Agnieszka Bodzenta

Home , Derived category, Derived functor, Exact functor, Exceptional object, Puppe sequence, Quotient category

June 12, 2019 HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY

AGNIESZKA BODZENTA

Contents

1. Categories, functors, natural transformations 2 1.1. Direct product, coproduct, fiber and cofiber product 4 1.2. Adjoint functors 5 1.3. Limits and colimits 5 1.4. Localisation in categories 5 2. Abelian categories 8 2.1. Additive and abelian categories 8 2.2. The category of modules over a quiver 9 2.3. Cohomology of a complex 9 2.4. Left and right exact functors 10 2.5. The category of sheaves 10 2.6. The long exact sequence of Ext-groups 11 2.7. Exact categories 13 2.8. Serre subcategory and quotient 14 3. Triangulated categories 16 3.1. Stable category of an exact category with enough injectives 16 3.2. Triangulated categories 22 3.3. Localization of triangulated categories 25 3.4. Derived category as a quotient by acyclic complexes 28 4. t-structures 30 4.1. The motivating example 30 4.2. Definition and first properties 34 4.3. Semi-orthogonal decompositions and recollements 40 4.4. Gluing of t-structures 42 4.5. Intermediate extension 43 5. Perverse sheaves 44 5.1. Derived functors 44 5.2. The six functors formalism 46 5.3. Recollement for a closed subset 50 1 2 AGNIESZKA BODZENTA

5.4. Perverse sheaves 52 5.5. Gluing of perverse sheaves 56 5.6. Perverse sheaves on hyperplane arrangements 59 6. Derived categories of coherent sheaves 60 6.1. Crash course on spectral sequences 60 6.2. Preliminaries 61 6.3. Hom and Hom 64 6.4. Serre duality 66 6.5. Derived functors in algebraic geometry 66 6.6. Grothendieck-Verdier duality 72 6.7. Spanning classes in the derived category 73 7. Full exceptional collections 79 7.1. Beilinson’s result 80 7.2. Equivalence of categories 80 7.3. Braid group action 81 7.4. Glued t-structure 82 7.5. Full exceptional collections on P2 and Markov numbers 83 7.6. Full exceptional collections on homogeneous spaces and toric varieties 83 7.7. Derived category under blow-up and projective bundles 84 8. Modern approach 90 8.1. The stable category of spectra 90 8.2. DG categories and DG enhancements 91 8.3. Infinity categories 92 8.4. Stable infinity categories 94 8.5. Examples of stable infinity categories 96 8.6. Derived algebraic geometry 98 8.7. Why do we care about derived algebraic geometry 99 References 100

1. Categories, functors, natural transformations

A category C is the data of a class of objects Ob(C) and a family of morphisms Mor(C).

Every morphism has a source and a target, we denote by HomC(C1,C2) the collection of objects whose source is C1 and target is C2. We assume that there exists an associative HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 3 composition HomC(C2,C3) × HomC(C1,C2) → HomC(C1,C3) and that every object has 0 00 the identity IdC ∈ HomC(C,C) such that for any f ∈ HomC(C,C ), g ∈ HomC(C ,C), f ◦ IdC = f and IdC ◦g = g. A subcategory D ⊂ C is a category such that Ob(D) ⊂ Ob(C) and Mor(D) ⊂ Mor(C).

A morphism f ∈ HomC(C1,C2) is an isomorphism if there exists g ∈ HomC(C2,C1) such that f ◦ g = IdC2 and g ◦ f = IdC1 . op The opposite category C has the same objects as C while HomCop (C1,C2) =

HomC(C2,C1). Examples of categories include the categories of Sets, (pointed) topological spaces, (abelian) groups, the category ∆:

Objects of ∆ are [n], for n = 0, 1,.... Hom∆([m], [n]) is the set of nonincreasing mappings from {0, . . . , m} to {0, . . . , n}. A (covariant) functor F : C → D is the data of a mapping ObC → ObD, C 7→ F (C) and a mapping Mor(C) → Mor(D), ϕ 7→ F (ϕ) such that F (ψϕ) = F (ψ)F (ϕ) and F (IdC ) = op IdF (C). A contravariant functor C → D is a functor C → D.

