilbert schemes, where in \(\hilb(X, p)\), the \(p\) denote the Hilbert polynomial. And we write \(X^{ [n]}\) for \(\hilb(X, n)\) for Hilbert scheme of points.

Referenced: Donaldson-Thomas theory, Hilb(A^2,n), Hilbert-Chow morphism, Simple resolution, Smoothness of Hilbert scheme, Symplectic structure of Hilbert scheme.

Homological dimension

[2021-09-08 Wed 20:29] Let \(M\) be a module of a local ring \(A\). The homological dimension denoted by \(\dimh(M)\) is defined as the minimal length of a projective resolution of \(M\). It is also the length of free resolution of \(M\).

Referenced: Auslander-Buchsbaum formula.

Hull

[2021-09-19 Sun 20:23] Let \(F\) be a Predeformation functor and \(\hat{F}\) its completeion. We say a pair \((R, \eta)\) with \(R\) a Noetherian Complete Local ring, and \(\eta \in \hat{F}( R)\), is a hull for \(F\) if the induced map \(\bar{h}_{R} \to F\) is smooth, and induces a bijection \(T_{\bar{h}_{R}} \ito T_{F}\) on tangent spaces.

Referenced: Schlessinger’s theorem.

TODO Hyper-Kähler quotient

Ind-scheme

[2021-11-14 Sun 14:23] An ind-scheme is object in \(\cat{Fun}(\cat{k-algebra}, \cat{Set})\) which can be written as \[X = \varinjlim_{I} X_{i}\] such that \(I\) is a direct diagram, \(X_{i}\) is a scheme, all maps \(X_{i} \to X_{j}\) are closed embedding. We call ind-affine, ind-projective, ind-finite type if all \(X_{i}\) are affine, projective, finite type.

Referenced: Loop space.

Inertia stack

[2021-12-28 Tue 21:11] The inertia stack of an Algebraic stack \(X\) is the fiber product

Isogeny

[2021-09-25 Sat 18:46] An isogeny is a surjective homomorphism between smooth connected Algebraic groups \(\phi: G \to G'\) with finite kernel \(K\). The degree of \(\phi\) is given by \(\abs{K}\).

See also Isogeny.

Referenced: Degree.

TODO Jacobian criterion

TODO K3 surfaces

Referenced: Intersection forms.

Kähler manifold with holomorphic symplectic structure

[2021-09-16 Thu 20:50] See [nakajima1999], Proposition 1.19.

The following proposition may be useful.

Cf. Calabi-Yau variety, Holonomy group.

Lattice

[2021-09-08 Wed 20:03] A lattice is a free abelian group of finite rank.

Referenced: Lattice ideal, Rational, Affine semigroup.

Lattice ideal

[2021-09-08 Wed 20:03] Let \(L \subset \ZZ^{s}\) be a sublattice. Then we call the ideal \(I_{L} = \langle x^{\alpha} - x^{\beta} \mid \alpha, \beta \in \NN^{s} \text{ and } \alpha - \beta \in L \rangle\) lattice ideal.

Cf. Lattice.

Referenced: Toric ideal.

TODO Linearly reductive

Loop space

[2021-11-14 Sun 14:27] We define the loop space to be functor \(Y((t))( R) = Y(R((t)))\).

Cf. Arc space.

Cf. Ind-scheme.

Referenced: Affine grassmannian.

TODO Luna’s slice theorem

TODO McKay correspondence

TODO Minimal model

Referenced: Castelnuovo theorem.

Minuscule

[2021-11-14 Sun 14:41] A coweight \(\mu\) is minuscule if \[\langle \alpha, \mu \rangle \leq 1\] for all \(\alpha\) positive. For minuscule \(\mu\), we have \({\rm Gr}_{\mu} = \overline{{\rm Gr}_{\mu}} = G/P_{\mu}\), where \(P_{\mu}\) is a partial flag.

TODO Moduli space of curves

Referenced: Canonical line bundle.

Canonical line bundle

[2021-09-13 Mon 19:47] Cf. Moduli space of curves.

We now consider the canonical line bundles on \(M_{g,n}\). We consider \(T_{i}^{* } \to \bar{M}_{g,n}\) the line bundle determined by the cotangent space at the \(i\)th point and denoted by \(\mathbb{L}_{i}\). We have geometric result on the canonical line bundles

We will use the standard notation \(\psi_{i} = c_{1}(\mathbb{L}_{i}) \in H^{2}(\bar{M}_{g,n})\).

For the forgetful map \[\pi: \bar{M}_{g,n + 1} \to \bar{M}_{g,n}\] we define \(k_{r} = \pi_{* } \psi_{n + 1}^{r + 1} \in H^{2r}(\bar{M}_{g,n})\). Under these assumptions, we have

Referenced: Descent integral.

Descent integral

[2021-09-13 Mon 21:09] Following the terminology in Canonical line bundle. We define \[\langle \tau_{1}^{k_{1}} \cdots \tau_{n}^{k_{n}} \rangle_{g,n} = \int_{\bar{M}_{g,n}} \psi_{1}^{k_{1}} \cdots \psi_{n}^{k_{n}}\] Here, we must make sure \(\sum k_{i} = 3g - 3 + n\) and \(2g - 2 + n \leq 0\). Or by simplify notation, we make \(n = \sum_{k_{i}} - 3g + 3\). These integrals are called descent integral.

Referenced: Witten’s Conjecture.

Witten’s Conjecture

[2021-09-13 Mon 20:43] Considering the Descent integrals, we have generating function \[F_{g}(t_{0}, t_{1}, t_{2}, \cdots) = \sum_{\{n_{i}\}} \prod_{i = 1}^{n} \frac{t_{i}^{n_{i}}}{n_{i}!} \langle \tau_{0}^{n_{0}}\tau_{1}^{n_{1}} \cdots \rangle_{g}\] where the sum on right hand side is by sum over sequences with only finitely many nonzero terms. It can also be written as \(F_{g} = \sum_{n = 0}^{\infty} \frac{\langle \phi^{n} \rangle_{g}}{n!}\), where \(\phi = \sum_{i = 0}^{\infty} t_{i} \tau_{i}\). \(F_{g}\) is the generating function that contains all integrals of \(\psi\) classes over all moduli spaces \(\bar{M}_{g,n}\).

We can put all data for all genera together by defining \[F(\lambda, t) = \sum_{g = 0}^{\infty} \lambda^{2g - 2} F_{g}\] We introduce another notation \[\langle \langle \tau_{k_{1}} \cdots \tau_{k_{n}} \rangle \rangle = \frac{\partial}{\partial t_{k_{1}}} \cdots \frac{\partial}{\partial t_{k_{n}}} F\]

We have following Witten’s conjecture and Kontsevich’s theorem

By taking \(U = \frac{\partial^{2} F}{\partial t_{0}^{2}}\) and setting \(n = 1\), we have KdV equation \[\frac{\partial U}{\partial t_{1}} = U \frac{\partial U}{\partial t_{0}} + \frac{1}{12} \frac{\partial^{3} U}{\partial t_{0}^{3}}\]

Referenced: Virasoro constraint.

TODO String equation

TODO Diliton equation

Virasoro constraint

[2021-09-13 Mon 21:34] Using the same generating function as in Witten’s Conjecture. We define partition function \(Z(\lambda, t) = \exp(F)\). Now we define a sequence of operators satisfying the Virasoro bracket. \[L_{n} = -\frac{(2n + 3)!!}{2 ^{n + 1}} \frac{\partial}{\partial t_{n + 1}} + \sum_{i = 0}^{\infty} \frac{(2i + 2n + 1)!!}{(2i - 1)!! 2 ^{n + 1}} t_{i} \frac{\partial}{\partial t_{i + n}} + \frac{\lambda^{2}}{2} \sum_{i = 0}^{n - 1} \frac{(2i + 1)!! (-2i +2n - 1)!!}{2 ^{n + 1}} \frac{\partial^{2}}{\partial t_{i} \partial t_{n - 1 - i}}\] Then we have that \[L_{n}(Z) = 0\] for \(n \geq -1\), which is called Virasoro constraint.

Moduli space of Stable Maps

The moduli space \(\overline{M}_{g,n}(X, \beta)\) is a Deligne-Mumford stack. And the moduli space admits Virtual fundamental class.

Referenced: Gromov-Witten theory.

Morphism

Referenced: Toric.

Étale

[2021-12-26 Sun 19:47] A morphism \(f: X \to Y\) is étale if one of the followings are satisfied

1. \(f\) is flat and unramified, that is for \(y \in Y\) scheme theoretic fiber \(X_{y}\) is isomorphic to a disjoint union of \(\spec K_{i}\) where \(K_{i}\) is separable field extension of \(\kappa(y)\).

2. \(f\) is flat and \(\Omega_{X/Y} = 0\).

3. For any surjection \(A \to A_{0}\) of rings with nilpotent kernel the following commutative diagram can be filled.

   

4. For every point \(x \in X\), there exist affine open neighborhoods \(\spec B\) of \(f(x)\) and \(\spec A \subset f^{-1}(\spec B)\) of \(x\) and an \(A\)-algebra isomorphism \[B \cong (A[x_{1}, \cdots, x_{n}]/(f_{1}, \cdots, f_{n}))_{g}\] for some \(f_{1}, \cdots, f_{n}, g \in A[x_{1}, \cdots, x_{n}]\) such that determinant \(\det(\frac{\dif f_{j}}{\dif x_{i}})_{1 \leq i,j \leq n} \in B\) is a unit.

In addition \(X\) and \(Y\) are locally of finite type over an algebraic closed field \(K\), then above is equivalent to

5. For all \\(x\\) closed point, the induced map \\(\hat{\mathcal{O}}\_{Y, f(x)} \to \hat{\mathcal{O}}\_{X, x}\\) on completions is an isomorphism.
6. If \\(X\\) and \\(Y\\) are smooth, the induced map \\(T\_{X,x} \to T\_{Y,f(x)}\\) on tangent spaces is an isomorphism.

A morphism is called étale at \(x\) if it is étale in a small neighborhood.

Toric

[2021-09-12 Sun 20:23] Let \(V_{i} = \spec (\CC[S_{i}])\) be the Affine toric variety coming from the affine semigroup \(S_{i}\), \(i = 1,2\). Then a Morphism \(\phi: V_{1} \to V_{2}\) is toric if the corresponding map of coordinate rings \(\phi^{* }: \CC[S_{2}] \to \CC[S_{1}]\) is induced by a Semigroup homomorphism \(\hat{\phi}: S_{2} \to S_{1}\).

A toric morphism is equivariant. That is \[\phi(t \cdot p) = \phi(t) \cdot \phi(p)\] for \(t \in T_{N}\) and \(p \in V_{1}\).

For Algebraic stack, a morphsim \(X \to Y\) is unramified or étale if \(X \to Y\) is representable by Deligne-Mumford stack and has the corresponding property.

Non-normal toric variety

[2021-09-08 Wed 20:08] Toric variety is not necessarily normal. For example, \(\spec \CC[x,y]/(x^{2} - y^{3})\).

Polyhedral cone

[2021-09-11 Sat 19:48] A convex polyhedral cone in a real vector space \(N_{\RR}\) is a set of the form \[\sigma = \cone(S) = \{\sum_{u \in S} \lambda_{u} u \mid \lambda_{u} \geq 0\}\] where \(S \subset N_{\RR}\) is finite. And we say that \(\sigma\) is generated by \(S\). In particular, we define \(\cone(\emptyset) = \{0\}\).

The dimension of polyhedral cone is the dimension of the minimal linear space containing the cone.

Referenced: Affine toric variety, Polytope, Dual cone, Supporting hyperplane, Face, Dual face, Relative interior space, Strongly convex, Separation lemma, Rational, Smooth, Simplicial, Saturated.

Polytope

[2021-09-11 Sat 19:57] A polytope in \(N_{\RR}\) is a set of the form \[P = \conv(S) = \{\sum_{u \in S} \lambda_{u}u \mid \lambda_{u} \geq 0, \sum_{u \in S} \lambda_{u} = 1\}\] where \(S \subset N_{\RR}\) is finite. We say that \(P\) is the convex hull of \(S\).

Cf. Polyhedral cone.

Dual cone

[2021-09-11 Sat 19:59] Given a Polyhedral cone \(\sigma \in N_{\RR}\), we have a dual cone in dual space \(M_{\RR}\) of \(N_{RR}\), \[\sigma^{\vee} = \{m \in M_{\RR} \mid \langle m, n \rangle \geq 0 \text{ for all } u \in \sigma\}\]

Obviously, we have \((\sigma^{\vee})^{\vee} = \sigma\).

Supporting hyperplane

[2021-09-11 Sat 20:01] For \(m \in M_{\RR}\), the dual space of \(N_{\RR}\), We define \(H_{m} = \{u \in N_{\RR} \mid \langle m,n \rangle = 0\}\), and \(H_{m}^{+ } = \{u \in N_{\RR} \mid \langle m, n \rangle \geq 0\}\). We call \(H_{m}\) a supporting hyperplane of a Polyhedral cone \(\sigma\) if we have \(\sigma \subset H_{m}^{+}\) and \(H_{m}^{ +}\) is then called supporting half-space.

Referenced: Face, Separation lemma.

Face

[2021-09-11 Sat 20:04] A face of a cone of the Polyhedral cone \(\sigma\) is \(\tau = H_{m} \cap \sigma\) for some \(m \in \sigma^{\vee}\) and \(H_{m}\) is the Supporting hyperplane. We may write as \(\tau \preceq \sigma\). If \(\tau \neq \sigma\), we called \(\tau\) the proper face and write \(\tau \prec \sigma\). A facet is a face of codimension \(1\) and an edge is a face of dimension \(1\).

The faces of the Strongly convex Rational Polyhedral cone gives affine open subset of the Affine toric variety. ([CLS2011] Proposition 1.3.16)

Referenced: Dual face.

Dual face

[2021-09-11 Sat 20:09] Given a Face \(\tau \preceq \sigma\), we have \[\tau^{\perp} = \{m \in M_{\RR} \mid \langle m,n \rangle = 0 \text{ for all } u \in \tau\}\] and the dual face is defined as \[\tau^{* } = \sigma^{\vee} \cap \tau^{\perp}\]

Cf. Polyhedral cone.

Relative interior space

[2021-09-11 Sat 20:11] The relative interior space of a Polyhedral cone \(\sigma\) is the interior of the cone with respect to the minimal linear space containing it. The terminology is useful, since we may consider the cone with less dimension than the ambient linear space.

Strongly convex

[2021-09-11 Sat 20:13] A Polyhedral cone is called strongly convex if the origin is a face of the cone. Or equivalently speaking we have \(\sigma \cap (- \sigma) = \{0\}\).

Referenced: Affine toric variety, Face, Smooth, Simplicial, Saturated.

Separation lemma

Cf. Supporting hyperplane, Polyhedral cone.

Rational

[2021-09-11 Sat 20:16] A Polyhedral cone is called rational if by equating \(N_{\RR} = N \otimes_{\ZZ} \RR\), where \(N\) is a Lattice, we have \(\sigma = \cone(S)\) for \(S \in N\).

For rational cone, \(\rho \subset \sigma\) an edge, the generator of \(\rho \cap N\) is called ray generator of \(\rho\).

