# Derived categories, perverse sheaves and intermediate extension functor

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1 Derived categories, perverse sheaves and intermediate extension functor Riccardo Grandi July 26, 2013 Contents 1 Derived categories 1 2 The category of sheaves 5 3 t-structures 7 4 Perverse sheaves 8 1 Derived categories We wish to give a brief description on how to construct derived categories. First, we need to have an abelian category, C, with some extra properties (e.g. having enough injective objects...). We then write C(C ) to denote the category of complexes in C : C 1 C 0 C 1 D 1 D 0 D 1 where morphisms satisfy the usual commutative squares. We further define C # (C ) C(C ), where # = b, +,, the subcategory of complexes bounded on both sides, bounded on the right, and bounded on the left, respectively. (By bounded on the right, we simply mean that, for some n, all C i = 0 for i n. A complex bounded on the left is defined similarly.) We give the following definitions: 1

2 Definition 1. A map f : X Y is said to be a quasi-isomorphism if H n (f) are isomorphisms, n Z. Definition 2. We also define [n] : C(C ) C(C ) to be the functor such that the k-th component of the new complex is the same as the (n + k)-th component of the original complex, i.e. (C [n]) k = C n+k. We call [n] the n-th shift functor. We also note that the differentials of the new complex are given by k [n] = ( 1) n k. Definition 3. We define the homotopy category, K # (C ), to be the category having for objects the same as in C # (C ), and having the same maps as in C # (C ) but modding out all the maps that are homotopic to 0. We note that K # (C ) is not abelian, however it remains additive. It is also triangulated; we have distinguished triangles given by some: X Y Z +1 X [1]. (E.g. X Id X 0 +1 X [1].) We have that H and Hom(, ) are cohomological functors in that they send triangles to long exact sequences. E.g. if we have: X Y Z X [1] in K # (C ), we get a long exact sequence in C : H 0 (X ) H 0 (Y ) H 0 (Z ) H 1 (X ). Definition 4. We say that a family of maps, S, is a multiplicative system, if: Id X S, f, g S f g S, We can always find X and t S such that the following homotopy commutes: g Y X t X f Y where s S and f and g are any morphisms, not necessarily in S. Similarly, s 2

3 Y s Y g f X t X For f, g Hom(X, Y ), it is equivalent to have: and s : Y Y such that s f = s g, t : X Xsuch that f t = g t. With such a multiplicative system it is possible to form fractions in a similar fashion to localisations on rings. We can now form the Derived category, D # (C ), with respect to S := family of quasi-isomorphisms: the objects are the same as in K # (C ) and maps are of the following kind: Hom D # (C )(X, Y ) = {(X s W 1 f Y ) s S}/, where the equivalence can be defined in terms of the following commutative diagram: W 1 f 1 s 1 X W 3 s 2 s 3 and composition can be defined looking at the following diagram: W 2 f 2 Y s W 1 u f W 3 h t W 2 X Y Z. (We then write f s 1 for any map in K # (C ).) There is a localisation functor Q : K # (C ) D # (C ), which leaves objects as is and on maps: (i.e. f f/1). f Q(f) = X Id X f Y, 3 g

4 Lemma 1. Let C 0 be a category with S 1 a multiplicative system. Let I 0 C 0 be a full subcategory such that for all X C 0 there is s : X J with s S 1, J I 0. Then I 0 S 1 is a multiplicative system and (I 0 ) I0 S 1 (C 0) S 1, an equivalence of categories. Remark: C D(C ) and D # (C ) D(C ) are fully faithful embeddings. Moreover, D # (C ) is triangulated and Q respects triangles. Definition 5. We say that an abelian subcategory C C is thick if whenever we have an exact X 1 X 2 X 3 X 4 X 5, with X 1, X 2, X 4, X 5 C, then X 3 C. Lemma 2. Let C C be a thick subcategory. Take D C (C ) D(C ) to be the full subcategory whose objects are complexes with cohomology in C. Then D # C (C ) is triangulated. Let F : C D be an additive functor between two abelian categories. Then there is an induced functor on the homotopy categories. We want to know when there exists an extension F : K # (C ) K#F K # (D) Q C D # (C ) F Q D D # (D). Definition 6. A derived functor for a left exact functor F is a pair (RF, τ), where RF is a δ-functor (triangle preserving) and τ Nat(Q D K # F, RF Q C ) is a natural transformation. Definition 7. We say that I C is F -injective if: X C we have an exact 0 X I with I I. If 0 X X X 0 is exact with X, X I then X I. F takes exact sequences in I to exact sequences in D. Theorem 1. If F : C D is left exact and there exists an F -injective subcategory of C, then RF : D # (C ) D # (D) exists for # = +, b. 4

