Constructible Derived Category
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1 Constructible Derived Category Dongkwan Kim September 29, Category of Sheaves In this talk we mainly deal with sheaves of C-vector spaces. For a topological space X, we denote by Sh(X) the abelian category of sheaves of C-vector spaces on X. We assume that readers are familiar with basic sheaf theory. (i.e. sheafification, Sh(X) is abelian, etc.) 1.1 Operations on Sheaves Here we recapitulate some operations (i.e. functors) on sheaves. First we define internal Hom and tensor product as follows. (1) Internal Hom. For F, G Sh(X), we define Hom(F, G) = Hom Sh(X) (F, G) Sh(X) as the sheaf defined by U Hom(F U, G U ) = Hom Sh(X) (F U, G U ). (2) Tensor product. For F, G Sh(X), we define F G = F C G Sh(X) as the sheaf associated to the following presheaf U F(U) G(U). Note that the natural map (Hom(F, G)) x Hom(F x, G x ) is in general neither injective nor surjective. However, we have (F G) x F x G x. Now for a continuous map f : X Y, we define operations of sheaves which corresponds to f as follows. (3) Push-forward or direct image. For F Sh(X), we define f F Sh(Y ) such that for open V Y we have f F : V F(f 1 (V )). In particular, if Y is a point, then we denote Γ(X, G) := f G as we identify Sh(Y ) with the category of C-vector spaces and call it the set of global sections of F on X. 1
2 (4) Pull-back or inverse image. For G Sh(Y ), we define f G Sh(X) to be a sheaf associated to the presheaf U lim G(V ). f(u) V Note that the natural map (f F) f(x) F x is not an isomorphism, however (f G) x G f(x). For a subspace f : Z X, using the functors above we have a way to restrict a sheaf F Sh(X) to Z still considered as an element in Sh(X) by taking f f F. We will define two other analogous restriction functors when the subspace is locally closed, which would coincide with f f in some special situations. (5) For a closed subset f : Z X and for F Sh(X), we define F Z := f f F. For an open subset U, we set F U := ker(f F X\U ). Finally, if A is locally closed, say A = U Z for U open and Z closed, then we define F A := (F U ) Z. 1 We briefly mention one of its properties, which would become obvious as we discuss further. Proposition 1. Suppose f : A X is locally closed and g : B = X \ A X. Then for any F Sh(X), f (F A ) = f F, g (F A ) = 0. (6) The sheaf of sections supported by a subspace. For a closed subset f : Z X and F Sh(Z), we let Γ Z (F) be the sheaf defined by V ker(f(v ) F(V \ (V Z))). For A = U Z locally closed with U open and Z closed, define Γ A (F) by Γ A (F)(V ) := Γ A (F U )(U V ). It is straightforward to check that for an open embedding f : U X, Γ U (F) = f f F. So far we defined Hom,, f, f, A, Γ A. If topological spaces are locally compact 2, there is one more functor which is of our interest. (7) Proper push-forward or direct image with proper support. Assume that X, Y are locally compact topological spaces and given a continuous map f : X Y and a sheaf F Sh(X). We denote by f! F the sheaf defined by U {s F(f 1 U) f supp(s) : supp(s) U is proper 3 }. 1 This is well-defined, that is this definition does not depend on the choice of U or Z. 2 In this talk locally compact spaces are always Hausdorff. 3 Here f being proper means that every inverse image of compact set is again compact. 2
3 In particular, if Y is a point then we denote Γ c (X, F) := f! (F) as we identify Sh(Y ) with the cateogory of C-vector spaces and call it the set of compactly (or properly) supported global sections of F on X. Note that we have a natural embedding f! F f F, which is clearly an isomorphism if f is a priori proper. Also we have the following property which would be apparent by the argument later on. It is not in general true for the usual push-forward. Proposition 2. [[KS, Proposition 2.5.2]] If f : X Y is a continuous map between locally compact spaces, then for y Y and F Sh(X) the canonical morphism (f! F) y Γ c (f 1 (y), F f 1 (y)) is an isomorphism. 