Referenced: Affine toric variety, Face, Smooth, Simplicial, Saturated.

Smooth

[2021-09-11 Sat 20:20] A Strongly convex, Rational Polyhedral cone is called smooth or regular if its minimal generators form part of a \(\ZZ\)-basis of \(N\).

The following theorem justify the above name. ([CLS2011] Theorem 1.3.12)

Simplicial

[2021-09-11 Sat 20:24] A Strongly convex, Rational Polyhedral cone is called simplicial if its minimal generators are linearly independent over \(\RR\).

Predeformation functor

[2021-09-19 Sun 20:14] A predeformation functor is a covariant functor \(F: \cat{Art}(\Lambda, k) \to \cat{Set}\) such that \(F(k)\) is the one-element set. The tangent space \(T_{F}\) is defined to be \(F(k[\epsilon]/(\epsilon^{2}))\). The objects of \(\cat{Art}(\Lambda, k)\) are local Artin \(\Lambda\)-algebra with residue field \(k\).

Cf. Tangent space.

Referenced: Deformation functor, Hull, Schlessinger’s theorem.

Presheaf

[2021-11-06 Sat 21:35] A presheaf on a category \(C\) is a functor \[F: C^{op} \to \cat{Set}\] We usually write \(\hat{C}\) for teh category of presheaves on \(C\).

Referenced: Separated, Sheaf.

Separated

[2021-11-06 Sat 21:36] A Presheaf \(F\) on \(C\) is called separated if for every \(U \in C\) and \(\{U_{i} \to U\}_{i \in I} \in \cat{Cov}(U)\) the map \(F(U) \to \prod F(U_{i})\) is injective.

Prestack

[2021-11-04 Thu 18:05] A prestack over \(\cat{Sch}\) is a category \(\mathfrak{X}\) together with functor \(p: \mathfrak{X} \to \cat{Sch}\) such that

1. for any diagram

   

   there exist a morphism \(a \to b\) over \(S \to T\).

2. for any diagram

   

   there exists a unique arrow \(a \to b\) over \(R \to S\) filling in the diagram.

Cf. Presheaf.

A morphism of prestacks \(f: \mathfrak{X} \to \mathfrak{Y}\) is a functor commutes with morphism to \(S\). For \(f,g\) two morphisms, a 2-morphism is a natural transformation \(\alpha: f \to g\) such that \(\alpha_{a}: f(a) \to g(a)\) in \(\mathfrak{Y}\) is over \(\id_{s}\).

Prorepresnetable

[2021-09-19 Sun 20:23] Let \(\widehat{\cat{Art}(\Lambda, k)}\) be the category of complete Noetherian local \(\Lambda\)-algebra with residue field \(k\). Set \(\hat{F} := \varprojlim_{n} F(R/ \mathfrak{m}^{n})\). We say that \(F\) is propresentable if and only if \(\hat{F}\) is representable.

Referenced: Schlessinger’s theorem.

Pure

[2021-09-08 Wed 20:29] \(E\) is pure of dimension \(d\) if \(\dim F = d\) for all non-trivial coherent subsheaves \(F \subset E\).

Referenced: Stable.

Rational surface

[2021-09-25 Sat 18:36] A rational surface is an algebraic surface birational equivalent to \(\CC P^{2}\).

Reflexive

[2021-09-08 Wed 20:39] A coherent sheaf \(E\) of codimension \(c\) is called reflexive if \(\theta_{E}: E \to E^{DD}\) is an isomorphism. \(E^{DD}\) is called the reflexive hull of \(E\).

Cf. Dual sheaf.

Regular sequence

[2021-09-08 Wed 20:47] Let \(M\) be a module over a local ring \(A\). An element \(a \in \mathfrak{m}\) of \(A\) is called \(M\)-regular if \(m \to am\) is injective morphism. A sequence \(a_{1}, \cdots, a_{l} \in \mathfrak{m}\) is a \(M\)-regular sequence if \(a_{i}\) is \(M/(a_{1}, \cdots, a_{i - 1}) M\) regular for all \(i\).

The definition can be generalized to sheaf theoretic language. Let \(X\) be a Noetherian scheme, let \(E\) be a coherent sheaf on \(X\) and \(L\) a line bundle on \(X\). A section \(s \in H^{0}(X, L)\) is called \(E\)-regular if and only if \(E \otimes L^{\vee} \stackrel{\cdot s}{\to} E\) is injective. A sequence \(s_{1}, \cdots, s_{l} \in H^{0}(X, L)\) is called \(E\)-regular similarly in the module theoretic language.

Referenced: Depth.

Residual gerbe

[2021-12-28 Tue 21:44] Let \(X\) be an Algebraic stack and \(x \in \abs{X}\) be a point. Choose a smooth presentation \((U, u) \to (X, x)\). The residual gerbe of \(x\) is the substack \(G_{x} \subset X\) defined as the stackification of the full subcategory \(G_{x}^{pre}\) consisting of object \(a \in X\) over \(S\) which factor as \(a: S \to \spec \kappa(u) \to X\).

TODO Riemann-Roch theorem

TODO Ring

TODO Local ring

Referenced: Hull, Complete.

Complete

[2021-09-19 Sun 20:14] A Local ring is called complete if it is complete with respect to the \(\mathfrak{m}\)-adic topology.

Referenced: Hull.

TODO Noetherian

Referenced: Hull.

Schlessinger’s theorem

[2021-09-19 Sun 20:24] The Schlessinger’s theorem is a criterion for existence of Hull or Prorepresnetibility.

Cf. Predeformation functor, Small thickening.

Referenced: Deformation functor.

Semigroup

Referenced: Toric, Affine semigroup, Irreducible.

Affine semigroup

[2021-09-08 Wed 20:17] An affine semigroup is a Semigroup where the binary operation is commutative, finite generated, and can be embedded in a Lattice \(M\).

Referenced: Affine toric variety, Pointed, Saturated.

Irreducible

[2021-09-11 Sat 20:26] An element in a Semigroup is called irreducible if \(m = m' + m''\) in the semigroup, then we have \(m' = 0\) or \(m'' = 0\).

Pointed

[2021-09-12 Sun 19:24] A pointed affine semigroup \(S\) is the Affine semigroup such that \(S \cap (-S) = \{0\}\).

The pointed affine semigroup is related to Affine toric variety by the following proposition.

Saturated

[2021-09-12 Sun 19:44] An Affine semigroup \(S \subset M\) is saturated if for all \(k \in \NN \backslash \{0\}\) and \(m \in M\), \(km \in S\) implies \(m \in S\). ([CLS2011] p.37)

One important source of the saturated affine semigroup is \(S_{\sigma} = \sigma^{\vee} \cap M\) where \(\sigma\) is a Strongly convex Rational Polyhedral cone.

The saturated affine semigroup is related to normal Affine toric variety. ([CLS2011] Theorem 1.3.5)

TODO Semismall map

Referenced: Small map.

TODO Serre duality

Sheaf

[2021-09-11 Sat 20:46] A Presheaf \(F\) on \(C\) is called a sheaf if for every object \(U \in C\) and covering \(\{U_{i} \to U\}\) the sequence \[F(U) \to \prod_{i} F(U_{i}) \rightrightarrows \prod_{i,j} F(U_{i} \times_{U} U_{j})\] is exact, where the two maps on the right are induced by the two projections \(U_{i} \times_{U} U_{j} \to U_{i}\) and \(U_{i} \times U_{j} \to U_{j}\). Here exact means equalizer.

We write \(C^{~}\) for the category of sheaves on \(C\).

Cf. Site.

The left adjoint \(F \to F^{a}\) is called sheafification and \(F^{a}\) is called the sheaf associated to \(F\).

For \(f: C' \to C\) functor between small categories, we have functor \(f_{*}: \hat{C} \to \hat{C}\) by \((f_{*}F)(X) := F(f(X))\). And we have left adjoint functor \(\hat{f}^{*}\). And if \(f\) is Continuous, then the induced functor between Topoi \(f_{*}: T \to T'\) also has a left adjoint.

If finite limits in \(C'\) are representable and that \(f\) commutes with all finite limits in \(C'\). Then \(f^{*}\) commutes with finite limits and hence \(f\) defines a morphism of topoi.

Referenced: Framed moduli space, Sheafification, Topos.

Euler characteristic

[2021-09-11 Sat 20:33] The Euler characteristic of a sheaf \(E\) is \[\chi(E) = \sum (-1)^{i} h^{i}(X, E)\]

Hilbert polynomial

[2021-09-11 Sat 20:34] The Hilbert polynomial is the function \[P(E,m): m \to \chi(E \otimes \OO(m))\]

For Hilbert polynomial \(P(E, m) = \sum_{i = 0}^{\dim(E)} \alpha_{i}(E) \frac{m^{i}}{i!}\), we define reduced Hilbert polynomial as \[p(E)= P(E)/ \alpha_{d}(E) \]

Referenced: Multiplicity, Rank, Stable, μ-stable.

Multiplicity

[2021-09-11 Sat 20:37] For Hilbert polynomial \(P(E, m) = \sum_{i = 0}^{\dim(E)} \alpha_{i}(E) \frac{m^{i}}{i!}\), and \(E \neq 0\), the leading coefficient \(\alpha_{\dim(E)}(E)\) is called multiplicity.

Rank

[2021-09-11 Sat 20:38] For Hilbert polynomial \(P(E, m) = \sum_{i = 0}^{\dim(E)} \alpha_{i}(E) \frac{m^{i}}{i!}\), and \(\dim(E) = \dim(X) = d\), \[\rk(E) := \frac{\alpha_{d}(E)}{\alpha_{d}(\OO_{X})}\]

Stable

[2021-09-11 Sat 20:40] We write \(p_{1} \leq p_{2}\) for two polynomials if \(p_{1}(x) \leq p_{2}(x)\) when \(x \gg 0\).

A coherent sheaf \(E\) of dimension \(d\) is semistable if \(E\) is Pure and for any proper subsheaf \(F \subset E\), one has Hilbert polynomial \(p(F) \leq p(E)\). And is stable if \(E\) is semistable and \(p(F) < p(E)\)

Referenced: Geometrically stable.

Geometrically stable

[2021-09-11 Sat 20:40] A coherent sheaf \(E\) is geometrically stable if for any base field extension \(X_{K} = X \times_{k} \spec(K) \to K\) the pull-back \(E \otimes_{k} K\) is Stable.

μ-stable

[2021-09-11 Sat 20:41] Let \(E\) a coherent sheaf of dimension \(d = \dim X\) with Hilbert polynomial \(P(E, m) = \sum_{i = 0}^{\dim(E)} \alpha_{i}(E) \frac{m^{i}}{i!}\). We define the degree of \(E\) by \[\deg(E) := \alpha_{d - 1}(E) - \rk(E) \cdot \alpha_{d - 1}(\OO_{X})\] and its slope by \[\mu(E) := \frac{\deg(E)}{\rk(E)}\]

A coherent sheaf \(E\) of dimension \(d\) is \(\mu\)-(semi)stable if \(T_{d - 2}(E) = T_{d - 1}E\) and \(\mu(F)(\leq)\mu(E)\) for all subsheaves \(F \subset E\) with \(0 < \rk(F) < \rk(E)\).

Sheafification

[2021-11-06 Sat 22:01] Cf. Sheaf.

Simple resolution

[2021-09-21 Tue 15:31] See [nakajima1999] p.47.

A simple singularity is a quotient space \(\CC^{2} / \Gamma\), where \(\Gamma\) is a finite subgroup of \(\SU(2)\). A simple resolution is a resolution of simple singularity.

The following theorem construct the simple resolution by Hilbert scheme of points.

Referenced: Simple singularity.

Simple singularity

[2021-09-21 Tue 15:32] See Simple resolution.

Site

[2021-11-06 Sat 21:24] A category with a Grothendieck topology is called a site.

Similarly, we can define fppf site (flat and locally of finite presentation) and smooth site.

Referenced: Continuous, Étale site, Fppf site, Sheaf, Smooth site, Topos.

TODO Small map

[2021-11-20 Sat 21:44] See Semismall map.

Small thickening

[2021-09-19 Sun 20:23] A surjective map \(f: A \to B\) in \(\cat{Art}(\Lambda, k)\) is a small thickening if \(\ker f \mathfrak{m}_{A} = 0\) and \(\ker f\) is principal.

Referenced: Schlessinger’s theorem.

Smooth

[2021-09-28 Tue 19:58] Let \(F, F': \cat{Art}(\Lambda, k) \to \cat{Set}\) be covariant functors, with a morphism \(\phi: F \to F'\). We say that \(\phi\) is smooth if for every surjection \(A \to B\) in \(\cat{Art}(\Lambda, k)\) the map \[F(A) \to F(B) \times_{F'(B)}F'(A)\] is surjective.

Smoothness of Hilbert scheme

[2021-09-16 Thu 20:44] See [nakajima1999], Section 1.3.

Cf. Hilbert scheme, Hilbert-Chow morphism.

Smooth site

[2021-11-06 Sat 21:32] See Site.

TODO Spectral sequence

Referenced: Eilenberg-Zilber theorem.

TODO Beilison spectral sequence

Splitting torus

[2021-09-16 Thu 20:02] A splitting torus is a torus of type \(k^{\times n}\).

Cf. Torus.

Stabilizer

[2021-12-28 Tue 21:08] If \(X\) is an Algebraic stack and \(x: \spec K \to X\) is a field-valued point, the stabilizer of \(x\) is defined as the group scheme \(G_{x} = Aut_{X(K)}(x)\).

By representability of the diagonal, \(G_{x}\) is a group algebraic space.

If \(X\) is a Deligne-Mumford stack and \(x \in \abs{X}\), we define the geometric stabilizer of \(x\) as the discrete group \(G = G_{\bar{x}}\) where \(\bar{x}: \spec k \to X\) is any representative of \(x\) with \(k\) algebraic closed.

Stack

[2021-11-04 Thu 18:06] A Prestack \(\mathfrak{X}\) over a Site \(\mathfrak{S}\) is a stack if the following conditions hold for all coverings \(\{S_{i} \to S\}\) of an object \(S \in \mathfrak{S}\):

1. For object \(a\) and \(b\) in \(\mathfrak{X}\) over \(S\) and morphism \(\phi_{i}: a|_{S_{i}} \to b\) such that \(\phi_{i}|_{S_{ij}} = \phi_{j}|_{S_{ij}}\), there exists a unique morphism \(\phi: a \to b\) with \(\phi|_{S_{i}} = \phi_{i}\).
2. For objects \(a_{i}\) over \(S_{i}\) and isomoprhisms \(\alpha_{ij}: \alpha_{i}|_{S_{ij}} \to \alpha_{j}|_{S_{ij}}\) satisfying \(\alpha_{ij}|_{S_{ijk}} \circ \alpha_{jk}|_{S_{ijk}} = \alpha_{ik}|_{S_{ijk}}\) on \(S_{ijk}\), then there exists an object \(a\) over \(S\) and isomorphisms \(\phi_{i}: a|_{S_{i}} \to a_{i}\) such that \(\alpha_{ij} \circ \phi_{i}|_{S_{ij}} = \phi_{j}|_{S_{ij}}\) on \(S_{ij}\).

Symmetric power of P^1

[2021-09-16 Thu 20:28] Fact, \(S^{n}(\PP^{1}) = \PP^{n}\).

