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1. Introduction 1 2. T-Structures on Triangulated Categories 1 3

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1. Introduction 1 2. T-Structures on Triangulated Categories 1 3 PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a stratifiable space. I give the definition of t structures, describe the simple perverse sheaves and examine when the 6 functors on the constructibe derived category preserve the subcategory of perverse sheaves. The main reference for this talk is [HTT]. Contents 1. Introduction 1 2. t-Structures on Triangulated Categories 1 3. Perverse t-Structure 6 4. Properties of the Category of Perverse Sheaves 10 4.1. Minimal Extensions 10 References 14 1. Introduction b Let X be a complex algebraic vareity and Dc(X) be the constructible derived category of sheaves on b X. The category of perverse sheaves P (X) is defined as the full subcategory of Dc(X) consisting of objects F ∗ that satifsy two conditions: 1. Support condition: dim supp(Hj(F ∗)) ≤ −j, for all j 2 Z. 2. Cosupport condition: dim supp(Hj(DF ∗)) ≤ −j, for all j 2 Z. Here, D denotes the Verdier duality functor. If F ∗ satisfies the support condition, we say that F ∗ 2 pD≤0(X) and if it satisfies the cosupport condition, we say that F ∗ 2 pD≥0(X): These conditions b actually imply that P (X) is an abelian category sitting inside Dc(X) but to see this, we first need to talk about t-structures on triangulated categories. 2. t-Structures on Triangulated Categories Let me begin by recalling (part of) the axioms of a triangulated category. The only examples we will really be considering will be derived categories of abelian categories so we could get away with not defining the abstract notion of a triangulated category but sometimes, it's easier to prove things by removing unnecessary properties. Let C be an additive category equipped with an automorphism T called the translation functor. By a triangle in C, we mean a sequence of morphisms F ! G ! H ! T F: Definition 2.1. A triangulated category is a triple (C; T; T ) where C; T is a pair of an additive category and a translation functor as above and T is a subset of the set of all triangles in T (called the set of distinguished triangles in C) such that the following hold: 1 (TR0) Any triangle isomorphic to a distinguished triangle is distinguished (where isomorphism of triangles means a commutative diagram with the top and bottom rows triangles and the vertical maps isomorphisms). (TR1) For any F 2 C, id F F 0 TF is distinguished. (TR2) Any morphism f : F ! G in C can be embedded into a distingusihed triangle F ! G ! H ! T F: (TR3) F ! G ! H ! TF is distinguished if and only if G ! H ! TF ! TG is dinstinguished, where if f is the morphism from F to G, then −T f is the morphism from TF to TG. (TR4) Given two dinstinguished triangles f1 F1 G1 H1 TF1 and f2 F2 G2 H2 TF2 and a commutative diagram f1 F1 G1 f2 F2 G2 we can embed them into a morphism of triangles f1 F1 G1 H1 TF1 f2 F2 G2 H2 TF2 (TR5) Octahedron axiom (see [HTT, p. 337]) Suppose A is now an abelian category and let C be the derived category of A, with T the shift functor F ! F [1]. Recall the mapping cone construction: given a morphism of complexes f : F ∗ ! G∗, define Cf as the complex n n+1 n Cf = F ⊕ G with differential given by the matrix d 0 F : f dG 2 This has an obvious map to TF given by projection onto the left factor and an obvious map from G given by inclusion of the right factor. We set T as the set of triangles isomorphic to mapping cone triangles. Proposition 2.2. The triple (C; T; T ) as above is a triangulated category. Proof. Axioms (TR0), (TR2) and (TR3) are obvious by construction. Axiom (TR1) follows from the long exact sequence of cohomology associated to the mapping cone construction. We omit the check of the octahedral axiom because it is tedious. We just need to check axiom (TR5) where the two triangles are mapping cone triangles. Given a commutative diagram f1 F1 G1 φ f2 F2 G2 n n+1 n n+1 n n+1 n we define a map ρ : Cf1 ! Cf2 by defining ρ : F ⊕ G1 ! F2 ⊕ G2 as φ ⊕ . Checking the resulting diagram commutes is now an easy exercise. So, if we view triangulated categories as generalizations