5 To compute RF (X ) for X D # (C ), we take a quasi-isomorphism X I where I is a complex of F -injectives. Then RF (X ) = K # F (I ). Remarks: If we have C C C with F, G left exact and I C is F -injective, I C is G-injective, and F (I ) I, then I is G F injective and R(G F ) = RG RF. We also recover the usual derived functor. That is, if we have 0 A B C 0 in C and define R i F so that we get an exact 0 F (A) F (B) F (C) R 1 F (A), then R i F (A) = H i (RF ([A])), where [A] is a complex with A in position 0. 2 The category of sheaves Assume all topological spaces are locally compact and Hausdorff. Let Sh(X) denote the category of sheaves on X and Mod(R) denote the category of R-modules on X (For a sheaf R of rings on X). Let f : X Y be continuous. Then, for F Sh(X), G Sh(Y ), we have the following functors: Direct image: f : Sh(X) Sh(Y ), (f F )(V ) = F (f 1 V ). f is left exact. We get Γ when Y = {pt}. Proper direct image: f! : Sh(X) Sh(Y ), (f! )(V ) = {s F (f 1 (V )) such that f Supp(s) : Supp(s) V is proper }. f! is left exact. We get Γ c for Y = {pt}. Inverse image: f 1 : Sh(Y ) Sh(X). Consider the presheaf U lim G(V ). V f(u) Then f 1 G is the sheafification. Note that (f 1 G) x = G f(x). This implies that f 1 is exact. Here Γ and Γ c correspond to the global section of the sheaf, or the global section with compact support. For F, G Sh(X), let Hom(F, G) Sh(X), given by U Hom(F U, G U ). This makes Hom(, ) into a left exact bifunctor. If R is sheaf of rings on Y and M Mod(R), N Mod(f 1 R) then Hom R (M, f N)) = Hom f 1 R(f 1 M, N)). 5

6 Thus (f 1, f ) are an adjoint pair. Important: Mod(R) has enough injectives. Let D # (R) := D # (Mod(R)). We will often look at Rf, Rf!, RΓ, RΓ c, f 1, RHom(, ), RHom(, ). One has that Hom D + (R)(L, Rf N ) = Hom D + (f 1 R)(f 1 L, N ), so that (f 1, Rf ) is an adjoint pair. Theorem: Poincaré-Verdier duality. Let A be a nice commutative ring. There exists a functor of triangulated categories, f! : D + (A Y ) D + (A X ), called the twisted or shrieck inverse image, such that Rf RHom AX (M, f! N ) = RHom AY (Rf! M, N ) Hom D + (A X )(M, f! N ) = Hom D + (A Y )(Rf! M, N ). Here A Y, A X denote the constant sheaf of A on X and Y respectively. We note that these relations are not defined on the level of sheaves but only on the derived categories. For c : X {pt}, we write ω X := c! (A ) {pt} D# (A X ) and call it the dualising complex. We also define D X (M ) := RHom AX (M, ω X ) and call it the Verdier dual. Definition 8. We say that a sheaf F on X is (algebraically) constructible if F Xα (F X an ) is a locally constant sheaf on X α α (Xα an ). Definition 9. We define D b c(x) to be the full subcategory of D b (Q X an) such that the cohomologies of the complexes are (algebraically) constructible. We have the following properties: D X D X = Id, when restricted to D b c(x). for f : X Y, we have f! ω Y = ω X, f! D Y = D X f 1, Rf D X = D Y Rf!. Let X be a complex analytic space or an algebraic variety. A stratification of X is X = α X α which is locally finite, with X α s smooth and X α is the union of some other X β s. The functors we saw work mostly fine on D b c(x). 6