TODO Symplectic resolution

Symplectic structure of Hilbert scheme

[2021-09-16 Thu 20:41] See [nakajima1999], Section 1.4.

Cf. Symplectic manifold, Hilbert scheme.

Topos

[2021-11-06 Sat 22:02] A topos is a category \(T\) equivalent to the category of Sheaves of sets on Site.

A morphism of topoi \(f: T \to T'\) is an adjoint pair \[f = (f^{* }, f_{* }, \phi)\] where \[f_{* }: T \to T', \quad f^{* }: T' \to T\] are functors and \(\phi\) is an isomorphism \[\phi: \Hom_{T}(f^{* }(-), -) \to \Hom_{T'}(-, f_{* }(-))\] of bifunctors and \(f^{* }\) commutes with finite limits.

Referenced: Continuous, Category of modules, Sheaf, Ringed, Point.

Ringed

[2021-11-07 Sun 10:12] A ringed Topos is a pair \((T, \Lambda)\), where \(\Lambda\) is a ring object in topos \(T\). A morphism between ringed topoi is a pair \((f, f^{\shar})\) consisting of a morphism of topoi \(f: T \to T'\) and a morphism of rings in \(T'\) \(f^{\shar}: \Lambda' \to f_{*} \Lambda\).

Point

[2021-11-07 Sun 15:17] Let \(\text{pt}\) denote the Topos of sheaves on the one-point space. A point of the topos \(T\) is a morphism of topoi \[x: \text{pt} \to T\] We say that the topos \(T\) has enough points if there exists a set of points \[\{x_{i}: \text{pt} \to T\}\] such that the induced functor \[T \to \cat{Set}^{I}, F \to \{x_{i}^{*}F\}\] is faithful.

Toric ideal

[2021-09-08 Wed 20:05] A prime Lattice ideal is called toric ideal.

We have following criterion for toric ideal.

Referenced: Affine toric variety.

TODO Toric variety

[2021-09-08 Wed 20:08] The toric variety is a field connecting algebraic geometry, combinatorial and mathematical physics. Our main reference is [CLS2011].

Cf. Affine toric variety

Referenced: Non-normal toric variety.

Torsion filtration

[2021-09-08 Wed 20:30] The torsion filtration of a coherent sheaf \(E\) is the unique filtration \[0 \subset T_{0}(E) \subset \cdots \subset T_{d}(E) = E\] where \(d = \dim(E)\) and \(T_{i}(E)\) is the maximal subsheaf of \(E\) of dimension \(\geq i\).

Torsion free

[2021-09-08 Wed 20:52] A coherent sheaf \(E\) is called torsion free if for each \(x \in X\) and \(s \in \OO_{X,x} \backslash \{0\}\) the multiplication by \(s\) induces a injective morphism \(E_{x} \to E_{x}\).

Torsor

[2021-12-26 Sun 20:39] Let \(S\) be a Site and \(G\) a sheaf of groups on \(S\). A \(G\)- torsor on \(S\) is a sheaf \(P\) of sets on \(S\) with left action \(\sigma: G \times P \to P\) of \(G\) such that

1. For every object \(X \in S\), there exists a covering \(\{X_{i} \to X\}\) such that \(P(X_{i}) \neq 0\), and
2. The action map \((\sigma, p_{2}): G \times P \to P \times P\) is an isomorphism.

Torus

[2021-09-08 Wed 19:51] Over \(\CC\), a torus is \(\CC^{* n}\). In general a scheme \(X\) over \(k\) is called a torus if \(X_{k^{a}}\) is of the form \((k^{a})^{\times n}\) where \(k^{a}\) is the algbraic closure of \(k\).

The following proposition is useful in toric geometry over \(\CC\). The image of a torus in a torus is also a torus.

A group subvariety of a torus is a torus.

Referenced: Affine toric variety, Character (of a torus), Splitting torus.

Unramified

[2021-12-28 Tue 20:41] See Étale.

Versal extension

[2021-09-15 Wed 21:37] An \(A\)-Extension \((P, f)\) is called versal if for every other \(A\)-extensions \((R', \phi)\) of \(R\), there exists a homomorphism of extensions \(r: (P, f) \to (R, \phi)\). For example, for \(R = P/I\) where \(P\) a polynomial ring over \(A\), \[0 \to I/I^{2} \to P/I^{2} \to R \to 0\] is a versal \(A\)-extension of \(R\).

Virasoro bracket

[2021-09-13 Mon 21:32] Virasoro bracket is a set of commutator relation for operators \(L_{n}\) such that \[[L_{n}, L_{m}] = (n - m) L_{n + m}\]

Referenced: Virasoro constraint.

TODO Virtual fundamental class

[2021-09-01 Wed 16:57] The analog of Fundamental class for stacks and orbifold.

Referenced: Donaldson-Thomas theory, Moduli space of Stable Maps.

TODO Yoneda product

Symplectic Geometry

Liouville vector field

[2021-09-02 Thu 10:51] Let \((M, \omega)\) is a Symplectic manifold. A vector field \(V: M \to TM\) is Liouville if \(L_{X} \omega = \omega\). Cf. Lie derivative.

The existence of a Liouville vector field implies that \((M, \omega)\) is exact as the one-form \(\lambda = i_{V} \omega\) satisfies that \(\dif \lambda = \omega\).

Referenced: Weinstein manifold.

Symplectic manifold

[2021-09-02 Thu 10:43] A symplectic manifold is a smooth Manifold \(X\) of even dimension \(\dim X = 2n\), equipped with a symplectic form a closed smooth \(2\)-form \(\omega \in \Omega_{cl}^{2}(X)\) such that \(\omega\) is non-degenerate. That is \(\omega^{n}\) has the maximal rank at every point \(p\), or equivalently that \((\wedge^{2} T_{p}^{* }X, \omega_{p})\) symplectic vector space for every \(p\).

Referenced: Symplectic structure of Hilbert scheme, Liouville vector field, Weinstein manifold.

Weinstein manifold

[2021-09-02 Thu 10:42] A Symplectic manifold \((M, \omega)\) is Weinstein if it comes with a distinguished Liouville vector field \(Y\), and a proper bounded below function \(h: M \to \RR\), such that \(\dif h(Y)\) is positive on a sequence of level set \(h^{-1}(c_{k})\) with \(\lim_{k}c_{k} = \infty\).

A Weinstein manifold is called of finite type if \(\dif h(Y) > 0\) outside a compact subset of \(M\) and is called complete if the flow of \(Y\) is defined for all times.

Geometric representation theory

[2021-08-29 Sun 15:26] In geometric representation theory, we tries to encode the algebra operation in geometric objects. The one of the general construction is the usage of correspondence, that is the cycles or sheaves etc in \(X_{1} \times X_{2}\). It is a nonlinear analog of matrices. To retain the linear information, we consider functors such as Equivariant cohomology and Equivariant K-theory.

Langlands dual pair

[2021-09-25 Sat 18:39] The Langlands dual is a pair of Algebraic groups \((X, Y)\) such that the root system of \(X\) is coroot of \(Y\) and vice versa. The center of \(X\) is the fundamental group of \(Y\) and vice versa.

For example \((\Sp(n), \SO(2n + 1))\), \((\SO(2n), \SO(2n))\) and \((\SL(n), \PGL(n))\).

Steinberg variety

[2021-08-29 Sun 15:41] For a general symplectic resolution \[\pi: X \to X_{0}\] we have the Steinerg variety \(X \times_{X_{0}} X\).

Referenced: Algebraic Geometry.

Geometric Topology

[2021-09-15 Wed 18:20] Named geometric topology, this section includes some result in \(4\)-manifold and knot theory and higher dimensional topology.

11/8 conjecture

By Freedman’s classification of topological 4-manifold, Rokhlin Theorem and Donaldson diagonalizable theorem, the only remaining case to determine the existence of smooth structure on topological \(4\)-manifold are those with \(Q_{X}\) even and indefinite. That is \(Q_{X} = mE_{8} \oplus n \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\). Since we have K3 surface with \(Q_{X} = 2E_{8} \oplus 3 \begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}\) and \(S^{2} \times S^{2}\) with \(Q_{X} = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}\), we can show that the for \(Q_{X}\) with \(m = 2k\) and \(n \geq 3k\) the smooth structure on the manifold exists. The conjecture is that that is the only possibility for the existence of smooth structure.

A theorem by Furuta says

The Seiberg-Witten invariant together with spin Dirac operator can give one proof. The proof suggests Bauer-Furuta invariant.

Affine braid group

[2021-09-21 Tue 15:56] We define the affine braid group to be \[\tilde{B_{n}} = \langle s_{1}, \cdots, s_{n -1}, t_{1}, \cdots, t_{n}\rangle\] with the same condition as Braid group with addition \[s_{i}t_{i + 1} = t_{i} s_{i}\]

TODO Alexander duality

TODO Alexander polynomial

Referenced: Fintushel-Stern knot surgery.

Anti-self-dual

[2021-10-15 Fri 20:21] A \(n\)-form \(w\) on \(2n\) manifold is called self-dual if \(* w = -w\). Cf. Self-dual.

Referenced: Self-dual.

TODO Bauer-Furuta invariant

Referenced: Geometric Topology.

Botany problem of 4-manifold

[2021-10-15 Fri 19:52] For \(b^{+} > 1\), \(\pi_{1} = 1\) irrducible case, by theorem of Fintushel-Stern, any such \(X\) admits \(\infty\) smooth structure.

For \(b^{+} = 1\), then \(\CC P^{2} \shar n \overline{\CC P}^{2}\) for \(n \geq 2\) has \(\infty\) smooth structure. \(S^{2} \times S^{2}\) and \(n < 2\) case is unknown.

\(b^{+} = 0\) case is completely unknown.

Braid

[2021-09-21 Tue 15:41] A braid is the isotopy class of a Geometric braid.

Theorem

The braids are in bijection with Redemeister classes of diagrams.

Referenced: Pure, Positive, Braid group.

Pure

[2021-09-21 Tue 15:46] A Braid is called pure if it connects \(\{x_{0}\} \times \{0\}\) to \(\{x_{i}\} \times \{1\}\).

Referenced: Braid group.

Positive

[2021-11-04 Thu 18:43] A Braid is called positive if it is a product of \(\sigma_{i}\) and not \(\sigma_{i}^{-1}\) in braid group. And a braid is quasi-positive if it has the form \(\prod_{k = 1}^{m} w_{k} \sigma_{ik} w_{k}^{-1}\) where \(w_{k}\) is any word in the braid group. A Knot positive (resp. quasi-positive) if it is a completion of a positive (resp. quasi-positive) braid.

Referenced: Quasi-positive, Knots bounds algebraic surface.

Quasi-positive

[2021-11-04 Thu 18:44] See Positive.

Referenced: Knots bounds algebraic surface.

Braid diagram

[2021-09-21 Tue 15:42] Consider the projection \(\pi: \RR^{2} \times [0,1] \to \RR \times [0,1]\). Then \(\hat{\beta}\) is called generic if \(\pi(\hat{\beta})\) has simple transversal singular points outside of \(\RR \times \{0,1\}\). Such \(\pi(\hat{\beta})\) is called a geometric braid diagram.

A braid diagram is a regular isotopy class of a geometric braid diagram.

Braid group

[2021-09-21 Tue 15:44] The Braids form a group called braid group where the composition is given by glueing two embeddings. Obviously, the Pure braids form a subgroup of braid group. The braid group is denoted by \(B_{n}\), and pure braid group by \(P_{n}\).

Theorem

\(B_{n}\) is generated by elements

with defining relations \[s_{i}s_{i+1}s_{i} = s_{i + 1}s_{i}s_{i + 1},\; i = 1, \cdots, n - 1\] \[s_{i}s_{j} = s_{j}s_{i}, \; \abs{i - j} > 1\]

The braid group can be realized as a Fundamental group of a manifold. Define \(C_{n} = \{x_{1}, \cdots, x_{n} \in \RR^{2} \mid x_{i} \neq x_{j}\}\). Then we have \(\pi_{1}(C_{n}) = P_{n}\). By considering the \(S_{n}\) action on \(C_{n}\), we have that \(B_{n} = \pi_{1}(C_{n}/ S_{n})\).

Referenced: Affine braid group, Hecke-Iwahori algebra, Representation of braids.

Characteristic class for 4-manifold

[2021-09-22 Wed 20:49] We may assume that \(X\) is a connected, closed, oriented, smooth \(4\)-manifold. We have a Signature theorem by Hirzebruch.

Cf. Pontryagin class.

And theorem by Wenjun Wu on Stiefel-Whitney class.

By above theorem, Freedman’s classification of topological 4-manifold, Donaldson diagonalizable theorem and algebraic classification of bilinear forms, we have

Clasp number

[2021-11-04 Thu 18:39] We define the clasp number of \(K\) to be minimal number of double points of the immersed disk in \(D^{4}\) with boundary \(K\) with only transverse double point.

We have \(c_{4}(k) \leq U(K)\) (Unknot number), for we can trade knot for double points. Also, we have \(g_{S}(K) \leq c_{4}(K)\) (Slice genus) since we can trade double points for genus.

Clifford algebra

[2021-10-15 Fri 19:56] Let \(H\) be a real vector space. The tensor algebra \(T(H)\) modulo ideal generated by \(\{v \otimes v + \norm{v}^{2}\}\) is called the Clifford algebra.

Referenced: Clifford module.

Clifford module

[2021-10-15 Fri 20:00] A Clifford module of Clifford algebra over \(H\) is a Hermitian complex vector space \(V\) equipped with a Clifford multiplication \(\gamma: H \to \End(V)\) such that

• If \(\norm{e} = 1\), then \(\gamma(e)^{2} = -1\).
• If \(e_{1} \perp v_{2}\), then \(\gamma(e_{1}) \cdot \gamma(e_{2}) + \gamma(e_{2}) \cdot \gamma(e_{1}) = 0\).
• \(\gamma(e)^{* } = - \gamma(e)\).

That is a Clifford module is a representation of Clifford algebra.

The irreducible Clifford module is competely identified.

Referenced: Dirac operator, Spin^C structure.

Corbordism

[2021-11-20 Sat 21:38] Let \(\mathscr{C}\) denote one of the following categories: smooth / topological / piecewise linear category. Let \(M_{0}, M_{1}\) be oriented \(\mathscr{C}\)-manifold. A cobordism \(W\) form \(M_{0}\) to \(M_{1}\) is a \(n + 1\) dimensional \(\mathscr{C}\)-manifold such that \(\partial W = \bar{M_{0}} \coprod M_{1}\).

We can define the cobordism category \(\cat{Cob}_{n}\), where the objects are closed \(n\) dimensional manifolds and the morphisms are isotopic classes of cobordisms.

Referenced: h-Cobordism.

TODO Cork

Critical point

[2021-09-30 Thu 15:16] The critical point of \(f: X \to \RR\) is the map \(\dif f_{p}: T_{p}X \to T_{f(p)} \RR\) is trivial. And \(r = f(p)\) is the critical value, otherwise \(r\) is called the regular value.

Cf. Regular point.

Referenced: Non-degenerate, Critical value, Morse inequality.

Non-degenerate

[2021-09-30 Thu 15:19] We say \(p\) Critical point non-degenerate if \(\det({\rm Hess} f_{p}) \neq q\). The index of the \(p\) is the number of negative eigenvalues of Hessian of \(f\) at \(p\).