of derived categories, then distinguished triangles are generalizations of mapping cone triangles. Since, cohomology takes mapping cone triangles to long exact sequences, we have the following definition. Definition 2.3. If A is an abelian category and C is triangulated, then a functor F : C ! A is cohomological if it converts distinguished triangles into long exact sequences (note the distinguished triangles can be viewed as a doubly infinite sequence of morphisms). Example 2.4. The functors HomC (F; −) and HomC (−;F ) are cohomological. Remark. The translation functor T is a generalization of the shift functors in derived categories. Hence, we will use the notation F [n] to denote T n(F ): As we have seen above, we have a way to get a triangulated category from an abelian category, by taking the corresponding derived category. A t-structure allows you to make the reverse construction i.e. it gives you an abelian category from a triangulated category. The way in which this works mimics the reconstruction of the abelian category from its derived category as the complexes concentrated in degree 0. Definition 2.5. Let (C; T; T ) be a triangulated category. A t-structure on C is the data of two full subcategories C≤0 and C≥0 that satifsfy the following properties: (T1) If C≤n = C≤0[−n] and C≥n = C≥0[−n], then C≤−1 ⊆ C≤0;C≥1 ⊆ C≥0: ≤0 ≥1 (T2) For any F 2 C ;G 2 C , HomC (F; G) = 0: (T3) For any F 2 C, there exists a distinguished triangle F0 ! F ! F1 ! F0[1] ≤0 ≥1 such that F0 2 C and F1 2 C . Definition 2.6. Given a triangulated category C with t-structure, the heart of the t-structure is the full subcategory C≤0 \ C≥0: 3 If C is the derived category of A, then we get a t-structure on C by setting C≤0 to be the complexes concentrated in degrees less than or equal to 0 (and similarly for C≥0). Axioms (T1) and (T3) are obvious and (T2) follows by taking truncations. The heart of this t-structure is the original abelian category A. It turns out that the heart of a t-structure on a triangulated category is always an abelian category. But to see this, we need to do some more work. We first construct truncation functors that generalize the truncation functors in the derived category setting. These will be constructed as adjoints to inclusion functors. Proposition 2.7. Denote by i : C≤n ! C and i0 : C≥n ! C the inclusions of the full subcategories. Then, there exist functors τ ≤n : C ! C≤n and τ ≥n : C ! C≥n such that for any Y 2 C≤n, Y 0 2 C≥n and any X 2 C, we have isomorphisms ≤n HomC≤n (Y; τ X) ! HomC (i(Y );X) and ≥n 0 HomC≥n (τ X; Y ) ! HomC (X; i (Y )): Proof. (see [HTT, Prop 8.14]) Let me do the proof for ≤ n. The proof for ≥ n is analogous. We can assume n = 0 by using the translation functor. It suffices by the Yoneda lemma, to prove that for each X 2 C, there exists some Z 2 C≤0 such that for Y 2 C≤n HomC (Y; Z) = HomC (Y; X): To construct this Z for fixed X, we take X0 as in (T3). Since HomC (Y; −) is cohomological, applying it to the distinguished triangle X0 ! X ! X1 ! X0[1] gives us an exact sequence HomC (Y; X1[−1]) ! HomC (Y; X0) ! HomC (Y; X) ! HomC (Y; X1): ≥1 But, X1;X1[−1] 2 C and so applying (T2) gives the desired result. ≤0 Remark. The proof shows that for X 2 C, the X0;X1 in (T3) are functorial because they are τ X and τ ≥1X respectively. We now prove some properties of the truncation functors that are simple consquences of the construction and the fact that Hom is a cohomological functor. Proposition 2.8. The canonical morphism from τ ≤nX ! X is embedded into a distinguished triangle τ ≤nX ! X ! τ ≥n+1X Proposition 2.9. The following conditions on X 2 C are equivalent: (1) X 2 C≤n (resp. C≥n) (2) The canonical map τ ≤nX ! X (resp. X ! τ ≥nX) is an isomorphism. (3) τ ≥n+1X = 0 (resp. τ ≤n−1X = 0:) Lemma 2.10. Let X0 ! X ! X00 be a distinguished triangle in C. If X0;X00 2 C≤0 (or C≥0), then so does X. Proposition 2.11. Let a; b 2 Z. (i) If b ≥ a, then τ ≤b ◦ τ ≤a =∼ τ ≤a ◦ τ ≤b =∼ τ a and τ ≥b ◦ τ ≥a =∼ τ ≥a ◦ τ ≥b =∼ τ b: 4 (ii) If a > b, then τ ≤b ◦ τ ≥a =∼ τ ≥a ◦ τ ≤b =∼ 0: (iii) τ ≥a ◦ τ ≤b =∼ τ ≤b ◦ τ ≥a: I leave the proof of this proposition to be looked up in [HTT, Prop 8.18].
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