7 3 t-structures Definition 10. Let D be a triangulated category, D 0, D 0 be full subcategories of D. We say that (D 0, D 0 ) is a t-structure if: D 1 D 0 and D 1 D 0, If X D 0 and Y D 1 then Hom(X, Y ) = 0, For all X D there is a triangle: with X 0 D 0, X 1 D 1, X 0 X X 1 +1, where D n = D 0 [ n] and D n = D 0 [ n] for all n Z. Example: Standard t-structure on D(C ): D 0 (C ) = {X D(C ) H j (X) = 0 for j < 0}, D 0 (C ) = {X D(C ) H j (X) = 0 for j > 0}. We note that here C = D 0 D 0, and in general, we call D 0 D 0 the heart of the t-structure. Hence using a different t-structure gives us a different heart. Proposition 1. Let i : D n D be the inclusion.then there exists a (truncation) functor, τ n, such that for all Y D n and X D, we have So (i, τ n ) is an adjoint pair. Hom D n(y, τ n X) = Hom D (i(y ), X). Proposition 2. Let i : D n D be the inclusion.then there exists a (truncation) functor, τ n, such that for all Y D n and X D, we have So (τ n, i) is an adjoint pair. Hom D n(τ n X, Y ) = Hom D (X, i(y )). Proposition 3. The heart is an abelian category. 7

8 Example: 1 d 1 Let s look at X X 0 d0 X 1. Then, Similarly, we define: τ 0 (X ) = 0 Imd 1 X 0 d 0 X 1 d 1. τ 0 (X ) = X 1 d 1 X 0 d 0 Coker(d 1 ) 0. From the third axiom, there is a triangle: τ n X X τ n+1 X +1. Here, we see that H n (X ) = τ 0 τ 0 X[n]. In general, we define cohomological functors to be H n (X ) := τ 0 τ 0 X[n], for different t-structures. So in general, if D = D(C ), then it is not necessarily true that H n (X ) = H n (X ), where the right hand side is the cohomology of X. Suppose we have a functor F : D 1 D 2 and t-structures (C 1, D 0 1, D 0 1 ) and (C 2, D 0 2, D 0 2 ), where the C i denote their respective hearts. Then we have p F : C 1 i D 1 F D 2 H0 C 2. (1) Definition 11. F is left t-exact if F (D 0 1 ) D 0 2 ; it is right t-exact if F (D 0 1 ) D 0 2. Proposition 4. If F is left (respectively right) t-exact then p F is a left (respectively right) exact functor of abelian categories. Proposition 5. If D i D i, F : D 1 D 2, G : D 2 D 1. With (F, G) an adjoint pair, then: If F (D 1 ) D 2 and F (D 0 1 ) D d 2, then G(D 0 2 ) D d 1 (whenever G takes an object of D 0 2 to D 1 ). If G(D 0 2 ) D d 1, then F (D 0 1 ) D d 2. 4 Perverse sheaves Inside D b c(x), we define subcategories: p Dc 0 (X) : F p D 0 c (X) if dim Supp H j (F ) j, j Z. 8

