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Short Lecture on Sheaves and Derived Categories

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Short Lecture on Sheaves and Derived Categories SHORT LECTURE ON SHEAVES AND DERIVED CATEGORIES STEPHANE´ GUILLERMOU Abstract. Extended notes of the first lecture of a minicourse in Montreal, June 2019. 1. Sheaves In this section k is a given ring. 1.1. Definition. A presheaf P (of k-modules) on a topological space X is the data of k-modules P (U) for all open subsets U of X together V with linear maps rU : P (U) −! P (V ) for all inclusions V ⊂ U such that W V W rV ◦ rU = rU for W ⊂ V ⊂ U. For a section s 2 P (U) we usually V set sjV = r (s). A sheaf F is a presheaf such that, for any covering S U U = i2I Ui and sections si 2 F (Ui) satisfying sijUij = sijUij , there exists a unique s 2 F (U) such that si = sjUi . We often set Γ(U; F ) = F (U). The stalk of a presheaf at x 2 X is P = lim P (U), where U runs x −!U2X over the open neighborhoods of x. A morphism of presheaves f : P −! P 0 is the data of groups mor- phisms f(U): P (U) −! P 0(U) which commute with the restriction 0 maps, that is, rV;U ◦ f(U) = f(V ) ◦ rV;U , for all V ⊂ U ⊂ X.A morphism of sheaves is a morphism of the underlying presheaves. We denote by Mod(kX ) the category of sheaves of k-modules on X. Examples 1.1. (i) The constant sheaf kX on X is defined by kX (U) = ff : U −! k; f is locally constantg. If N is a k-module, we define NX , the constant sheaf with stalks N, in the same way. (ii) If Z ⊂ X is a closed subset, we define kX;Z (or kZ if X is under- stand) by kX;Z (U) = ff : U \ Z −! k; f is locally constantg. A morphism u in Mod(kX ) is an isomorphism if and only if ux is an isomorphism for all x 2 X. Lemma 1.2 (Associated sheaf of a presheaf). Let X be a topological space and let P 2 P(X). There exist a sheaf P a and a morphism of 1 2 STEPHANE´ GUILLERMOU a presheaves u: P −! P such that ux is an isomorphism, for each x 2 X. Moreover the pair (P a; u) is unique up to isomorphism. Any morphism v in Mod(kX ) has a kernel, given by U 7! ker v(U), a and a cokernel, given by (U 7! coker v(U)) . The category Mod(kX ) is abelian (which means that it is additive, kernel and cokernel exist and are well-behaved in the sense \ker(coker(v)) ' coker(ker(v))"). We can u v also check: a sequence F −! G −! H in Mod(kX ) is exact if and only if ux vx the sequences of stalks Fx −! Gx −! Hx are exact for all x 2 X. 1.2. Operations. Proposition 1.3. Let Fi, i 2 I, be a family of sheaves in Mod(kX ). Q L Then the product Fi and the sum Fi exist in Mod(kX ). The i2I Q i2I Q product is the sheaf defined by Γ(U; i2I Fi) = i2I Γ(U; Fi) for any open subset U. The sum is the sheaf associated with the presheaf U 7! L i2I Γ(U; Fi). For any x 2 X we have a canonical isomorphism M M (1.1) ( Fi)x ' (Fi)x: i2I i2I Definition 1.4. For F; G 2 Mod(kX ) we define a sheaf Hom(F; G) 2 Mod(kX ), the internal hom sheaf, by Γ(U; Hom(F; G)) = HomMod(kU )(F jU ;GjU ): We define the tensor product F ⊗kX G as the sheaf associated with the presheaf U 7! F (U) ⊗k G(U). We can prove (1.2) (F ⊗kX G)x ' Fx ⊗k Gx; for all x 2 X. Lemma 1.5. The functor Hom(·; ·) is left exact in both arguments. The functor · ⊗kX · is right exact in both arguments, and exact if k is a field. Let f : X −! Y be a continuous map between topological spaces. Definition 1.6. For F 2 Mod(kX ) we define a sheaf f∗F 2 Mod(kY ) −1 by (f∗F )(V ) = F (f (V )) for any open subset V ⊂ Y , with the re- striction maps naturally given by those of F (it is clear that f∗F is a presheaf and it is easy to check that it is actually a sheaf). If u: F −! G is a morphism in Mod(kX ), we define f∗u: f∗F −! −1 f∗G by (f∗u)(V ) = u(f (V )). We obtain a functor f∗ : Mod(kX ) −! Mod(kY ). Lemma 1.7. For any continuous map f : X −! Y , the functor f∗ is left exact. SHORT LECTURE ON SHEAVES AND DERIVED CATEGORIES 3 y Definition 1.8. For G 2 Mod(kY ) we define a presheaf f G on X by (f yG)(U) = lim G(V ), where V runs over the open neighbor- −!V ⊃f(U) hoods of f(U) in Y . The restriction maps are naturally induced by those of G. We set f −1G = (f yG)a. A morphism u: F −! G induces morphisms on the inductive limits, (f yu)(U):(f yF )(U) −! (f yG)(U), for all U 2 Op(X), which are com- patible and define f yu: f yF −! f yG. We set f −1u = (f yu)a. We thus −1 obtain a functor f : Mod(kY ) −! Mod(kX ). −1 Lemma 1.9. The functor f is left adjoint to f∗. In particular there −1 exist natural isomorphisms Hom(f G; F ) ' Hom(G; f∗F ) for all F 2 Mod(kX ), G 2 Mod(kY ). When f : X −! Y is an embedding we often write −1 GjX := f G: If f is the inclusion of an open set, we have (GjX )(U) = G(U), for all U 2 Op(X). Example 1.10. Let X be a Hausdorff topological space and Z ⊂ X a compact subset. Then, for any F 2 Mod(kX ) and V 2 Op(Z), we have (F j )(V ) ' lim F (U), where U runs over the open neighborhoods Z −!U⊃V of V in X. Lemma 1.11. Let f : X −! Y be a continuous map and let x 2 Y . For −1 any F 2 Mod(kY ) we have a natural isomorphism (f F )x ' Ff(x). Since the exactness of a sequence of sheaves can be checked in the stalks we deduce: Lemma 1.12. For any continuous map f : X −! Y , the functor f −1 is exact. 1.3. Locally closed subsets. A subset W of X is locally closed subset if we can write W = U \ Z with U open and Z closed. Lemma 1.13. Let W ⊂ X be a locally closed subset and F 2 Mod(kX ). Then there exists a unique sheaf FW 2 Mod(kX ) such that FW jW ' F jW and FW jXnW ' 0. Moreover we have FW ' F ⊗ (kX )W . We set for short kX;W = (kX )W and even kW = kX;W when it is clear that we consider sheaves on X. Example 1.14. If W is closed in X, the sheaf kW is already defined in Example 1.1. In general we have kW (U) ' ff : U \ W −! k; f is locally constant and fx; f(x) 6= 0g is closed in Ug. 4 STEPHANE´ GUILLERMOU Lemma 1.15 (Excision). Let W ⊂ X be a locally closed subset and let W 0 ⊂ W be a closed subset of W . Then W 0 and W n W 0 are locally closed in X and we have an exact sequence: 0 −! kW nW 0 −! kW −! kW 0 −! 0: Lemma 1.16 (Mayer-Vietoris). Let Z1;Z2 ⊂ X be closed subsets and U1;U2 ⊂ X open subsets. We have exact sequences 0 −! kZ1[Z2 −! kZ1 ⊕ kZ2 −! kZ1\Z2 −! 0; 0 −! kU1\U2 −! kU1 ⊕ kU2 −! kU1[U2 −! 0: 1.4. Proper direct image. A topological space X is locally compact if, for any x 2 X and any neighborhood U of x, there exists a compact neighborhood of x contained in U. Now we assume X; Y are Hausdorff and locally compact. Then a map f : X −! Y is proper if the inverse image of any compact subset of Y is compact. Definition 1.17. Let f : X −! Y be a continuous map of Hausdorff and locally compact spaces. For F 2 Mod(kX ) we define a subsheaf f!F 2 Mod(kY ) of f∗F by −1 (f!F )(V ) = fs 2 (f (V )); fjsupp s : supp(s) −! V is properg for any open subset V ⊂ Y . If u: F −! G is a morphism in Mod(kX ), the morphism f∗u: f∗F −! f∗G sends f!F to f!G. We obtain a functor f! : Mod(kX ) −! Mod(kY ). ∼ If the map f itself is proper, then we have f! −! f∗. Lemma 1.18. The functor f! is left exact. For F 2 Mod(kX ) and U 2 Op(X) we set Γc(U; F ) = fs 2 F (V ); supp(s) is compact.g We have Γc(U; F ) ' a!(F jU ), where a is the projection U −!fptg. Proposition 1.19. Let f : X −! Y be as in Definition 1.17. For any F 2 Mod(kX ) and y 2 Y we have −1 (f!F )y ' Γc(f (y); F jf −1(y)): Example 1.20. In the situation of Lemma 1.13 let j : Z −! X be the −1 inclusion.
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