Referenced: Index.

Critical value

[2021-09-30 Thu 15:18] See Critical point.

Referenced: Regular value.

Diagram

The image of a generic geometric representative \(\hat{L}\) of Link \(L\) from \(\RR^{3}\) to \(\RR^{2}\) is a diagram of \(L\).

For equivalent class of diagrams, we have the following theorem by Reidemeister.

Theorem

Equivalence class of diagrams with respect to the local moves

We can also consider the diagram of Framed links in \(\RR^{3}\). And we have equivalence class of regular isotopy class of framed diagrams on \(\RR^{2}\) is given by

Referenced: Skein relations, Unknot number.

Dirac operator

[2021-10-15 Fri 20:05] Given a Clifford module \(S\) over \(H\), we define the Dirac operator \(\dirac: C^{\infty}(H, S) \to C^{\infty}(H, S)\) by \[\dirac \phi = \sum_{i = 1}^{n} \gamma(e_{i}) \cdot \frac{\partial \phi}{\partial e_{i}}\]

\(\dirac\) is a self-adjoint operator with \(\dirac^{2} = \Delta\).

On manifold, given a Spin^C connection \(A\), we can also define the Dirac operator \[\dirac_{A} = \rho \circ \nabla_{A}: \Gamma(s) \to \Gamma(s)\] For decomposition \(S = S^{+} \oplus S^{-}\), we have \(\dirac_{A} = \dirac_{A}^{ +} + \dirac_{A}^{-} \)\(\dirac_{A}^{ +}: \Gamma(S^{ +}) \to \Gamma(S^{-})\) and \(\dirac_{A}^{-}: \Gamma(S^{-}) \to \Gamma(S^{ + })\).

On manifolds, the Dirac operator is also self-adjoint. And we call \(\dirac_{A}^{2}\) the Dirac Laplacian. We have \[\dirac_{A}^{2} \phi = \Delta \phi + \rho(F_{A^{\tau}}^{+ }) \phi + \frac{s}{4} \phi\] where \(\Delta\) is the usual Laplacian and \(F_{A^{\tau}}^{+}\) is the Self-dual part of curvature of the connection induced by \(A\) on \(\det(S^{ +})\)

Donaldson diagonalizable theorem

Cf. Intersection forms.

One of the proofs is given by Seiberg-Witten invariant and following algebra result.

Referenced: Geometric Topology, Characteristic class for 4-manifold.

TODO Dot notation

Elliptic fibration

[2021-09-25 Sat 18:31] A map \(q: X \to \Sigma\) is an elliptic fibration if there exists a finite set \(S \subset \Sigma\) such that \(X \backslash q^{-1}(S) \to \Sigma \backslash S\) is a fiber bundle whose fiber is a smooth curve of genus \(1\), that is the elliptic curve. We call \(q^{-1}(t)\) a regular fiber if \(t \not \in S\) and a singular fiber if \(t \in S\).

Cf. Vector bundle.

Exotic R^4

[2021-09-15 Wed 18:51] \(\RR^{4}\) has infinitely many smooth structures. While other Eucliean space has unique smooth structure by Stallings.

The construction follows from the oberservation for \(X = \CC P^{2} \shar 9 \overline{\CC P}^{2}\), we have \(Q_{X} \cong (-E_{8}) \oplus (-1) \oplus (1)\). Then if all \(\RR^{4}\) has same smooth structure, some kind of surgery gives impossible \(4\)-manifolds. By examing the construction carefully, one shows that there exists a parameter such that every two construction gives different smooth structures.

Exotic \(R^{4}\) can also be constructed using Knot theory.

TODO Exotic surfaces

TODO Fintushel-Stern knot surgery

[2021-11-14 Sun 15:02] The following theorem by Fintushel-Stern gives many exotic smooth structures.

Framed

[2021-09-21 Tue 16:01] See Framing.

Referenced: Jones-Markov trace.

Framing

Framing of a Link in \(\RR^{3}\) is an isotopy class of a continuous section of a normal bundle to \(L \subset \RR^{3}\).

Referenced: Diagram, Framed.

Freedman’s classification of topological 4-manifold

Cf. Manifold, Kirby-Sieberman invariant, Whitehead theorem, Intersection forms, Signature.

By Freedman’s result, the Chern number \(c(X) = 2 \chi(X) + 3 \sigma(X)\), and Holomorphic Euler number \(\chi_{h}\) and \(t(X) = \begin{cases} 0 & Q_{X} \text{ is even.} \\ 1 & Q_{X} \text{ is odd.} \end{cases}\).

Referenced: Geometric Topology, Characteristic class for 4-manifold, Geography of 4-manifold.

Gabai’s lightbulb theorem

[2021-11-14 Sun 15:13] Given \(F \hookrightarrow X\), we say an embedded sphere \(\Sigma \hookrightarrow X\) is a geometric dual of \(F\) if \(\Sigma \cdot \Sigma = 0\) and \(F \pitchfork \Sigma = \{pt\}\). And \(F\) is \(\Sigma\)-essential if \(\pi_{1}(F \backslash \Sigma) \to \pi_{1}(X \backslash \Sigma)\) is trivial. The Gabai’s lightbulb theorem states

Geography of 4-manifold

[2021-10-15 Fri 19:33] By Freedman’s classification of topological 4-manifold, the geography of \(4\)-manifold is the question which triple \((c, \chi_{h}, t)\) can be realized as smooth, Irreducible, simply connected \(X\).

The following conjecture relates the irreduciblity of \(4\)-manifold and Holomorphic Euler number.

Also, we have following theorem.

For complex geography problem, we have inequalities for surface of general type, Bogomolov-Miyaoka-Yau inequality\(c \leq 9 \chi_{h}\) and Noether inequality \(c \geq 2 \chi_{h} - 6\). If \(X\) is minimal, simply conected then \(X \cong_{diff} \CC P^{2}\) or \(X \cong_{diff} S^{2} \times S^{2}\). If \(X\) is elliptic, then \(c = 0\). The currenst status of complex geography problem is in following diagram.

The current status of smooth geography problem is deduced from complex case by rational blow down.

The current status of symplectic geography problem is in following picture.

The \(c < 0\) case is deleted by the theorem of Taubes.

There exists symplectic BMY conjecture.

Geometric braid

[2021-09-21 Tue 15:38] A geometric braid in \(\RR^{2} \times [0,1]\) is an embedding \(\hat{\beta} : [0,1]^{\coprod n} \to \RR^{2} \times [0,1]\) connecting \(\{x_{0}, \cdots, x_{n}\} \times \{0\}\) and \(\{x_{0}, \cdots, x_{n}\} \times \{1\}\) componentwise.

Referenced: Braid.

Geometric tangle

[2021-10-19 Tue 18:02] A geometric tangle is an embedding of \([0, 1]^{\coprod k} \coprod (S^{1})^{\coprod m}\) into \(\RR^{2} \times [0,1]\) such that images of endpoints of intervals are on \(\RR^{2} \times \{0\}\) and \(\RR^{2} \times \{1\}\), intersection of intervals is transversal to the planes and images of cirvles are strictly inside of \(\RR^{2} \times [0,1]\).

A based tangle is the regular isotopy class, preserving boundary points of a geometric tangle. And a standardized tangle is a based tangle with endpoints being in a line \(\RR \subset \RR^{2}\).

TODO Handle decomposition

Referenced: Monotone, Handle move, Morse complex, Morse theory.

Monotone

[2021-09-30 Thu 15:49] A Handle decomposition \(X_{0}| \cdots | X_{k}\) is called monotone if \(X_{i}\) only consists of \(i\)-handles for all \(i\).

Cf. Self-indexed Morse function.

Referenced: Handle move, Morse complex.

TODO Handle move

[2021-09-30 Thu 16:44] Any two Monotone Handle decomposition of \(X\) are related by the following handle moves

• handle slide
• creation / cancellation

h-Cobordism

[2021-09-15 Wed 18:42] With the same notation as in Corbordism, we say that \(W\) is h-cobordism if \(M_{0} \hookrightarrow W\) and \(M_{1} \hookrightarrow M\) are all homotopy equivalences.

The following theorem by Smale, Kirby-Siebenmann show that h-cobordism is powerful classification tool in high dimensional topology.

The key lemma in the proof of the last theorem is by Whitney, which gives the mysterious dimension bound in the theorem.

Referenced: Botany problem of 4-manifold, Poincarè conjecutre, Wall Theorem.

Hecke-Iwahori algebra

[2021-09-21 Tue 15:53] For Braid group \(B_{n}\), the elements \((s_{i} - q)(s_{i} + q^{-1})\) generates an ideal in the group algebra \(\CC[B_{n}]\). The Hecke-Iwahori algebra is the quotient algebra with respect to the ideal, i.e., \[H_{n}(q) = \langle s_{1}, \cdots, s_{n} \rangle\] such that \[s_{i}s_{i+1}s_{i} = s_{i + 1}s_{i}s_{i + 1},\; i = 1, \cdots, n - 1\] \[s_{i}s_{j} = s_{j}s_{i}, \; \abs{i - j} > 1\] \[(s_{i} - q)(s_{i} + q^{-1}) = 0\] The interesting case is that \(q\) be the root of \(1\).

\(H_{n}(q)\) is a deformation of \(\CC[S_{n}]\).

Heegaard diagram

[2021-10-15 Fri 19:03] A genus \(g\) Heegaard diagram is \((\Sigma_{g}, \alpha_{1}, \cdots, \alpha_{g}, \beta_{1}, \cdots, \beta_{g})\) consists of a genus \(g\) Orientable surface \(\Sigma_{g}\), two families of simple closed curves \(\alpha_{* }\), \(\beta_{* }\) such that \(\alpha_{i} \cap \alpha_{j} = \beta_{i} \cap \beta_{j} = 0 \) for \(i \neq j\) and \(\Sigma_{g} - \coprod \alpha_{i}, \Sigma_{g} - \coprod \beta_{i}\) are both connected.

Cf. Heegaard splitting.

Referenced: Trisection diagram.

Heegaard splitting

[2021-10-15 Fri 19:03] A genus \(g\) Heegaard splitting of \(3\)-manifold is a decomposition of \(Y\) into two genus \(g\) handle bodies.

Referenced: Heegaard diagram.

Hessian

[2021-09-30 Thu 15:21] The Hessian is the well-defined map \({\rm Hess} f_{p}: T_{p}X \otimes T_{p}X \to \RR\) defined by \(\left(\frac{\partial^{2}f}{\partial x_{i} \partial x_{j}}\right)\).

Referenced: Non-degenerate.

Holomorphic Euler number

[2021-10-15 Fri 19:26] We define the holomorphic Euler number by \(\chi_{h}(X) = \frac{\chi(X) + \sigma(X)}{4}\).

Referenced: Freedman’s classification of topological 4-manifold, Geography of 4-manifold.

HOMFLY invariant

[2021-09-13 Mon 18:37] Since Skein relations are compatible with Reidemeister moves, for every Link, we may consider its diagram \(D_{L}\) and we define \([D_{L}] = \langle L \rangle [O]\). Here \(O\) is one loop and \(\langle L \rangle \in \ZZ[A, B^{\pm 1}, C]\). \(\langle L \rangle\) depends only on \(L\) but not on representative diagram \(D_{L}\). \(\langle L \rangle\) is the HOMFLY invariant of framed links.

For unframed links, we also have HOMFLY invariants by consider \(L = L_{1} \coprod \cdots \coprod L_{n}\) as framed links and consider \(\langle \langle L \rangle \rangle = B^{- \sum_{i} w(L_{i})} \langle L \rangle\) where \(w(L_{i})\) is the winding number.

Homotopic slice

[2021-12-05 Sun 17:34] A Knot \(K \subset S^{3}\) is called homotopy slice if \(K\) bounds a smooth \(i:D^{2} \hookrightarrow X\) with \(\partial X = S^{3}\) such that \(i\) is homotopic to \(i': D^{2} \to \partial X\) relative to \(\partial D^{2} = K\).

Kronheimer and Mrowka has the following conjecture.

Cf. Rasmussen invariant.

The above conjecture is proved for \(X = \CC P^{2}\).

Index

[2021-09-30 Thu 15:54] See Non-degenerate.

Referenced: Morse inequality.

Integer homology 3-sphere

[2021-10-15 Fri 19:05] The integer homology 3-sphere is a \(3\)-manifold \(Y\) with \(H_{1}(Y; \ZZ) = 0\).

Freedman proved following result.

Cf. Contractible.

Intersection forms

[2021-09-17 Fri 21:48] For \(4\)-manifold, we have nondegenerated bilinear form \[Q_{X}: H^{2}(X; \ZZ)/(tors) \times H^{2}(X; \ZZ)/(tors) \to \ZZ\] defined by \((\alpha \cup \beta)[X]\) called intersection form. It is a symmetric, unimodular form.

As an example of the intersection form, we have the intersection form of K3 surfaces \(Q_{X} = 2E_{8} \oplus 3 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)

Cf. K3 surfaces, E8.

We can generalized above intersection form to manifold with boundary. The interesection form is defined on \(V = \im(H_{2}(X) \to H_{2}(X, \partial X))/(torsion)\). \(Q_{X}(i_{* }(\alpha), i_{* }(\beta)) = (\pd(\alpha) \cap \pd(\beta)) \cdot [X]\).

For a manifold with boundary, the intersection form is still nondegenerate but not necessarily unimodular. In fact, we have the following proposition

Referenced: Lefschetz fixed point theorem, Donaldson diagonalizable theorem, Freedman’s classification of topological 4-manifold, Signature.

Irreducible

[2021-10-15 Fri 19:22] A simply connected, closed, oriented \(4\)-manifold \(X\) is irreducible if \(X \not \cong_{diff} X_{0} \shar X_{1}\) with \(X_{0}, X_{1} \not \cong S^{4}\).

Referenced: Geography of 4-manifold, Seiberg-Witten invariant.

Jones-Markov trace

[2021-09-21 Tue 15:58] There exists a Jones-Markov trace \(\tr: \tilde{B}_{n} \to k = \ZZ[q, q^{-1}][\frac{1}{(q - q^{-1})}]\) such that \[\tr(\beta \gamma \beta^{-1}) = \tr(\gamma)\] \[\tr(\beta s_{n}) = \tr(\beta t_{n}) = z \tr(\beta)\]

It defines invariants of Framed Links. To get a invariant of links, we consider Writhe number, \(\omega(\beta)\). We have \(\tr(\beta)z^{w(\beta)}\) is an invarinat of links.

TODO Jones polynomial

Referenced: Jones-Markov trace.

TODO Jones polynomial

Jones twist

[2021-09-21 Tue 16:15] A map \(\mu: V \to V\) is called Jones twist if \[(\mu \otimes \mu) R = R(\mu \otimes \mu)\] \[(\id \otimes \tr)(\id \otimes \mu) PR = z \id_{V}\]

TODO Khovanov homology

Referenced: Lee homology.

TODO Kirby calculus

TODO Kirby move

Kirby-Sieberman invariant

[2021-09-17 Fri 22:00] We define the set \({\rm Top}_{m} = \{\text{ homomorphism } f: \RR^{m} \to \RR^{m} \text{ such that } f(0) = 0\}\). And \({\rm PL}_{m} = \{\text{ piecewise linear homeomorphism} f: \RR^{m} \to \RR^{m} \text{ such that } f(0) = 0\}\).