4 If f : X Y is proper then f f = f! f, which is true especially when f is a closed embedding. One might already expect that we have the following property. Proposition 3. If X is locally compact and f : Z X is a closed embedding, thus in particular Z is also locally compact, then for F Sh(X) we have F Z = f! f F. 1.2 Exactness, Adjunction Formula, and Base Change We state, but omit proofs of, the exactness properties of the operations defined above. Proposition 4. Let f : X Y be a continuous map between topological spaces. Then, (1) Hom : Sh(X) op Sh(X) Sh(X) is left exact. (2) : Sh(X) Sh(X) Sh(X) is exact. 5 (3) f : Sh(X) Sh(Y ) is left exact. Thus in particular, Γ is left exact. (3 ) f : Sh(X) Sh(Y ) is exact if f is a closed embedding. (4) f : Sh(Y ) Sh(X) is exact. (5) F F X : Sh(Y ) Sh(Y ) is exact if f is a locally closed embedding. (6) Γ X : Sh(Y ) Sh(Y ) is left exact if f is a locally closed embedding. 4 This also has an étale analogue which is difficult to prove. See the comments on [M, p.117] after Theorem 17.7 and for more infomation. 5 In general if we consider tensor product over a sheaf of rings then it is right exact. Here we only consider sheaves of C-vector spaces, thus every sheaf is flat. 3
4 (7) f! : Sh(X) Sh(Y ) is left exact if X and Y are locally compact. (7 ) f! : Sh(X) Sh(Y ) is exact if Y is locally compact and f is a locally closed embedding. Also we have the following adjunction formulae. Proposition 5. Let f : X Y be a continuous map between topological spaces and let F, F, F Sh(X), G, G Sh(Y ). Then (a) Hom(F F, F ) Hom(F, Hom(F, F )) (b) Hom(G, f F) f Hom(f G, F) (c) Hom(G X, G ) Hom(G, Γ X (G )) if f is a locally closed embedding. If we take the set of global sections on each formula, we get the usual adjunctions. Meanwhile, suppose we are given the following diagram of locally compact spaces. A g f B g C f D Then for any F Sh(C) we have a well-defined morphism f! F f! g g F by adjunction. Now for U D, we have f! g F(U) = {s F( g 1 f 1 U) f : supp( g s) U is proper} g f! F(U) = {s F( f 1 g 1 U) f : supp(s) g 1 U is proper} In other words, if s f! g F(U) then s F( g 1 f 1 U) = F( f 1 g 1 U) and there exists Z C such that supp(s) g 1 Z and f Z : Z U is proper. Now suppose further that the diagram above is cartesian. Then f : g 1 Z g 1 U is also proper, thus so is supp(s) g 1 U. It defines a natural morphism f! g g f!, and f! F f! g g F g f! g F. Again by adjunction we have g f! F f! g F. Now for x B, we have (g f! F) x = (f! F) g(x) = Γ c (f 1 (g(x)), F f 1 (g(x))) Thus we proved the following theorem = Γ c ( f 1 (x), ( g F) f 1 (x) ) = ( f! g F) x. Theorem 6 (Proper Base Change). If the diagram above is cartesian, we have a natural isomorphism g f! f! g. Now we see that Proposition 1 and 2 are special cases of this theorem together with Proposition 3. 4
5 1.3 Constructible Sheaves The main object with which we are concerned is (cohomologically) constructible sheaves. We briefly discuss the corresponding definitions from now on. Let X be a complex analytic space and P = (X i ) i I be a partition of X which satisfies the following properties. Each X i is nonempty, locally closed and connected. P is locally finite. Each X i \ X i is a union of elements in P. For each X i P, X i and X i \ X i are closed complex analytic subspaces in X. Then we say F Sh(X) is constructible if there exists a partition P which satisfies conditions above such that for each X i P, F Xi is a local system, i.e. a locally constant sheaf of C-vector spaces of finite dimension. In this case we also say that F is constructible with respect to P. We denote by Sh c (X) the full subcategory of constructible sheaves on Sh(X) and by Sh c (X, P) its full subcategory of constructible sheaves with respect to a fixed P. Then it is easy to show that Sh c (X) is an abelian category. Also, it is easy to check that internal Hom and tensor product preserve constructibility. For an analytic morphism f : X Y between complex analytic varieties, one can also easily show that f preserves constructibility. Push-forward is a little tricky and does not preserve the property in general. However, if f is indeed algebraic, then f preserves it using Chevalley constructibility theorem. Similarly, for F Sh c (X) if f is proper on supp F, then f F Sh c (Y ). We postpone going into details about it until we introduce a constructible derived category. Example 7. Suppose X = C and i : Z = {0} X, j : U = X \ Z X. We let F Sh(U) be the local system with monodromy matrix A GL n (C) which has Jordan blocks A 1,, A r with eigenvalues λ 1,, λ r such that λ 1 = = λ s = 1. Then one can easily see that i j F C s, i j! F = Some Questions So far we discussed definitions and some properties of operation on sheaves. Now these questions arise naturally. Is there any description for Γ Z similar to F Z f! f F? Is there any adjunction formula for f!? For a local system, we can easily take its dual. Can we expect such a thing in general? To answer these questions properly, we shall introduce the notion of derived category. 5