A natural transformation η : F → G of functors C → D is the data of ηC ∈

HomD(F (C),G(C)), for any C ∈ Ob(C). Morphisms ηC induce commutative diagrams

ηC2 F (C2) / G(C2) O O F (ϕ) G(ϕ)

ηC1 F (C1) / G(C1) for any ϕ ∈ HomC(C1,C2). We say that a functor F : C → D is faithful if the map F : Mor(C) → Mor(D) is injective. F is full if the map is surjective. It is essentially surjective if every object in D is isomorphic to F (C), for some C ∈ Ob(C). A subcategory D ⊂ C is full if the embedding functor D → C is fully faithful. Functor F : C → D is an equivalence if there exists G: D → C and natural transformations η : IdC → G ◦ F , ν : IdD → F ◦ G such that ηC and νD are isomorphisms, for all C ∈ Ob(C), D ∈ Ob(D).

Exercise 1.1. Show that a functor F : C → D is an equivalence if and only if it is fully faithful and essentially surjective.

Examples of functors

C op C • Any object C ∈ C deﬁnes functors h : C → Set and hC : C → Set via h (C1) = C HomC(C,C1), hC (C1) = HomC(C1,C). Functors h , hC are representable. 4 AGNIESZKA BODZENTA

• Let ∆op Set be the category of simplicial sets. Its objects of ∆op Set are functors ∆op → Set. Morphisms are natural transformations of functors. The geometric realisation is a functor | − |: ∆op Set → Top. To a simplicial F∞ set X = {Xn = X([n])} it assigns |X| = n=0(∆n × Xn)/R where ∆n is the geometrical n-dimensional simplex

n n+1 X ∆n = {(x0, . . . , xn) ∈ R | xi = 1, xi ≥ 0} i=0

and the equivalence relation R is deﬁned as follows: (s, x) ∈ ∆n × Xn is identiﬁed

with (t, y) ∈ ∆m × Xm if there exists f ∈ Hom∆([m], [n]) with y = X(f)x and F∞ s = ∆f t. The topology on |X| is the weakest for which n=0 Xn × ∆n → |X| is

continuous. The map ∆f : ∆m → ∆n is the unique linear mapping which sends

vertex ei ∈ ∆m to ef(i) ∈ ∆n. • Another example of a functor is the Singular simplicial set Sing: Top → ∆op Set.

For a topological space Y , Sing(Y )(n) is the set of continuous maps ∆n → Y For

f ∈ Hom∆([n], [m]) the map Sing(Y )(f) maps ϕ: ∆m → Y to ϕ ◦ ∆f : ∆n → Y . • A presheaf of sets is a functor (TopY )op → Set. Here, Y is a topological space and TopY is the category whose objects are open subsets of Y and morphisms are inclusions U → V .

1.1. Direct product, coproduct, ﬁber and coﬁber product.

Let X, Y be objects of a category C. The direct product X × Y is the object Z representing the functor

C 7→ hX (C) × hY (C)

The direct sum X ⊕ Y is the object Z representing the functor

C 7→ hX (C) ∪ hY (C).

Let S be an object of C. Deﬁne category CS whose objects are pairs (C, ϕ) of objects of C and morphisms ϕ: C → S. Morphisms CS (C1, ϕ1) → (C2, ϕ2) in CS are such f ∈ HomC(C1,C2) that ϕ2 ◦ f = ϕ1.

Let now X, Y be objects of CS, i.e. assume ﬁxed ϕ: X → S, ψ : Y → S. The ﬁber product X ×S Y of X and Y over S is the direct product of (X, ϕ), (Y, ψ) in CS considered as an object of C.

Exercise 1.2. Write down the universal property of a ﬁber product. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 5

Given morphisms αX : S → X, αY : S → Y in a category C the coﬁber product X tS Y of X and Y along S is an object Z ∈ C together with βX : X → Z, βY : Y → Z satisfying the following universal property: given C in C and ϕX : X → C, ϕY : Y → C such that

ϕX ◦αX = ϕY ◦αY , there exists unique ϕ: Z → C such that ϕ◦βX = ϕX and ϕ◦βY = ϕY .

Exercise 1.3. Present coﬁber product as a coproduct in an appropriate category.

1.2. Adjoint functors.

Let C, D be categories and F : C → D, G: D → C functors. Functor F is left adjoint to G, F a G if there exist natural transformations ε: FG → IdD, η : IdC → GF , called the adjunction counit and unit, such that maps

HomC(C,G(D)) → HomD(F (C),D), ϕ 7→ εD ◦ F (ϕ),

HomD(F (C),D) → HomC(C,G(D)), ψ 7→ G(ψ) ◦ ηC are inverse to each other.