The Kirby and Sieberman has following deep and mysterious theorem.

So to have PL structure on \(X\), \(\dim X \geq 5\) we need to overcome the obstruction called Kirby-Sieberman invariant \(KS(X) \in H^{4}(X; \ZZ/2)\). So \(X\) has a PL-structure then \(\ks(X) = 0\). And for \(\dim X = 4\), \(\ks(X) = 0\) if and only \(X \times \RR\) has a PL-structure.

Referenced: Freedman’s classification of topological 4-manifold.

TODO Knizhnik–Zamolodchikov equation

Knot

[2021-09-13 Mon 18:13] A knot (in \(S^{3}\)) is an isotopy class of an Immersion \(\hat{K}: S^{1} \to S^{3}\), \(K = [\hat{K}]\).

Referenced: Positive, Link, Homotopic slice, Knots bounds algebraic surface, Linking number, Poincarè conjecutre, Unknot number.

Knots bounds algebraic surface

[2021-11-04 Thu 18:45] By Seiberg-Witten invariant, it is natural to ask when a Knot bounds an algebraic surface. The answer is the following theorem.

Cf. Quasi-positive, Slice genus.

For Positive knot, we have \(U_{K} = g_{S}(K)\) which is not true for quasi-positive knot.

TODO Lee homology

[2021-11-28 Sun 14:33] Cf. Khovanov homology.

Referenced: Rasmussen invariant.

Lefschetz hyperplane theorem

Lickorish-Wallace theorem

Cf. Link, Orientation, Surgery.

Link

[2021-09-13 Mon 18:13] A link is an isotopy class of an Immersion \(\hat{L}: (S^{1})^{\times K} \to S^{3}\), \(L = \hat{L}\).

Cf. Knot.

Referenced: Diagram, Framing, HOMFLY invariant, Jones-Markov trace, Skein relations, Lickorish-Wallace theorem, Seifert surface.

Linking number

[2021-10-08 Fri 13:41] Let \(K_{1}, K_{2}\) be Oriented Knots in \(S^{3}\). We define the linking number \(lK(K_{1}, K_{2}) \in \ZZ\) by \[[K_{2}] = lK(K_{1}, K_{2})[m_{1}] \in H_{1}(S^{3} \backslash K_{1}) \cong \ZZ\] where \(m_{1}\) is the meridian of \(K_{1}\). The following is a sketch graph of meridian.

An alternative definition is consider \(\Sigma_{1}\) Seifert surface of \(K_{1}\) and then define \(lK(K_{1}, K_{2}) = \Sigma_{2} \cdot K_{1}\). Or we consider embeddings \(\Sigma_{1} \to D^{4}\), \(\Sigma_{2} \to D^{4}\) such that \(\partial \Sigma_{1} = K_{1}\), \(\partial \Sigma_{2} = K_{2}\) and define \(lK(K_{1}, K_{2}) = K_{1} \cdot K_{2}\).

TODO Mazur manifold

Milnor conjecture (theorem)

[2021-11-04 Thu 18:24] The Milnor conjecture is the following. (now a theorem)

One can use local Thom conjecture (theorem) to prove.

Morse complex

[2021-09-30 Thu 16:38] Now we consider the complex constructed from Morse theory. We have \(X = X_{0} | X_{1} | \cdots | X_{n}\) Monotone. And for \(H_{k} \subset X_{k}\) and \(H_{k + 1} \subset X_{k + 1}\), we define the pairing \(\langle H_{k}, H_{k + 1}\rangle\) the intersection number of the belt sphere of \(H_{k}\) and attaching sphere of \(H_{k + 1}\). Then we have Morse complex \(C_{k}^{n}(X) = \ZZ \langle H_{k}^{1}, \cdots, H_{k}^{a_{k}}\rangle\) and define the differential to be \[\dif H_{k}^{i} = \sum_{j} \langle H_{k - 1}^{j}, H_{k}^{i}\rangle H_{k - 1}^{j}\] We have that \(\dif^{2} = 0\).

Cf. Handle decomposition.

Referenced: Morse inequality.

Morse function

[2021-09-30 Thu 15:39] A Morse function is a funciton if all critical points are non-degenerate. If the origin manifold \(X\) has boundary, then \(f^{-1}(\max f) = \partial X\).

Referenced: Monotone, Self-indexed.

Self-indexed

[2021-09-30 Thu 15:54] A Morse function is called self-indexed if for all \(x \in {\rm crit}(f)\), we have \(f(x) = {\rm index}(x)\).

Referenced: Monotone.

Morse inequality

[2021-09-30 Thu 16:41] From Morse complex, we have that

Cf. Critical point, Index.

Morse theory

Let \(f\) be a smooth real valued function on manifold M. Let \(a < b\) and suppose that the set \(f^{-1}[a,b]\) is compact and contain no critical points of \(f\). Then \(M^{a}\) is diffeomorphic to \(M^{b}\). Furthermore, \(M^{a}\) is a deformation retract of \(M^{b}\), inclusion map \(M^{a} \to M^{b}\) is a homotopy equivalence.

The theorem supplies the condition when the retraction is possible.

Given a Morse function, we have a Handle decomposition.

Poincarè conjecutre

We have following generalized Poincarè conjecture which is proved by various people in various dimensions.

For \(n \geq 6\), we can give a proof by h-Cobordism.

However, the smooth structure on \(S^{n}\) may not be unique. \(S^{2k + 1}\) has unique smooth structure if and only if \(2k + 1 = 1,3,5,61\).

The dimension \(4\) smooth Poincaré conjecture is still open. A possible approach is suggested by Freedman-Gompf-Morrison-Walker using Knot on \(S^{4}\) to do some surgery and making contradiction.

TODO Rasmussen invariant

[2021-11-28 Sun 14:33] Cf. Lee homology.

Referenced: Homotopic slice, Slice genus.

TODO Rational blow down

Realization of Finite Presnetation Group as Fundamental Group of Manifolds

[2021-09-15 Wed 18:20] We have following theorem by Markov, showing that any finite presentation group can be realized as a fundamental group of a manifold.

Together with the following theorem of Adyan-Rubin, we show that a complete classification of high dimensional manifold is impossible.

Regular value

[2021-09-30 Thu 15:18] See Critical value.

Representation of braids

[2021-09-21 Tue 16:10] The following proposition relates the representation of Braid group to Yang-Baxter equation.

Rokhlin Theorem

Cf. Signature.

Referenced: Geometric Topology.

Seiberg-Witten equation

[2021-10-23 Sat 15:28] The Seiberg-Witten equation is the following equations for \((A, \phi \in \Gamma^{S^{+}})\) \[ \begin{cases} F_{A^{\tau}}^{+} - \rho^{-1}(\phi^{*} \phi)_{0} = 0 & \\ \dirac_{A}^{+} \phi = 0 & \end{cases} \] Here \((\phi^{* } \phi)_{0}: \psi \to \langle \psi, \phi> \cdot \phi - \frac{\abs{\phi}^{2}}{2} \psi\). We have \(((\phi^{* } \phi)_{0})^{* } = (\phi^{* } \phi)_{0}\) and \(\tr (\phi^{* } \phi)_{0} = 0\).

Referenced: Seiberg-Witten moduli space.

Seiberg-Witten invariant

[2021-10-23 Sat 15:51] For simply connected \(4\)-manifold \(X\), we have a natural \(S^{1}\) bundle \(P\) on Seiberg-Witten moduli space \(\mathscr{M}_{SW}\) coming from the Coulomb guage description. So we define the Seiberg-Witten invariant by \[SW(X, S) = c_{1}(P)^{d(S)/2} \cdot [\mathscr{M}_{SW}] \in \ZZ\] where \(d(S)\) is the dimension of the Seiberg-Witten moduli space. When \(d(S)\) is odd, we simply define the Seiberg-Witten invariant to be \(0\).

We have following conjecture for Seiberg-Witten invariants called simple type conjecture

The symplectic case is proved by Taubes.

Let \(\Char(X)\) denote the set \(\{K \in H^{2}(X; \ZZ) \mid K \cdot [\Sigma] = \Sigma \cdot \Sigma \pmod 2\}\), and we define the \(SW\) as a map \[SW(\alpha) = \sum_{S \in \Spin^{\CC}(X), c_{1}(S) = K} SW(X, S)\] The Seilberg-Witten invariants have following properties:

1. \(SW_{X}(K) = 0\) for all but finitely many \(K \in \text{char}(X)\). Those \(K\) are called basic classes.
2. If \(X\) has postive scalar curvature, then \(SW_{X}(K) = 0\) for every \(K\).
3. \(SW_{X}(-K) = (-1)^{b_{2}^{+}(X) - b_{1}(X) + 1} SW_{X}(K)\).
4. If \(X \cong_{diff} X_{1} \shar X_{2}\) with \(b_{2}^{+}(X_{1}), b_{2}^{+}(X_{2}) > 0\), then \(SW_{X}(K) = 0\) for every \(K\).
5. Blow-up formula: If \(X \cong_{diff} X' \shar \overline{\CC P}^{2}\), then basic classes of \(X\) are \(\{K \pm \pd E \mid K \text{ is a basic class of } X'\}\). Moreover, \(SW_{X}(K \pm \pd E) = SW_{X'}(K)\).
6. If \(X\) is an algebraic surface, then \(SW_{X}(c_{1}(TX)) = \pm 1\). (Witten)
7. If \(X\) is symplectic, and \(J\) compatible almost complex structure, then \(SW_{X}(c_{1}(TX, J)) = \pm 1\). (Taubes)
8. Adjunction inequality. (See below)

As an application of Taubes result on symplectic \(4\)-manifold, we consider the following corollary.

Some more works show the following theorem by Kotschick, Taubes.

Cf. Irreducible.

We state the adjunction inequality by Kronheimer-Mrowka, Ozsrath-Szabo, Fintushel-Stern.

Since the symplectic \(4\)-manifold is simple type, so we can deduce the following symplectic Thom conjecture.

Referenced: Geometric Topology, Donaldson diagonalizable theorem, Fintushel-Stern knot surgery, Knots bounds algebraic surface, Thom conjecture (theorem).

Seiberg-Witten moduli space

[2021-10-23 Sat 15:35] By Seiberg-Witten equation, the solution forms Seiberg-Witten moduli space, \[\mathscr{M}_{SW} = \{\text{Seiberg-Witten solutions}\}/ \mathscr{G}\] where \(\mathscr{G} = \Gamma(X, \End(S^{+}, \rho)) = C^{\infty}(X, U(1))\) acts on \(\{(A, \phi)\}\) by \(u \cdot (A, \phi) = (A - u^{-1} \dif u, u \cdot \phi)\).

Or we can formulate in another way by fixed the Coulomb gauge with respect ot a base Spin^C connection \(A_{0}\) by setting \(\dif^{* }(A - A_{0}) = 0\). \[\mathscr{M}_{SW} = \{\text{Coulomb gauge Seiberg-Witten solutions}\}/ \mathscr{G}_{h}\] where \(\mathscr{G}_{h} = \{\text{harmonic map }X \to S^{1}\} \cong H^{1}(X; \ZZ) \times S^{1}\).

The following result is important and magic by Clifford Taubes.

The proof relies on the estimate of the \(\abs{\phi}\) and elliptic boosting technique for elliptic partial differential equations.

However, in general the moduli space is not a manifold, but in \(b^{+ }_{2} > 0\) case, by perturbing the equation a little, we can get an Oriented manifold of dimension \(\frac{c_{1}(S)^{2} - \sigma(X)}{4} + b^{1}(X) - b_{2}^{+}(X)\). The result depend on Sard-Smale theorem which is an analog of Sard theorem in infinite dimensional case. For \(b^{ +} > 1\), all such moduli space is equivalent, but in \(b_{2}^{ +} = 1\) case, we have \(2\) choices.

Referenced: Seiberg-Witten invariant.

Seifert genus

[2021-10-08 Fri 13:37] See Seifert surface.

Referenced: Thom conjecture (theorem).

Seifert surface

[2021-10-08 Fri 13:35] For any Oriented Link \(L \to S^{3}\), there exists an embedded orientable surface \(\Sigma \to S^{3}\) with \(\partial \Sigma = L\). Such surface is called Seifert surface.

The Seifert genus \(g(K)\) is the minimal genus of Seifert surface. The slice genus \(g_{S}(K)\) is the minimal genus of orientable \(\Sigma \to D^{4}\) with \(\partial \Sigma = K\).

Obviously, we have \(g_{S}(K) \leq g(K)\).

Referenced: Linking number, Seifert genus, Slice genus.

Self-dual

[2021-10-15 Fri 20:21] A \(n\)-form \(w\) on \(2n\) manifold is called self-dual if \(* w = w\).

Any \(n\)-form can be decomposed to a sum of self-dual part and Anti-self-dual part.

Referenced: Anti-self-dual, Dirac operator.

Signature

[2021-09-17 Fri 21:50] For Intersection forms \(Q_{X}\), we have \(b^{+ }(X)/b^{- }(X)\) denote the number of the positive and negative eigenvalues of \(Q_{X}\). And we define the signature to be \(\sigma(X) = b^{+ }(X) - b_{2}^{-}(X)\).

Referenced: Characteristic class for 4-manifold, Freedman’s classification of topological 4-manifold, Rokhlin Theorem.

Skein relations

We now consider a vector space \(V\) of formal \(\ZZ\) linear combinations of Diagrams of Links modulo Reidemeister moves. The Skein relations are defined by

where \(A, B, C\) are formal variables and \(B\) is invertible. We have \[C = \frac{B - B^{-1}}{A}\]

Also, we Skein is related to Reidemeister moves.

Referenced: HOMFLY invariant.

Slice genus

[2021-10-08 Fri 13:40] See Seifert surface.

The slice genus is bounded below by Rasmussen invariant.

Referenced: Clasp number, Knots bounds algebraic surface, Slice ribbon conjecture.

Slice ribbon conjecture

[2021-11-04 Thu 18:33] We say a knot \(K\) is slice if \(g_{S}(K) = 0\) (Slice genus). And say \(K\) is ribbon if \(K\) bounds immersed \(D^{2} \subset S^{3}\) such that self intersection of \(D^{2} = \coprod \text{arcs}\). We know that ribbon is a slice, the conjecture is the converse also holds.

Smale conjecture

[2021-12-05 Sun 17:34] The Smale conjecture for dimension \(n\) means

The conjecture is true for \(n \leq 3\) and false for \(n \geq 4\). The last case \(n = 4\) was proved in around 2018.

Spin^C connection

[2021-10-23 Sat 15:15] A unitary connection \(A\) on \(S\) is spin\(^{\CC}\) connection if \[\nabla_{A}(\rho(v) \cdot s) = \rho(v) \cdot \nabla_{A} s + \rho(\nabla_{LC} v) \cdot s\] for every \(v \in \Gamma(T_{* }X), s \in \Gamma(s)\). Here \(\nabla_{LC}\) is the Levi-Civita connection which relies on Riemannian metric.

Referenced: Dirac operator, Seiberg-Witten moduli space.