9 p Dc 0 (X) : F p D 0 c (X) if dim Supp H j (D X F ) j, j Z. Perv(Q X ) := p D 0 c (X) p D 0 (X). c Proposition 6. Take F Dc(X) b and X = α X α a stratification such that the X α s are connected and i 1 X α F and i! X α F have locally constant cohomology sheaves for all α, where i Xα : X α X is the inclusion map. Then, 1. F p D 0 c (X) iff H j (i 1 X α F ) = 0, α, j > dim X α. 2. F p D 0 c (X) iff H j (i! X α F ) = 0, α, j < dim X α. Proof. We prove 1. We have dim Supp H j (F ) j sup α {dim(x α supph j (F ))} j X α Supp H j (F ) =, whenever dim X α > j H j (F Xα ) = 0, whenever dim X α > j. It can be shown that ( p D 0 c (X), p D 0 c (X)) is a t-structure, and hence the associated perverse cohomology groups: p H n (A ) = p τ 0 p τ 0 (A [n]). We can define, as in (1), the perversifications p Rf, p Rf!, p f 1, and ( p f 1, p Rf ), ( p Rf!, p f! ) are adjoint pairs. Proposition 7. Let f : Y X be a continuous function with dim(f 1 (x)) d for x X. Then: 1. If F p D 0 c (X) then f 1 F p D d c (Y ). 2. If F p D 0 c (X) then f! F p D d c (Y ). Proof. 1. Let F p D 0 c (X). We have dim SuppH j (f 1 F [d])) = dim(f 1 (SuppH j+d (F )) dim SuppH j+d (F ) + d j d + d = j. So f 1 F [d] p D 0 c (Y ), i.e. f 1 F p D d c (Y ). 2. f! = D Y f 1 D X. If i : Z X is locally closed then Ri ( p D 0 (Z)) p D 0 (X); and Ri! ( p D 0 (Z)) p D 0 (X). 9

10 If j : U X is open then j 1 = j! is t-exact. If i : Z X is closed then i! = i. For instance, if we write Perv Z (Q X ) for the perverse sheaves on X supported on Z, then we have: Perv Z (Q X ) p i 1 = p i! Perv(Q Z ); Perv(Q Z ) p i = p i! Perv Z (Q X ). Intermediate extension: We write i, j as above where Z = X \ U. We have j! F j F. Then we have passing through p H 0 : p j! F p j F. p j! F Then p j! F is called the intermediate extension. We have the following: D X p j! F = p j! D U F, Given mild conditions on U U X, the extension is transitive. Characterisations: The following are equivalent: for F Perv (Q U ), 1. G = p j! F, 2. G Perv(Q X ) satisfies G U = F, i 1 G p D 1 c (Z), i! G p D 1 c (Z). 3. G Perv(Q X ) and G U = F and G has neither subobjects nor quotients supported on Z. If F Perv(Q U ) is simple then p j! F is simple in Perv(Q X ). We also have that p j! preserves monomorphisms and epimorphisms, but is not exact. If U X is open and smooth, let L Loc(U), local system on U; in this case L[d U ] Perv(Q U ), where d U := dim U. If L is irreducible, then IC(X, L) := p j! (L[d U ]) is simple in Perv(Q X ). If Z X is closed, Z 0 Z open, then for L Loc(Z 0 ), i Z ( p j Z0! L[d Z ]) = (i Z ) IC(Z, L). The structure theorem says that simple objects in the category Perv(C X ) are all of this form. 10

11 Definition 12. If L is an irreducible local system on Z 0 Z, closed, then IC(Z, L) is simple, so i Z IC(Z, L) is a simple perverse sheaf on X. We call these sheaves Deligne-Goreski-MacPherson complexes (DGM-complexes). Theorem: Structure theorem for perverse sheaves (BBD). Let X be a variety. We have: Perv(Q X ) is a full subcategory of D b c(x) which is abelian, stable by extensions and Verdier duality. The simple objects are DGM-complexes. All objects are finite successive extensions of simple objects (Perv(Q X ) is artinian and noetherian.) References [1] Hotta, Takeuchi and Tanisaki: D-Modules, Perverse Sheaves, and Representation Theory [2] De Cataldo and Migliorni: The decomposition theorem, perverse sheaves and the topology of algebraic maps 11

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