Exercise 1.4. Show that ε: FG → IdD, η : IdC → GF yield F a G if and only if the F η ηG compositions F −→ F GF −→εF F , G −→ GF G −→Gε G are the identity transformations.

1.3. Limits and colimits.

Consider a category I and a functor F : I → C. A cone to F is an object N of C together with ψi : N → F (i), for any i ∈ I, such that for every α ∈ HomI (i, j), F (α) ◦ ψi = ψj.A limit of F : I → J is a cone (L, ϕi) such that given any other cone (N, ψi) there exists a unique morphism u: N → L such that ϕi ◦ u = ψi.

A cocone to F is an object W of C together with ψi : F (i) → W , for any i ∈ I, such that for every α ∈ HomI (i, j), ψj ◦ F (α) = ψi.A colimit of F : I → J is a cocone (T, ϕi) such that given any other cocone (W, ψi) there exists a unique morphism u: T → W such that u ◦ ϕi = ψi.

1.4. Localisation in categories.

The reference for this section is [Sta13, section 4.26]. Let C be a category. A set of arrows S in C is called a left multiplicative system if it has the following properties LMS 1 The identity of every object of C is in S and the composition of two composable elements in S is in S; 6 AGNIESZKA BODZENTA

LMS2 Every solid diagram

g X / Y

t s f Z / W with t ∈ S can be completed to a commutative dotted square with s ∈ S; LMS3 For every f, g : X → Y and t ∈ S such that f ◦ t = g ◦ t there exists a s ∈ S such that s ◦ f = s ◦ g. A set of arrows S in C is called a right multiplicative system if it has the following properties RMS 1 The identity of every object of C is in S and the composition of two composable elements in S is in S; RMS2 Every solid diagram

g X / Y

t s f Z / W with s ∈ S can be completed to a commutative dotted square with t ∈ S; RMS3 For every f, g : X → Y and s ∈ S such thats ◦ f = s ◦ g there exists a t ∈ S such that f ◦ t = g ◦ t. A set of arrows is a multiplicative system if it is both left and right multiplicative system. Let C be a category and S a left multiplicative system. We deﬁne a new category S−1C of left fractions whose objects are objects of C. Morphisms X → Y in S−1C are 0 0 equivalences classes of pairs (f : X → Y , s: Y → Y ) with s ∈ S. Two pairs (f1 : X → 0 0 0 0 Y1 , s1 : Y → Y1 ), (f2 : X → Y2 , s2 : Y → Y2 ) are equivalent if there exists a third pair 0 0 (f3 : X → Y3 , s3 : Y → Y3 ) and morphisms u: Y1 → Y3, ν : Y2 → Y3 ﬁtting into the commutative diagram

Y1 ? _ f1 u s1 X f2 / Y3 o s2 Y O f3 ν s3 Y2

The composition of the equivalence classes of the pairs (f : X → Y 0, s: Y → Y 0) and (g : Y → Z0, t: Z → Z0) is deﬁned as the equivalence class of a pair (h ◦ f, u ◦ t) where h HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 7 and u ∈ S are chosen to ﬁt into a commutative diagram

Y g / Z0

s u 0 00 Y h / Z which exists by assumption. The identity morphism is the equivalence class of the pair (Id, Id).

Proposition 1.5. Let C be a category and S a left multiplicative system of morphisms of C. (1) The rules X 7→ X, (f : X → Y ) 7→ (f : X → Y, Id: Y → Y ) deﬁne a functor Q: C → S−1C which commutes with ﬁnite colimits. (2) For any s ∈ S the morphism Q(s) is an isomorphism in S−1C. (3) If G: C → D is a functor such that G(s) is invertible for every s ∈ S, then there exists a unique functor H : S−1C → D such that H ◦ Q = G.

Similarly, one deﬁnes the category S−1C for a right multiplicative system S of morphisms of C.

Lemma 1.6. Let C be category and S a multiplicative system. The category of left fractions and the category of right fractions are canonically isomorphic.

Proof. The universal property implies existence of mutually inverse functors Cleft → Cright and Cright → Cleft. We say that a multiplicative system S is saturated if it satisﬁes MS4 given three composable morphisms f, g, h, if fg ∈ S, gh ∈ S then g ∈ S.