6 2 Derived Category of Sheaves 2.1 Derived Category of Abelian Categories We assume that we are already familiar with the language of a derived category, but we briefly mention some properties. For an abelian category A, we denote by K(A) the homotopy category of chain complexes in A. The derived category of A, usually denoted by D(A), is the category obtained from K(A) by inverting quasi-isomorphisms, which is a morphism of chain complexes which induces isomorphisms on their cohomology. This process is called localization. One of its major properties is that it is a triangulated category. Even though D(A) is no longer abelian, it has an analogue of an exact sequence in an abelian category, called a distinguished triangle. Then a functor between triangulated category is called exact if it is additive, commutes with translations, and sends distinguished triangles to distinguished ones. Note that K(A) is also triangulated and the localization functor K(A) D(A) is exact. Also we may consider at first the category of chain complexes of bounded cohomology, and we let D b (A) be its derived category. Likewise, D + (A), D (A), respectively, is the derived category of bounded below complexes, bounded above complexes, respectively. Sometimes D b (A) is the same as the category of bounded chain complexes if we have some finiteness condition on A. 2.2 Derived Category of Sheaves For a topological space X we may define D(Sh(X)), the derived category of sheaves on X. We assume we already know that Sh(X) has enough injectives. Therefore, it is possible to derive functors which are left exact on Sh(X). Let f : X Y be a continuous map between topological spaces. Then we have RHom : D (Sh(X)) op D + (Sh(X)) D + (Sh(X)) Rf : D + (Sh(X)) D + (Sh(Y )) RΓ X : D + (Sh(Y )) D + (Sh(Y )) if f is a locally closed embedding. Rf! : D + (Sh(X)) D + (Sh(Y )) if X, Y are locally compact. Also, since f,, Z (where Z is a locally closed subset) are already exact, they automatically define functors on the derived category of sheaves. Meanwhile, in general Sh(X) does not have enough projectives as the next example shows. Example 8. Suppose X = C with the usual topology. Then we claim that the skyscraper sheaf C 0 supported by the origin does not admit an epimorphism from a projective sheaf. Thus suppose otherwise and let P C 0 be such a morphism. We will show that P(U) C 0 (U) is a zero map for all neighborhood U of the origin. Choose ε > 0 such that D 2ε U, where D 2ε = {x C x < 2ε} is a disk of radius 2ε. Since we have a surjection C Dε C 0, we should be 6
7 able to lift P C 0 to obtain P C Dε. (Note that C Dε is a constant sheaf on D ε extended by zero on C \ D ε.) However, then C Dε (U) = 0 since U is strictly bigger than D ε, which means P(U) C 0 (U) is a zero map. In other words, P 0 (C 0 ) 0 is zero, which contradicts the assumption. 2.3 Adjunction and Base Change From adjunction of operations on sheaves, we easily deduce the following formulae. For the sake of simplicity, we restrict our situation to the bounded derived category of sheaves. Proposition 9. Let f : X Y be a continuous map between topological spaces and let F, F, F D b (Sh(X)), G, G D b (Sh(Y )). Then (a) RHom(F F, F ) RHom(F, RHom(F, F )) (b) RHom(G, Rf F) Rf RHom(f G, F) (c) RHom(G X, G ) RHom(G, RΓ X (G )) if f is a locally closed embedding. One advantage to consider derived categories is that we may also find adjunction for f!. Theorem 10. Suppose f : X Y is a continuous map between locally compact spaces and f! has finite cohomological dimension. Then we have a well-defined functor f! : D b (Sh(Y )) D b (Sh(X)) such that for F D b (Sh(X)) and G D b (Sh(Y )) the following adjunction formula holds. RHom(Rf! F, G) Rf RHom(F, f! G) Note that f! is unique up to isomorphism by Yoneda lemma. Also in general f! is not a derived functor of a functor on sheaves and only well-defined on the derived category. 