Spin^C structure

[2021-10-15 Fri 20:16] A Spin\(^{\CC}\) structure on \(X\) is a Hermitian bundle \(S \to X\) with bundle map \(\rho: T_{* }X \to \End(S)\) such that for all \(x\), \(\rho_{x}: T_{x}X \to \End(S_{x})\) is an irreducible Clifford module. Consider \(\pi = \rho(e_{1} \cdots e_{n})\), we have \(\pi^{2} = -1\). So \(S = S^{+} \oplus S^{-}\) by \(\pi\) action.

Spin\(^{\CC}\) structure is useful in \(4\)-manifold by following theorem.

which result from the following proposition

Cf. Stiefel-Whitney class.

Spin structure

[2021-09-17 Fri 22:16] Let \(\SO(n) \to {\rm Fr} \to X\) be the frame bundle of a smooth Manifold. A spin structure is a lift of \({\rm Fr}\) to a \(\Spin(n)\) bundle.

We have following lemma on the existence of spin structure.

TODO Surgery

Referenced: Lickorish-Wallace theorem.

Tangle

[2021-10-19 Tue 18:02] A tangle is the regular isotopy class of a standardized Tangle with respect to deformation of endpoints.

Referenced: Tangle.

Thom conjecture (theorem)

[2021-11-04 Thu 18:24] The Thom conjecture is the following, now a theorem.

Cf. Seiberg-Witten invariant, Seifert genus.

The following corolloary called local Thom conjecture is useful.

Referenced: Milnor conjecture (theorem).

Trisection

[2021-10-15 Fri 19:03] Let \(X\) be a closed smooth Oriented connected \(4\)-manifold. For \(0 \leq k \leq g\), a \((g, K)\)- trisection of \(X\) is a decomposition \(X = X_{1} \cap X_{2} \cap X_{3}\) such that

1. \(X_{i} \cong_{diff} \shar^{k}(S^{1} \times B^{3})\)
2. \((X_{i} \cap X_{j}) = H_{ij} \cong_{diff} \shar^{g}(S^{1} \times B^{2})\)
3. \(X_{1} \cap X_{2} \cap X_{3} = \Sigma_{g}\)

The following theorem of Gay and Kirby shows that trisection exists for any Orientable \(4\)-manifold and unique in some sense.

Trisection diagram

[2021-10-15 Fri 19:03] A trisection diagram is a set of three curves \(\{\alpha_{i}\}, \{\beta_{i}\}, \{\gamma_{i}\}\) on \(\Sigma_{g}\) with \(1 \leq i \leq g\) such that any two collection is a Heegaard diagram for \(\shar^{k}(S^{1} \times S^{2})\).

Unknot number

[2021-11-04 Thu 18:38] Given a Knot \(K\), we define the unknotting number \(U(K)\) the minimal number that a knot Diagram of \(K\) can be made into an unknot.

Referenced: Clasp number.

Wall Theorem

Whitehead theorem

[2021-09-17 Fri 21:55] The Whitehead theorem is of the following form:

The proof relies on the following lemma of Hopf.

Referenced: Freedman’s classification of topological 4-manifold.

Writhe number

[2021-09-21 Tue 16:02] We define the Writhe number of \(\beta\) to be

Referenced: Jones-Markov trace.

Yang-Baxter equation

An solution is given by \[R(e_{i} \otimes e_{j}) = \begin{cases} q e_{i} \otimes e_{i} & i = j \\ e_{i} \otimes e_{j} & i < j \\ e_{i} \otimes e_{j} + (q - q^{-1})e_{j} \otimes e_{i} & i > j \end{cases} \]

Referenced: Representation of braids.

Quantum Fields and Strings

TODO Cobordism conjecture

TODO Extended topological quantum field theory

Super algebra

A super algebra over \(k\) is a Super vector space \(A\), with a given morphism \(A \otimes A \to A\). \(A\) is associative if \((xy)z = x(yz)\). A unit is an even element \(1\) such that \(1 x = x 1 = x\). \(A\) is commutative if \[xy = (-1)^{p(x)p(y)}yx\] for homogeneous elements.

We can define the module for algebra \(A\). And we make left module right module by \[m \cdot a := (-1)^{p(m)p(a)} a \cdot m\]

We also have opposite algebra \(A^{o}\) of \(A\) by product \[x \cdot_{opp} y = (-1)^{p(x)p(y)}y \cdot x\]

We can define the tensor product of modules and algebras naturely.

Super vector space

We have parity reversing functor \(\Pi\) such that \((\Pi V)_{0} = V_{1}\) and \((\Pi V)_{1} = V_{0}\). Also we use the notation \(m_{0} \mid m_{1}\) to denote the dimension of \(V\) with \(\dim V_{0} = m_{0}\) and \(\dim V_{1} = m_{1}\).

The tensor product is defined by \((V \otimes W)_{k} = \oplus_{i + j = k} V_{i} \otimes W_{j}\). We define the commutativity isomorphism \[c_{V,W}: V \otimes W \to W \otimes V, v \otimes w \to (-1)^{p(v)p(w)} w \otimes v\] By above definition, we have the following diagram commutes.

Referenced: Super algebra.

TODO Topological quantum field theory

Chern-Weil theory

[2021-12-13 Mon 19:06] See Chern class.

Chern-Simons functional

[2021-12-13 Mon 19:50] For \(A\) a connection on \(SU(2)\) principal bundle on manifold \(M\). We usually denote \[CS(A) = \frac{1}{8 \pi^{2}}\int_{M} \tr(F_{A} \wedge F_{A}) \in \RR/\ZZ\] called Chern-Simons functional, where \(F_{A}\) is the Curvature.

Or on 3-manifold, we define for Connection \(A\) (matrix valued 1 form), \[CS(A) = \frac{1}{8 \pi^{2}} \int_{Y} \tr(A \wedge \dif A + \frac{2}{3}A \wedge A \wedge A) \in \RR/\ZZ\]

Critical poiont of Chern-Simons functional are Flat connections and corresponds to group homomorphism \(\rho: \pi_{1}(M) \to SU(2)\).

Yang-Mills equation

[2021-12-13 Mon 20:05] Let \(P\) be a \(SU(2)\) bundle over \(4\)-manifold \(M\), then we call \[F_{A}^{+} = 0\] Anti-self-dual equation. The connection satisfying the equation minimizing energy equaiton Yang-Mills functional \[YM(A) = \int_{M} \abs{F_{A}}^{2}\] Critical point of Yang-Mills functional is given by Yang-Mills equation

\begin{equation*} \begin{cases} d^{*}_{A}F_{A} = 0 & \\ d_{A} F_{A} = 0 & \end{cases} \end{equation*}

The second equation is Bianchi identity (Cf. Curvature) and is automatically satisfied. In fact, solutions of anti-self-dual equation is a special solutions of the Yang-Mills equation, because it takes globally minimal value.

Algebraic Number Theory

[2021-11-28 Sun 14:37] Apart from algebraic number theory, the theory of cohomology of groups is also put here. The main reference is [milne2020a] and [kedlaya2021], for detailed proof see these and references there.

Abstract class field theory

[2021-12-24 Fri 18:43] If \(L/K\) is a cyclic extension of local fields, then we have \(\shar H_{T}^{0}(\Gal(L/K), L^{*}) = [L : K], \shar H_{T}_{-1}(\Gal(L/K), L^{*}) = 1\). We can construct the reciprocity map from above two equality. Such theory is called abstract class field theory.

Adèle

[2021-11-28 Sun 14:39] Let \(K\) be a number field of degree \(n\). It then has \(n\) distinct embeddings \(\tau: K \to \CC\). The product embedding \[j: K \to \prod_{\tau} \CC, a \to (\tau(a))_{tau}\] induces an isomorphism of \(K_{\CC} = K \otimes_{\QQ} \CC\), which induces Hermitian inner product. By considering the involution by complex conjugation, then we have \(K_{\RR}\). We have embedding \(K\) into \(K_{\RR}\), \(\mathfrak{o}_{K}\) corresponds to a lattice in \(K_{\RR}\).

We define the ring of finite adèles \(\mathbb{A}_{\QQ}^{fin}\) as any of the following isomorphic objects:

1. the tensor product \(\hat{\ZZ} \otimes_{\ZZ} \QQ\).
2. the direct limit of \(\frac{1}{n} \hat{\ZZ}\) over all nonzero integers \(n\).
3. The Restricted topological product \(\prod_{p}' \QQ_{p}\), where we only allow tuples \((\alpha_{p})\) for which \(\alpha_{p} \in \ZZ_{p}\) for almost all \(p\).

We define the ring of adèles over \(\QQ\) as \(\mathbb{A}_{\QQ} = \RR \times \mathbb{A}_{\QQ}^{fin}\). Then \(\mathbb{A}_{\QQ}\) is a locally compact topological ring with a canonical embedding \(\QQ \to \mathbb{A}_{\QQ}\). We refer to the elements of \(\QQ\) as principal adèles with \(\mathbb{A}_{\QQ}\).

\(\QQ\) is discrete in \(\mathbb{A}_{\QQ}\).

for general number field \(K\). We define the ring of finite adèles \(\mathbb{A}_{K}^{fin}\) as following isomorphic objects:

1. the tensor product \(\hat{\mathfrak{o}_{K}} \otimes_{\mathfrak{o}_{K}} K\);
2. the direct limit of \(\frac{1}{\alpha} \hat{\mathfrak{o}_{K}}\) over all nonzero \(\alpha \in \mathfrak{o}_{K}\);
3. the restricted direct product of the pairs \((K_{\mathfrak{p}}, \mathfrak{o}_{K_{\mathfrak{p}}}\) over all primes \(\mathfrak{p}\) of \(K\).

The ring of adèles is \(K_{\RR} \times \mathfrak{A}_{K}^{fin}\).

We have \[\abs{x}_{K} = \prod_{v} \abs{x}_{v}\] where if \(v\) is real take real absolute value, \(v\) complex, square of the absolute value and \(v\) finite Place normalized so that \(\abs{p}_{v} = p^{-1}\).

Adelic reciprocity law

[2021-12-29 Wed 20:17] The adelic reprocity law is the following

We have similar existence theorem and norm limitation theorem

The following local-global compatibility is important to global class field theory. We define a map \(r_{K,v}:K_{v}^{*} \to G_{v} \subset G\) as local reprocity map if \(v\) is finite place, sign map if \(v\) is real place, trivial map if \(v\) is complex place. Then we define \[\tilde{r}_{K}: I_{K} \to G, (\alpha_{v}) \to \prod_{v} r_{K,v}(\alpha_{v})\]

Cf. Artin’s reciprocity law.

Approximation theorem

[2021-11-28 Sun 14:38] The following theorem is referred to as approximation theorem.

Cf. Adèle, Place.

Artin’s reciprocity law

[2021-11-28 Sun 16:18] Let \(L/K\) be a finite abelian extension of number fields. For each prime \(\mathfrak{p}\) of \(K\) that does not Ramify in \(L\), let \(\mathfrak{q}\) be a prime of \(L\) over \(K\), and put \(\kappa = \mathfrak{o}_{K} / \mathfrak{p}\) and \(\lambda = \mathfrak{o}_{L}/ \mathfrak{q}\). Then the residue field extension \(\lambda/ \kappa\) is an extension of finite fields, so it has a canonical generator \(\sigma\), the Frobenius element, which acts by rasing to \(q\)-th power. Since \(\mathfrak{p}\) does not ramify, the decomposition group \(G_{\mathfrak{q}}\) is isomorphic to \(\Gal(\lambda/ \kappa)\), so we get a canonical element of \(G_{\mathfrak{q}}\). We say it to be Frobenius of \(\mathfrak{p}\) since in abelian Galois group the conjugaton has no effect.

Now for \(\mathfrak{m}\) divisible by all primes of \(K\) which ramify in \(L\), define the homomorphism called Artin map \[J^{\mathfrak{m}}_{K} \to \Gal(L/K), \; \mathfrak{p} \to {\rm Frob}_{\mathfrak{p}}\]

The Artin’s reciprocity law is the following theorem

Augmentation ideal

[2021-12-22 Wed 20:39] Let \(M_{G}\) denote the maximal quotient of \(M\) on which \(G\) acts trivially. Then we have \(M_{G} = M / M I_{G}\), where \(I_{G}\) is the augmentation ideal of \(\ZZ[G]\) \[I_{G} = \{ \sum_{g \in G} z_{g} [g] \mid \sum_{g} z_{g} = 0\}\] We call \(M_{G}\) the group of \(G\)-coinvariants. \(M \to M_{G}\) is right exact but not left exact.

Brauer group

[2021-11-28 Sun 14:40] For any field \(K\), the group \(H^{2}(\Gal(\bar{K}/K), \bar{K}^{*})\) is called the Brauer group of \(K\), denoted by \({\rm Br}(K)\).

We have following isomorphism

The left hand side means \(H^{2}(\Gal(K^{ur}/K), (K^{ur})*)\).

We consider the map \({\rm inv}\) we have isomorphism \[H^{2}(K_{nr}/K) \xrightarrow{v} H^{2}(\hat{\ZZ}, \ZZ) \xrightarrow{\delta^{-1}} \Hom(\hat{\ZZ}, \QQ/\ZZ) \xrightarrow{\gamma} \QQ/\ZZ\]

Referenced: Brauer group.

Chebotarëv density theorem

[2021-12-21 Tue 18:57] The Caebotarëv density theorem is the following.

Cf. Dirichlet density.

Class field axiom

[2021-12-26 Sun 17:55] With notation of Weil group, we say \(A\) satisfies the class field axiom if forevery cyclic extension \(L/K\) of finite subextensions of \(\bar{k}/ k\), \[\shar H^{i}_{T}(\Gal(L/K), A_{L}) = \begin{cases} [L:K] & i = 0 \\ 1 & i = -1 \end{cases} \]

Cohomology of local fields

[2021-12-24 Fri 19:11] We collect some result of cohomology of local fields here.

For Unramified case,

For cyclic case,

For general case,

Cohomology of group

[2021-11-28 Sun 14:39] We define the cohomology of the group to be \(H^{i}(G, M)\) the derived functor of \((\bullet)^{G}\). To compute it, we can use either Injective resolution or F-acyclic resolution.

Given a \(G\)-module \(M\), we can define \(G\)-module \(N^{i}\) for \(i \geq 0\) as the set of functions \(\phi: G^{i + 1} \to M\) with \(G\)-action \[(\phi^{g})(g_{0}, \cdots, g_{i}) = \phi(g_{0}g^{-1}, \cdots, g_{i}g^{-1})^{g}\] We have \(N^{i} = \Ind_{1}^{G} N_{0}^{i}\) for \(N_{0}^{i}\) consists of \(\phi(g_{0}, \cdots, g_{i}) = 0\) when \(g_{0} \neq e\). So \(N^{i}\) is Induced. Also we define the map \(\dif^{i}: N^{i} \to N^{i + 1}\) by \[(\dif^{i} \phi)(g_{0}, \cdots, g_{i + 1}) = \sum_{j = 0}^{i + 1} (-1)^{j} \phi(g_{0}, \cdots, \hat{g_{j}}, \cdots, g_{i + 1})\] With above definition, we have \(0 \to M \to N^{0} \to N^{1} \to \cdots\) exact, which is in fact acyclic resolution by Shapiro’s lemma.

There are some functoriality of group cohomology.