Lemma 1.7. Let C be a category and S a left multiplicative system. Given any ﬁnite −1 0 collection gi : Xi → Y of morphisms of S C we can ﬁnd an element s: Y → Y of S and 0 a family of morphisms fi : Xi → Y such that each gi is the equivalence class of the pair

(fi, s).

Proof. Let (Xi → Yi, si : Y → Yi) be a representative of gi. The lemma follows if we can 0 0 ﬁnd s: Y → Y in S such that for each i there is ai : Yi → Y with ai ◦ si = s. If we have two indices i = 1, 2, we complete the square

Y s2 / Y2

s1 t2 0 Y1 a1 / Y 8 AGNIESZKA BODZENTA with t2 ∈ S. Then s = t2 ◦ s2 ∈ S works. If we have n > 2 morphisms, we use the above trick to reduce to the case of n − 1 and proceed by induction.

2. Abelian categories

2.1. Additive and abelian categories.

Category C is additive if

(A1) Each set HomC(C1,C2) is endowed with a structure of an abelian group, the composition of morphisms is bi-additive with respect to these structures,

(A2) There exists a zero object 0 ∈ Ob(C) such that HomC(0, 0) is the zero group.

(A3) For any pair of objects C1, C2 in C the direct sum and the direct product of X and Y exist and the canonical map X ⊕ Y → X × Y is an isomorphism.

Let k be a ﬁeld (or a commutative ring). Category C is k-linear if HomC(C1,C2) are endowed with a structure of a k-module and the composition factors via HomC(C2,C3)⊗k

HomC(C1,C2) → HomC(C1,C3). Consider category C which satisﬁes axioms A1 and A2 (it is sometimes called preadditive). Let α: X → Y be a morphism in C. The kernel of α is an object K ∈ C together with the map k : K → X such that, for any C ∈ C and any ϕ: C → X such that α ◦ ϕ = 0 there exists a unique ϕ: C → K such that k ◦ ϕ = ϕ. The cokernel of α is an object Q ∈ C together with the map c: Y → Q such that, for any C ∈ C and any ψ : Y → C such that ψ ◦ α = 0 there exists unique ψ : Q → C such that ψ ◦ c = ψ. We say that an additive category A is abelian if

(A4) for any morphism ϕ ∈ HomA(X,Y ) there exists a sequence

j K −→k X −→i I −→ Y −→c Q

such that – j ◦ i = ϕ, – (K, k) is the kernel of ϕ,(Q, c) is the cokernel of ϕ, – (I, i) is the cokernel of k and (I, j) is the kernel of c. Consider diagrams

βX S / X X

βY γX

γY Y Y / S HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 9 in an abelian category A. Then the cokernel of S → X ⊕ Y is the coﬁber product X tS Y and the kernel of X ⊕ Y → S is the ﬁber product X ×S Y .

2.2. The category of modules over a quiver.

A quiver Q consists of the set of vertices Q0 and arrows Q1 together with the source and target maps s, t: Q1 → Q0. Let k be a field. The path algebra k[Q] of a quiver Q is a k-algebra whose k-basis consists of paths in Q, i.e. sequences of arrows p = an . . . a1 such that s(ai+1) = t(ai) for i = 1, . . . , n − 1. The source of a path p is defined as the source of a1 and the target of p is t(an). Algebra k[Q] contains also length-zero paths ei at any vertex of i ∈ Q0. The composition in k[Q] is defined as ( p1p2 if s(p1) = t(p2) p1 ◦ p2 = 0 otherwise.

Let I be an ideal in k[Q] generated by a ﬁnitely many linear combinations of paths with the same source and target. We say that (Q, I) is a quiver with relations and k[Q]/I is the path algebra of (Q, I). Example of a quiver with relations is

(1) x • y / • z 7 g with x2 = 0, zy = y.

A right module over a quiver (Q, I) is a right module over the path algebra of (Q, I).

Let M be a right module over (Q, I) and let Mi = Mei. For an arrow a with s(a) = i and t(a) = j we have ejaei = a, hence a can be thought of as a map Mj → Mi.

The right module over the quiver (1) consists of V1, V2 and maps x: V1 → V1, y : V2 → V1 2 and z : V2 → V2 such that x = 0, y ◦ z = y.