6 Now it is a direct consequence from adjunction that if f : X Y is a locally closed embedding then RΓ X Rf f!. For more information one may refer to [KS, Section 3.1]. Meanwhile, we have a derived version of proper base change. Suppose the following diagram is cartesian as before. A f B g g C f D Then we proved that f! g g f! under suitable conditions. Since pull-back is exact, if we derive both sides we get R f! g g Rf!, say as a morphism form D b (Sh(C)) to D b (Sh(B)). (Here we need to verify the fact that g sends injective objects to f! -acyclic objects.) Also by adjunction, it also implies that R f g! g! Rf. 6 If f! were a left adjoint on the category of sheaves, then it should be right exact. 7
8 2.4 Constructible Derived Category In this section we assume X is a complex analytic space. Then we may also consider D b (Sh c (X)), the bounded derived category of constructible sheaves. However, since we mainly deal with the cohomology of a chain complex on the derived category rather than each term on the complex, it is reasonable to consider D b c(sh(x)), the bounded derived category of sheaves of constructible cohomology, or the constructible derived category on X. A striking fact is that when X is a complex algebraic variety, then they are the same 7, which is proved in [N]. 8 From now on we only deal with D b c(sh(x)). The operations we introduced so far preserve cohomological constructibility. In other words, we have the following Proposition 11. Let f : X Y be a morphism between complex analytic spaces and F, F D b c(sh(x)), G D b c(sh(y )). Then, (a) f G, f! G D b c(sh(x)). (b) Rf F, Rf! F D b c(sh(y )) if f is algebraic or f supp F is proper. (c) RHom(F, F ), F F D b c(sh(x)). We omit the proof. 3 Poincare-Verdier Duality 3.1 Duality Functor We will define a duality functor on D b (Sh(X)), which would be a generalization of taking dual of a local system. Thus it would be a contravariant endofunctor on D b (Sh(X)) and it would be nice if it is also an involution. To that end, among the operations we have it is natural to choose RHom(, ) for some D b (Sh(X)). Now we give the definition of Verdier duality as follows. Definition 12. Let f : X Y be a continuous map of finite cohomological dimension, so that f! is well-defined. Then we call ω X/Y := f! C Y D b c(sh(x)) D b (Sh(X)) the relative dualizing complex on X over Y. If Y is a single point, then we set ω X := ω X/Y and call it the dualizing complex on X. Definition 13. Keep assuming the situation above. We define a contravariant endofunctor D = D X : D b (Sh(X)) D b (Sh(X)) : F RHom(F, ω X ) and call it the Verdier duality on D b (Sh(X)). 7 It is not in general true if we fix a partition P on X and consider Sh c(x, P) instead, or X is just a complex analytic space 8 The perverse analogue is proved by Beilinson. 8
9 Thus in particular, ω X = DC X. We investigate some interesting properties of Verdier duality. First, we need the following lemmas. Lemma 14 (Projection Formula). For continuous f : X Y where X, Y are locally compact and F D b (Sh(X)), G D b (Sh(Y )), we have Rf! F G Rf! (F f G). Lemma 15. Keep the setting above and assume further that f has finite cohomological dimension. Then for G, G D b (Sh(Y )) we have f! RHom(G, G ) RHom(f G, f! G ). Proof. For any F D b (Sh(X)) we have Hom(F, f! RHom(G, G )) = Hom(Rf! F, RHom(G, G )) from which the result follows. = Hom(Rf! F G, G ) = Hom(Rf! (F f G), G ) = Hom(F f G, f! G ) = Hom(F, RHom(f G, f! G )) Proposition 16. Keep the setting above, then we have f! D = Df. Proof. For any G D b (Sh(Y )), we have f! DG = f! RHom(G, ω Y ) = RHom(f G, f! ω Y ) = RHom(f G, ω X ) = Df G. Thus the result follows. Proposition 17. Keep the setting above, then we have DRf! = Rf D. Proof. It follows from Hom(G, DRf! F) = Hom(Rf! F G, ω Y ) = Hom(Rf! (F f G), ω Y ) and Yoneda lemma. = Hom(F f G, f! ω Y ) = Hom(F f G, ω X ) = Hom(f G, DF) = Hom(G, Rf DF) From now on we investigate properties of the dualizing complex. Suppose f : X Y is a morphism of finite cohomological dimension between locally compact spaces, and G, G D b (Sh(Y )). Then by adjunction we have a natural morphism Rf! f! G G, from which we also have Rf! f! G G G G. By projection formula it is equivalent to Rf! (f! G f G ) G G, thus by adjunction again we have f! G f G f! (G G ). If we set G = C Y, then f! G = ω X/Y, thus we have ω X/Y f G f! G. Now we are ready to state the following theorem. 9