1. Cohomology group has no nontrivial \(G\)-action.
2. If \(H\) is a subgroup of \(G\), \(M\) a \(G\)-module, \(M'\) is \(M\) with \(H\)-action only, we have restriction homomorphism \[{\rm Res}:H^{i}(G, M) \to H^{i}(H, M')\]
3. Let \(M\) be Ga \(\)-module, then we have corestriction map \[{\rm Cor}:H^{i}(H, M) \cong H^{i}(G, \Ind_{H}^{G} M) \to H^{i}(G, M)\]
4. \({\rm Cor} \circ {\rm Res}\) acts as multiplication by \([G:H]\) on each cohomology group.
5. Let \(H\) be a normal subgroup of \(G\). We have \(G/H\) acts on \(M^{H}\), and therefore inflation homomorphisms \[{\rm Inf}: H^{i}(G/H, M^{H}) \to H^{i}(G, M)\]

the following proposition is called inflation-restriction exact sequence

The underlying theory is Spectral sequence.

Cf. G-module.

Coinvariant

[2021-12-22 Wed 20:41] See Augmentation ideal.

Conductor

[2021-12-19 Sun 14:26] In Kronecker-Weber theorem, the smallest number \(K \subset \QQ(\zeta_{n})\) is called the conductor of \(K/ \QQ\).

In Artin’s reciprocity law, the conductor of \(L/K\) is the smallest formal product \(\mathfrak{m}\) for which the reciprocity law holds.

Corestriction

[2021-12-22 Wed 21:02] See Cohomology of group.

TODO Dedekind domain

Referenced: Fractional ideal.

Decomposition group

[2021-12-21 Tue 19:25] Let \(L/K\) be field extension \(G\) the Galois group. Let \(\mathfrak{p}\) prime ideal of \(K\) and \(\tau\) prime ideal of \(L\) over \(\mathfrak{p}\). Then the decomposition group is the group \(G_{\tau} \subset G\) which consists of the stabilizer of \(\tau\) under the action of \(G\) on the primes above \(\mathfrak{p}\).

Dirichlet density

[2021-12-21 Tue 19:13] A set of primes \(S\) in a number field \(K\) has Dirichlet density \(d\) if \[\lim_{s \to 1^{+}} \frac{\sum_{\mathfrak{p} \in S} \nm(\mathfrak{p})^{-s}}{\log \frac{1}{s - 1}} = d\] This persumes the existence of the limit; otherwise, we have lower Dirichlet density and upper Dirichlet density.

Cf. Norm.

Discrete valuation

[2021-11-28 Sun 14:37] The Valuation is a discrete valuation if there is a \(\delta > 0\) such that \(1 - \delta < \abs{\alpha} < 1 + \delta\) implies \(\abs{\alpha} = 1\). After taking log and do some normalization, we get a morphism \[v: k^{\times} \to \ZZ\]

TODO Discriminant

Fractional ideal

[2021-11-28 Sun 16:53] A fractional ideal of a Dedekind domain \(A\) is a nonzero \(A\)-submodule \(\mathfrak{a}\) \(K\) such that \[d \mathfrak{a} \subset A\] for some nonzero \(d \in A\). With product, all the fractional ideal of \(A\) form a group called fractional ideal group.

Referenced: Norm.

Frobenius element

[2021-11-28 Sun 16:40] Let \(L\) be a finite Unramified extension of \(K\). Then \(L\) is Galois over \(K\) and \(\Gal(L/K)\) on \(\mathscr{O}_{L}\) defines an isomorphism \(\Gal(L/K) = \Gal(l/k)\) where \(l/k\) are the residue field. Therefore \(\Gal(L/K)\) is a cyclic group and generated by \(\sigma\) with \(\sigma \alpha \equiv \alpha^{q} \pmod{\mathfrak{m}_{L}}\) for \(\alpha \in \mathscr{O}_{L}\). \(\sigma\) is called Frobenius element and denoted by \(\Frob_{L/K}\).

In general, we have to choose a prime \(\mathfrak{p}\) of \(K\)and \(\mathfrak{q}\) prime above \(\mathfrak{p}\), then the element of the Decomposition group \(G_{\mathfrak{p}}\) such that \(gx \equiv x^{\shar(\mathfrak{o}_{K}/ \mathfrak{p})} \pmod{\mathfrak{q}}\). In general, Frobenius element only defined up to conjugation.

Frobenius reciprocity law

[2021-12-22 Wed 19:56] The following is Frobenius reciprocity law.

Cf. Induction.

Generalized ideal class group

[2021-12-20 Mon 19:31] Let \(\mathfrak{m}\) be a formal product of places of \(K\). Let \(J^{\mathfrak{m}}_{K}\) be the group of Fractional ideals of \(K\) which are coprime to each finite place of \(K\) occuring in \(\mathfrak{m}\). Let \(P_{K}^{\mathfrak{m}}\) be the group of principal fractiona ideals generated by elements \(\alpha \in K\) such that

1. for \(\mathfrak{p}^{e} \mid \mathfrak{m}\) finite, \(\alpha \equiv 1 \pmod \mathfrak{p}^{e}\).
2. For every real place \(\tau\) in \(\mathfrak{m}\), \(\tau(\alpha) > 0\).

Then the ray class group \({\rm Cl}^{\mathfrak{m}}(K)\) is defined as the quotient \(J^{\mathfrak{m}}_{K} / P^{\mathfrak{m}}_{K}\). A quotient of a ray class group is called a generalized ideal class group.

Cf. Ideal class.

G-extension

[2021-12-19 Sun 14:32] If \(G\) a group, a G-extension of a field \(K\) is a Galois extension of \(K\) with Galois group \(G\).

G-module

[2021-12-21 Tue 19:54] Let \(G\) be a finite group. A (right) G-module is an abelian group \(A\) equipped with a right \(G\)-action.

Given a \(G\)-module \(M\), we denote by \(M^{G}\) the abelian group of \(G\)-invariant elements of \(G\). The functor \(M \to M^{G}\) is left exact.

For \(G\)-module of Profinite group \(G\), we require that \(M\) is topological group and the action \(G \times M \to M\) is continuous.

Induced

[2021-12-22 Wed 20:11] A \(G\)-module is called induced if it is of this form \(\Ind_{1}^{G} N\). for some \(N\).

The following is a corollary for Shapiro’s lemma.

Herbrand quotient

[2021-11-28 Sun 14:40] Let \(G\) be a finite cyclic group and let \(M\) be a \(G\)-module. If the group \(H_{T}^{i}(G, M)\) are finite, we define the Herbrand quotient as the ratio \[h(M) = \shar H^{0}_{T}(G, M) / \shar H_{T}^{-1}(G, M)\]

By periodicity theorem of Tate cohomology group, for exact sequence \(0 \to M' \to M \to M'' \to 0\), we have \(h(M) = h(M') h(M'')\).

A interesting fact is the following.

Henselian valuation

[2021-12-26 Sun 17:53] With the notation of Weil group, a henselian valuation of \(A_{k}\) with respect to \(d\) is a homomorphism \(v: A_{k} \to \hat{Z}\) such that

1. the group \(Z = \im(v)\) contains \(\ZZ\) and staisfies \(Z/nZ \cong \ZZ/ n \ZZ\) for all positive integer \(n\);
2. for every finite extension \(K\) of \(k\), \(v(\nm_{K/k}A_{K}) = f_{K/k} Z\).

for any finite subextension \(K\) of \(\bar{k}/k\), define the unit group \(U_{K}\) as the set of \(u \in A_{K}\) with \(v_{K}(u) = 0\). We say \(\pi \in A_{K}\) is a uniformizer for \(K\) if \(v_{K}(\pi) = 1\).

Hilbert class fields

[2021-12-19 Sun 15:10] Let \(L\) be the maximal Unramified abelian extension of a number field \(K\). Then \(L/K\) is finite, and its Galois group is isomorphic to the ideal class group \({\rm Cl}(K)\) of \(K\). The field \(L\) is called Hilbert class field of \(K\).

Homology of group

[2021-12-22 Wed 20:44] We define the homology of groups to be the left derived functor of Coinvariant.

Cf. Cohomology of group.

TODO Ideal class

Referenced: Finiteness of class group.

Idèle

[2021-11-28 Sun 14:39] Let \(K\) be a number field and let \(\mathbb{A}_{K}\) be the ring of Adèles associated to \(K\). We define the group of idèles \(I_{K}\) associated to \(K\) as the gorup of units of the ring \(\mathbb{A}_{K}\). In fact \(I_{K}\) is Restricted topological product of \((K_{v}^{*}, \mathbb{o}_{K_{v}}^{*})\).

We have \(I_{K} = \cup_{S} I_{K,S}\), where \(S\) is finite set and \(I_{K,S}\) denote the element \(x_{v} \in \mathbb{o}_{K_{v}}^{*}\) for each finite place \(v \in S\). Elements of \(I_{K,S}\) called S-units. (Cf. S-integer)

For each \(\alpha \in K^{*}\), the principal adèle \(\alpha \in \mathbb{A}_{K}\) is an idèle, so we have an embedding \(K^{*} \to I_{K}\). We refer to elements of this embedding principal idèles. Define the idèle class group of \(K\) as the quotient \(C_{K} = I_{K}/K^{*}\).

We have well defined norm \(\abs{\cdot}:C_{K} \to \RR_{+}^{*}\) derived from \(\mathbb{A}_{K}\). Let \(C_{K}^{0}\) denote the kernel of this map.

The following is the famous Dirichlet’s unit theorem

Induction

[2021-12-22 Wed 19:49] If \(H\) is a subgroup of \(G\) and \(M\) is an \(H\)-module, we define the induction of \(M\) from \(H\) to \(G\) to be \(\Ind_{H}^{G} M = M \otimes_{\ZZ[H]} \ZZ[G]\). We may also identify \(\Ind_{H}^{G} M\) with the set of functions \(\phi: G \to M\) such that \(\phi(gh) = \phi(g)^{h}\) for \(h \in H\).

Inflation map

[2021-12-22 Wed 21:02] See Cohomology of group.

Inertial degree

[2021-12-26 Sun 17:50] See Weil group.

Inertia group

[2021-11-28 Sun 16:13] See Weil group.

Kronecker-Weber theorem

[2021-12-19 Sun 14:22] The Kronecker-Weber theorem is the following

We also have a local version, called local Kronecker-Weber theorem.

Kummer theory

[2021-12-19 Sun 14:30] The following theorem is key in Kummer theory of ableian extensions.

L-function

[2021-12-21 Tue 19:06] Let \(\K\) be a number field \(\mathfrak{m}\) formal product of Places of \(K\). Let \(\chi_{\mathfrak{m}}: {\rm Cl}^{\mathfrak{m}}(K) \to \CC^{*}\) be a character. Then we can extend \(\chi_{\mathfrak{m}}\) to a function on all ideal of \(K\) by declaring its value to be \(0\) on ideals not coprime to \(\mathfrak{m}\). Then we define the L-function \[L(s, \chi_{\mathfrak{m}}) = \prod_{\mathfrak{p} \not \div \mathfrak{m}}(1 - \chi(\mathfrak{p}) \nm(\mathfrak{p})^{-1})^{-1}\] which convergent absolutely for \({\rm Re}(s) > 1\). In fact, we have the following theorem

We also have nonvanishing theorem

Cf. Norm.

Langlands correspondence

Langlands prgram

[2021-11-28 Sun 17:36] As far as I know there are several parts of Langlands program, namely reciprocity conjecture and functoriality conjecture. The following is the reciprocity conjecture

And the functoriality conjecture

Local existence theorem

Cf. Norm group.

Local reciprocity map

Here \(\phi_{K}\) and \(\phi_{L/K}\) are called local Artin maps or local reciporicity maps and \(\phi_{L/K}\) is also called norm residue map or symbol.

By local existence theorem, we have local existence theorem

To see \(\nm_{L/K} L^{*}\) is norms of which abelian extension, we have norm limitation theorem.

Maximal abelian extension

on p. 172

Norm

[2021-11-28 Sun 14:37] Let \(L/K\) be finite field extension, for \(\alpha \in L\), we have norm \(\nm(\alpha)\) in \(K\) defined as \(\det(a_{ij})\), where \(\alpha e_{i} = \sum a_{ij}e_{j}\) and \(\{e_{1}, \cdots, e_{n}\}\) is a basis for \(L\) over \(K\).

We can also consider the norm map \(\nm: \id(B) \to \id(A)\), where \(\id\) denote the Fractional ideal group.

In group theory, we define norm to be \[\nm_{G}(m) = \sum_{g \in G} m^{g}\] which induces a homomorphism \[\nm_{G}: H_{0}(G, M) \to H^{0}(G, M)\]

In abstract class field theory, with the notation of Weil group, we define the norm map \(\nm_{L/K}: A_{L} \to A_{K}\) by \(\nm_{L/K}(a) = \prod_{g} a^{g}\), where \(g\) runs over a set of right coset representatives of \(G_{L}\) in \(G_{K}\).

For Adèles, we can also have norm map, for \(L/K\) an extension of number fields, we define the norm map \(\nm_{L/K}: I_{L} \to I_{K}\) by \[\nm_{L/K}(x) \to \prod_{g} x^{g}\] where \(g\) runs over coset representatives of \(\Gal(\bar{K}/L)\) in \(\Gal(\bar{K}/K)\). Similar for trace map \(\tr_{L/K}: \mathbb{A}_{L} \to \mathbb{A}_{K}\).

Norm group

[2021-11-28 Sun 18:06] A subgroup of \(K^{\times}\) is called a norm group if it is of the form \(\nm(L^{\times})\) for some finite abelian extension \(L\).

Referenced: Local existence theorem.

Place

[2021-12-19 Sun 15:06] A place of a number field \(K\) is a equivalence class of nontrivial Absolute values on \(K\).

Principal ideal theorem

[2021-12-20 Mon 19:59] Principal ideal theorem concerns the ideals of Hilbert class fields.

Profinite group

[2021-11-28 Sun 14:40] A profinite group is a topological group which is Hausdorff and compact, and which admits a basis of neighborhoods of the identity consisting of normal groups. More explicitly, a profinite group is a group \(G\) plus a collection of subgroups of \(G\) of finite index as open subgroups, such that the intersection of two open subgroups is open, but the intersection of all of the open subgroup is trivial.

The profinite group can also be defined as limits. Suppose we are given a partially ordered set \(I\), a family \(\{G_{i}\}_{i \in I}\) of finite groups and a map \(f_{ij}: G_{i} \to G_{j}\) for each pair \((i, j) \in I \times I\) such that \(i > j\). Then there exists a profinite group \(G\) with open subgroups \(H_{i}\) for \(i \in I\) such that \(G/H_{i} \cong G_{i}\) in a manner compatible with the \(f_{ij}\), the limit of system.

So if \(L/K\) Galois extension but not necessarily finite, we make \(G = \Gal(L/K)\) into a profinite group by declaring that the open subgroups of \(G\) are precisely \(\Gal(L/M)\) for all finite extensions \(M\) of \(K\). In fact, we have following Galois correspondence

Ramification degree

[2021-12-26 Sun 17:49] See Inertia group.

Ramification index

[2021-11-28 Sun 14:38] Let \(L/K\) be an extension of number fields. Let \(\mathfrak{p}\) be a nonzero prime ideal in the integer of rings \(\mathfrak{o}_{K}\) of \(K\), and \[\mathfrak{p} \mathfrak{o}_{L} = \prod_{i=1}^{n} \mathfrak{P}_{i}^{e_{i}}\] for some prime ideal \(\mathfrak{P}_{i} \in \mathfrak{o}_{L}\). Then the natural number \(e_{i}\) is called the ramification index of \(\mathfrak{P}_{i}\) over \(\mathfrak{p}\), denoted by \(e(\mathfrak{P}_{i}/ \mathfrak{p})\). If \(e_{i} > 1\), we say that ideal \(\mathfrak{p}\) is ramified in \(L\).

Ramification index can be generalized to local fields.

Ramified

[2021-11-28 Sun 14:38] See Ramification index.

Ray class field

[2021-12-20 Mon 19:46] We say \(L/K\) is the ray class field corresponding to the product \(\mathfrak{m}\) if \(L/K\) has Conductor dividing \(\mathfrak{m}\) and the map \(J_{K}/J_{K}^{\mathfrak{m}} \to \Gal(L/K)\) is an isomorphism.

There is the existence theorem for ray class field.

Cf. Ray class group.

Ray class group

[2021-12-20 Mon 19:46] See Generalized ideal class group.

Reciprocity map

[2021-12-26 Sun 18:03] If \(A\) satisfies Class field axiom, \(v\) is henselian valuation of \(A_{k}\) with respect to \(d\). Let \(L/K\) be a Galois extension of finite subextensions of \(\bar{k}/k\). Let \(H\) be the semigroup of \(g \in \Gal(L^{ur}/K)\) such that \(d_{K}(g)\) is a positive integer. Define the map \[r' : H \to A_{K} / \nm_{L/K} A_{L}\] by for \(g \in \Gal(L^{ur}/K)\), \(M\) the fixed field of \(g\) set \(r'(g) = \nm_{M/K}(\pi_{M})\) for some uniformizer \(\pi_{M}\) of \(M\). \(r'\) induces a homomorphism \(r = r_{L/K}: \Gal(L/K) \to A_{K} / \nm_{L/K} A_{L}\) and is called reciprocity map.

TODO Residue class degree

Restricted topological product

[2021-11-28 Sun 14:38] Let \(I\) be an index set. For each \(i \in I\), let \(G_{i}\) be a set and let \(H_{i}\) be a set of \(G_{i}\). The restricted direct product \(G\) of the pairs \((G_{i}, H_{i})\) is the set of tuples \((g_{i})_{i = 1}^{\infty}\) such that \(g_{i} \in H_{i}\) for all but finitely many indices \(i\).

Restriction

[2021-11-28 Sun 14:39] See Cohomology of group.

Riemann zeta function

[2021-08-31 Tue 13:37] \[\zeta(s) = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \cdots = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}\] The function can be extend as a meromorphic function on \(\CC\).

Rosseta stone in mathematics

The similarity over number fields, function fields, algebraic curves over \(\CC\).

Referenced: Personal Thoughts.

Shapiro’s lemma

[2021-12-22 Wed 19:54] The following is Shapiro’s lemma.

Cf. Induction.

S-unit

[2021-11-28 Sun 14:39] Cf. S-integer.

Tamely ramified

[2021-12-19 Sun 15:09] See Unramified.

Tate cohomology group

[2021-12-22 Wed 20:51] We define the Tate cohomology groups to be \[H^{i}_{T} = \begin{cases} H^{i}(G, M) & i > 0 \\ M^{g} / \nm_{G}(M) & i = 0 \\ \ker(\nm_{G})/M I_{G} & i = -1 \\ H_{-i - 1}(g, M) & i < -1 \end{cases} \]

It is called cohomology group because it induces long exact sequence.

We have periodicity theorem for cohomology of cyclic groups.

Tate’s theorem

[2021-12-25 Sat 13:44] Tate’s theorem refers to the following

Cf. Tate cohomology group, Cohomology of local fields.

Totally ramified

[2021-12-19 Sun 15:06] See Unramified.

Transfer map

[2021-12-20 Mon 20:22] Let \(G\) be a finite group and \(H\) subgroup (not necessarily normal). Let \(g_{1}, \cdots, g_{n}\) be left coset representatives of \(H\) in \(G\). Let \(\phi(g) = g_{i}\) if \(g \in g_{i}H\). Then we define \(V(g) = \prod_{i = 1}^{n} \phi(g g_{i})^{-1}(g g_{i})\). The induced map \(V: G^{ab} \to H^{ab}\) is called the tranfer map.

Uniformizer

[2021-12-26 Sun 17:55] See Henselian valuation.

Unit group

[2021-12-26 Sun 17:54] See Henselian valuation.

If \(A\) satisfies Class field axiom, \(v\) is henselian valuation of \(A_{k}\) with respect to \(d\). Then we have following proposition.

Cf. Uniformizer, Unramified.

Unramified

[2021-12-19 Sun 15:02] Let \(L/K\) be an extension of finite extensions of \(\QQ_{p}\). Let \(\mathfrak{o}_{L}, \mathfrak{o}_{K}\) be the integral closure of \(\ZZ_{p}\) in \(K, L\). We say \(L/K\) is unramified if the maximal ideal of \(\mathfrak{o}_{K}\) generates the maximal ideal of \(\mathfrak{o}_{L}\).

For every extension \(L/K\) there exists a maximal subextension of \(L/K\) which is unramified, denoted by \(U\). If this is \(K\), we say \(L/K\) is totally ramified.

If \([L:U]\) is not divisible by \(p\), then \(L/K\) is tamely ramified.

The above definition can be generalized to global fields by specifiying the prime ideal.

In class field theory, the phrase \(L/K\) is unramified means that \(L/K\) is unramified over all finite places and every real embedding of \(K\) extends to a real embeddings of \(L\).

Valuation

[2021-11-28 Sun 14:38] A valuation \(\abs{\cdot}\) on a field \(k\) is a function defined on \(k\) with values in the non-negative real numbers satisfying the following axioms.

1. \(\abs{\alpha} = 0\) if and only if \(\alpha = 0\).
2. \(\abs{\alpha \beta} = \abs{\alpha} \abs{\beta}\).
3. There is a constant \(C\) such that \(\abs{1 + \alpha} \leq C\) whenever \(\alpha \leq 1\).

There always exists trival valuation that \(\abs{\alpha} = 1\) for all \(\alpha \neq 0\).

Two valuations are called equivalent if there \(c > 0\) such that \(\abs{\alpha}_{2} = \abs{\alpha}_{1}^{c}\).

Referenced: Discrete valuation.

Weil group

[2021-12-26 Sun 17:41] Let \(k\) be field, and \(\bar{k}\) an algebraic extension. Put \(G = \Gal(\bar{k}/k)\) and \(A\) be a \(G\)-module. Then for subextension \(K\) of \(\bark{k}/k\), define \(A_{K} = A^{\Gal(\bar{k}/K)}\).

Let \(d: G \to \hat{\ZZ}\) be a continuous surjective homomorphism. The Weil group of \(k\) is the subgroup \(d^{-1}(\ZZ)\) of \(G\).

The inertia group \(I_{k}\) is the kernel of \(d\). And maximal unramified extension \(k^{ur}\) of \(k\) to be the fixed field of \(I_{k}\). For subextension \(K\) of \(\bar{k}/k\), put \(I_{K} = G_{K} \cap I_{k}\) and let \(K^{ur} = k^{ur} K\). We say an extension \(L/K\) of subextension of \(\bar{k}/k\) is unramified if \(L \subset K^{ur}\).

Put \(d_{K} = \frac{1}{ [\hat{\ZZ}: d(G_{K})]} d\), then \(d_{K}\) induces isomorphism \(\Gal(K^{ur}/K) \cong \hat{ZZ}\). Given a finite extension \(L/K\) of subextensions of \(\bar{k}/k\), define the inertia degree \[f_{L/K} = [d(G_{K}) : d(G_{L})]\] and the ramification index \[e_{L/K} = [I_{K}: I_{L}]\]

we have \[e_{L/K} f_{L/K} = [L:K]\]

Zeta functions

[2021-12-21 Tue 18:58] Let \(K\) be a number field. The Dedekind zeta function \(\zeta_{K}(s)\) is defined for \({\rm Re}(s) >1\) \[\zeta_{K}(s) = \prod_{\mathfrak{p}}(1 - \nm(\mathfrak{p})^{-s})^{-1}\] where \(\mathfrak{p}\) runs over the nonzero prime ideals of \(\mathfrak{o}_{K}\).

Cf. Riemann zeta function, Norm.

Arithmetic Geometry

Absolute height

[2021-10-19 Tue 16:42] See Height.

Absolute value

[2021-10-19 Tue 16:26] An absolute value on a field \(k\) is a real-valued function \[\abs{\cdot}: k \to \linterval{0}{\infty}\] with the following properties

1. \(\abs{x} = 0\) if and only if \(x = 0\).
2. \(\abs{xy} = \abs{x} \abs{y}\).
3. \(\abs{x + y} \leq \abs{x} + \abs{y}\).

The absolute value is said to be nonarchimedean if it satisfies \[\abs{x + y} \leq \max \{ \abs{x}, \abs{y}\}\]

Referenced: Degree formula, Local degree, Nonarchimedean, Normalized absolute value.

Degree formula

[2021-10-19 Tue 16:29] Let \(k' / k\) be an extension of number fields, and let \(v \in M_{k}\) be an Absolute value on \(k\). Then \[\sum_{w \in M_{k'}, w | v} [k'_{w} : k_{v}] = [k' : k]\] where \(k_{v}\) is the \(v\) completion of \(k\).

Dirichlet theorem

The constant on the right hand side can be refined to \(\frac{1}{\sqrt{5}q^{2}}\), which can not be further required for \(\alpha = \frac{-1 + \sqrt{5}}{2}\).

Referenced: Roth’s theorem.

Elliptic curve

[2021-10-22 Fri 20:50] An elliptic curve \(E\) over \(K\) is a nonsingular projective curve of genus \(1\), which comes equipped with a rational point \(\OO \in E(K)\) the origin. A morphism \(E \to E'\) between elliptic curves over \(K\) is a morphism of \(K\)-schemes which preserves the origin. It is an isomorphism if it is an isomorphism of \(K\)-schemes.

The following theorem shows the elliptic curve can always be realized as a plane cubic curve.

Referenced: Origin, Group law, Isogeny.

Origin

[2021-10-22 Fri 21:02] See Elliptic curve.

Group law

[2021-10-22 Fri 20:50] We write \(\pic^{\circ} X\) for the subgroup of divisor classes of degree \(0\).

Cf. Elliptic curve.

The group is defined by \(P + Q = R\), where \((P) + (Q) = ( R) + (O)\).

TODO Dual

Isogeny

[2021-10-22 Fri 21:16] An isogeny is a morphism between Elliptic curves which is finite.

Referenced: Isogeny.

Riemann hypothesis

Tate module

[2021-10-22 Fri 21:19] Let \(l\) be a prime number unequal to the characteristic of \(K\). The Tate module of \(E\) is the inverse limit \[T_{l}E = \varprojlim E(\bar{K})[l^{n}]\] A free \(\ZZ_{l}\)-module of rank \(2\). The rational Tate module is \(V_{l}E = T_{l}E \otimes_{\ZZ_{l}} \QQ_{l}\). Here \(E(\bar{K})[m]\) denote the number of \(m\) torsion points of \(E(\bar{K})\).

Referenced: Tate’s isogeny theorem.

Tate’s isogeny theorem

Cf. Tate module.

Mordell-Weil theorem

[2021-10-22 Fri 21:43] The first version is weak Mordell-Weil theorem.

Then we lead to (full) Mordell-Weil theorem.

Finiteness of class group

Cf. Ideal class.

Finiteness of point under height

[2021-10-19 Tue 16:42] The following theorem shows some finiteness of points less than a given height.

This implies the Kronecker’s theorem.

Finiteness of unit group

Cf. S-integer.

Height

[2021-10-19 Tue 16:36] Let \(k\) be a number field and let \(P = (x_{0}, x_{1}, \cdots, x_{n}) \in \PP^{n}(k)\) be a point whose homogeneous coordinates are chosen in \(k\). The height is the quantity \[H_{k}(P) = \prod_{v \in M_{k}} \max \{\norm{x_{0}}_{v}, \norm{x_{1}}_{v}, \cdots, \norm{x_{n}}_{v}\}\] And define the logarithmic height by \[h_{k} = \log H_{k}(P)\] Cf. Normalized absolute value.

We have preceding formula for height in extension field

By above lemma, we can define the absolute height on \(\PP^{n}\) by \[H: \PP^{n}(\bar{Q}) \to \linterval{0}{\infty}\] \[H(P) = H_{k}(P)^{1/[k: \QQ]}\] where \(P \in \PP^{n}(k)\). The absolute logarithmic height is defined similarly.

We can extend the definition of height to varieties. Let \(\phi: V \to \PP^{n}\) is a morphism, then we can define \[h_{\phi}(P) = h(\phi(P))\] The definition of this height is dependent on the choice of the morphism \(\phi\).

Referenced: Absolute height.

TODO Jacobian variety

Referenced: Group law.

Local degree

[2021-10-19 Tue 16:31] Let \(v \in M_{k}\) be an Absolute value on number field \(k\). The local degree of \(v\) is the number \[n_{v} = [k_{v} : \QQ_{v}]\]

Nonarchimedean

[2021-10-19 Tue 16:29] See Absolute value.

Normalized absolute value

[2021-10-19 Tue 16:32] Let \(v \in M_{k}\) be an Absolute value on number field \(k\). The normalized absolute value of \(v\) associated to \(v\) is \[\norm{x}_{v} = \abs{x}_{v}^{n_{v}}\]

For normalized absolute value, we have the product formula, for \(x \in k^{* }\), we have \[\prod_{v \in M_{k}} \norm{x}_{v} = 1\]

Referenced: Height.

Roth’s theorem

Cf. Dirichlet theorem.

S-integer

[2021-10-19 Tue 16:34] The ring of S-integers of \(k\) is defined to be \[R_{S} = \{x \in k \mid \abs{x}_{v} \leq 1 \text{ for all } v \in M_{k}, v \not \in S\}\]

We can also characterize the rings of integers of \(k\) by \(M_{k}^{\infty}\)-integers of \(k\).

Referenced: S-unit, Finiteness of unit group.

Bibliography

[steinberg2012] Steinberg, Representation Theory of Finite Groups: An Introductory Approach, Springer (2012). ↩

[kirillov2016] Kirillov, Quiver Representations and Quiver Varieties, American Mathematical Society (2016). ↩

[dieck2008] tom Dieck, Algebraic Topology, European Mathematical Society (2008). ↩

[CLS2011] Cox, Little & Schenck, Toric Varieties, American Mathematical Society (2011). ↩

[milne2017] Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press (2017). ↩

[sernesi2006] Sernesi, Deformations of Algebraic Schemes, Springer Berlin Heidelberg (2006). ↩

[nakajima1999] Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, American Mathematical Society (1999). ↩

[milne2020a] @miscMilne2020a, title = Exercises of Algebraic Number Theory (v3.08), author = Milne, J. S., year = 2020 ↩

[kedlaya2021] Kedlaya, Note on Class Field Theory, (2021). ↩

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