10 Theorem 18. Suppose f : X Y is a topological submersion with fiber dimension d. Then, (a) H k (ω X/Y ) = 0 if k d and H d (ω X/Y ) is a local system of rank 1. When Y is a point it is called the orientation sheaf of X. (b) The natural morphism ω X/Y f G f! G above is an isomorphism. (c) If X and Y are orientable manifolds, then ω X/Y C X [d]. Thus in particular f [d] f!. Proof. To show (a) it suffices to consider f : R n {pt}. For (b) we need to keep track of the construction of this morphism. (c) is more complicated, see [KS, Lemma 3.3.7]. Example 19 (Poincaré duality). Suppose X is a topological manifold. Then for f : X {pt} we apply DRf! Rf D to C X to get DRf! C X Rf DC X, or DRΓ c (X, C X ) RΓ(X, ω X ). Thus by taking ( i)-th cohomology, we have H i c(x, C X ) H i (X, ω X ). If furthermore X is orientable, then ω X C X [d] where d = dim X, thus we have which is the Poincaré duality of X. H i c(x, C X ) H d i (X, C X ) Example 20 (Alexander duality). Suppose X is an orientable topological manifold and Z X is a closed subset. Then if we denote the closed embedding of Z to X by f : Z X, we have RΓ Z D Rf f! D Rf Df DRf! f Thus if we apply this to C X ω X [ d] then we have RΓ Z C X [d] DC Z. By taking ( i)-th cohomology on both sides we have which is the Alexander duality. H d i Z (X, C X) Hc(Z, i C Z ) 3.2 Verdier Duality on Constructible Derived Category Unfortunately, D is not a duality on the whole D b (Sh(X)). However, we have Theorem 21. Suppose X is a complex analytic space. Then the restriction of D is a contravariant endofunctor on D b c(sh(x)). Furthermore, D is an involution on D b c(sh(x)). We omit the proof. 10
11 3.3 Formalism of Six Functors Suppose f : X Y is an analytic morphism between complex analytic spaces. So far we defined the following functors. (1) RHom : D b c(sh(x)) op D b c(sh(x)) D b c(sh(x)) (2) : D b c(sh(x)) D b c(sh(x)) D b c(sh(x)) (3) Rf : D b c(sh(x)) D b c(sh(y )) (4) Rf! : D b c(sh(x)) D b c(sh(y )) (5) f : D b c(sh(y )) D b c(sh(x)) (6) f! : D b c(sh(y )) D b c(sh(x)). Also we have the duality functor D : D b c(sh(x)) D b c(sh(x)) op which is an involution, i.e. DD = id. So far we showed that they satisfy the following properties. f Rf, Rf! f!, and Tensor-Hom adjunction The adjunctions above are functorial, thus each f Rf, Rf!, f, f! functor. is a There exists a natural morphism Rf! Rf which is an isomorphism if f is proper. If f is smooth of relative complex dimension d, then there exists a natural isomorphism f f! [ 2d] If the following diagram is cartesian, A f B g g C f D then there exist natural isomorphisms R f! g g Rf! and R f g! g! Rf. For F D b c(sh(x)) and G D b c(sh(y )), we have the projection formula F Rf! G Rf! (f F G). DRf! Rf D, DRf Rf! D, Df! f D, Df f! D. This is a typical example of so-called Grothendieck s six operations. For another example, which corresponds to D-module theory, we refer readers to [E]. 11
12 References [CD] Denis-Charles Cisinski and Frédéric Déglise, Triangulated categories of mixed motives, arxiv: [math.ag] (2009), available at [CG] Neil Chriss and Victor Ginzburg, Representation Theory and Complex Geometry, Modern Birkhäuser Classics, Birkhäuser, [D] Alexandru Dimca, Sheaves in Topology, Springer, [E] Pavel Etingof, Formalism of six functors, available at ~etingof/dmodfactsheet.pdf. [GW1] Iordan Ganev and Robin Walters, The derived category of constructible sheaves (2014), available at [GW2], The derived category of constructible sheaves (2014), available at [I] Birger Iversen, Cohomology of Sheaves, Springer-Verlag, [KS] Masaki Kashiwara and Pierre Schapira, Sheaves on Manifolds, Grundlehren der mathematischen Wissenschaften, vol. 292, Springer-Verlag Berlin Heidelberg, [M] J.S. Milne, Lectures on Etale Cohomology (2013), available at org/math/coursenotes/lec.pdf. [N] Madhad V. Nori, Constructible Sheaves, Algebra, arithmetic and geometry, Part I, II(Mumbai, 2000), 2002, pp
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