2.3. Cohomology of a complex.

Let A be an abelian category. A (cohomological) complex over A is a sequence

i−1 i (2) ... → Ai−1 −−→d Ai −→d Ai+1 → ... of objects in A such that di ◦ di−1 = 0. Given a complex (2) the fact that di ◦di−1 = 0 implies that di−1 uniquely factors via the kernel Ki of di. The i-th cohomology Hi(A·) of (2) is the cokernel of the map Ai−1 → Ki. 10 AGNIESZKA BODZENTA

We say that a complex (2) is exact if Hi(A·) = 0 for all i. In particular, an exact complex

0 → A1 → A2 → A3 → 0 is called a short exact sequence.

i Exercise 2.1. Let 0 → A1 −→ A2 → A3 → 0 be a short exact sequence and let ϕ: B3 → A3 be a morphism. Let Q = A2 ×A3 B3 and let α: A1 → Q be induced by i: A1 → A2 and α 0: A1 → B3. Then 0 → A1 −→ Q → B3 → 0 is a short exact sequence in A.

i d Exercise 2.2. Let 0 → A1 −→ A2 −→ A3 → 0 be a short exact sequence and let ψ : A1 → B1 be a morphism. Let B2 = B1 tA1 A2 and θ : B2 → A3 be induced by d: A2 → A3, θ 0: B1 → A3. Then 0 → B1 → B2 −→ A3 → 0 is a short exact sequence in A.

2.4. Left and right exact functors.

Let now A, B be abelian categories. A functor F : A → B is left exact if for any exact sequence 0 → A1 → A2 → A3 in A the sequence 0 → F (A1) → F (A2) → F (A3) is exact in B. Functor F is right exact if for any exact sequence A1 → A2 → A3 → 0 in A the sequence F (A1) → F (A2) → F (A3) → 0 is exact in B. Functor F is exact if it is both left and right exact.

Exercise 2.3. Functor F : A → B is exact if and only if it maps any short exact sequence in A to a short exact sequence in B.

2.5. The category of sheaves.

Let X be a topological space. We deﬁne the category UX whose objects are open subsets U ⊂ X and morphism U → V correspond to inclusions U → V . op Let A be an abelian category. A presheaf on X with values in A is a functor UX → A.

Presheaves form a category PShA,X in which morphisms are natural transformations of functors. S A presheaf F is a sheaf if for any open U ⊂ X and any open covering U = Ui the diagram / F (U) / Q F (U ) Q F (U ∩ U ) i / i j is an equalizer.

Sheaves form a full subcategory of presheaves denoted ShA,X . HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 11

Assume that category A has colimits. Then the embedding functor ι: ShA,X → PShA,X has left adjoint s. For a presheaf F, s(F) = F ++, where F + is a presheaf whose value on an open U ⊂ X is deﬁne as F +(U) = lim H0(U, F). −→op JU op Here, JU is the category whose objects are open covers of U and morphisms are given op by inclusions of open covers. For U ∈ JU

0 Y H (U, F) = {(fi) ∈ F(Ui) | fi|Ui∩Uj = fj|Ui∩Uj for all pairs i, j}.

Functor ι is left exact, hence kernels of sheaves are kernels of presheaves. A cokernel of a morphism ϕ of sheaves is s(cokerϕ), where coker ϕ is the cokernel in the category of presheaves.

2.6. The long exact sequence of Ext-groups.

An object P ∈ A is projective if the functor hP is exact. An object I ∈ A is injective if hI is exact. 1 Given objects A1,A2 of an abelian category A the group ExtA(A2,A1) is the group whose elements are classes of extensions of A2 by A1, i.e. short exact sequences 0 →

A1 → A → A2 → 0 modulo isomorphisms

0 / A1 / A / A2 / 0 O O O Id ' Id

0 0 / A1 / A / A2 / 0

1 The zero element of ExtA(A2,A1) is the trivial extension 0 → A1 → A1 ⊕ A2 → A2 → 0.

Given ζ = {0 → A1 → A → A2 → 0}, ξ = {0 → A1 → B → A2 → 0} the sum ζ + ξ is the top row in the diagram

0 / A1 / C / A2 / 0 O O O ∆

0 / A1 ⊕ A1 / Q / A2 / 0

∆ 0 / A1 ⊕ A1 / A ⊕ B / A2 ⊕ A2 / 0

where ∆’s are the morphisms induces by Id: Ai → Ai, Q = A2 ×A2⊕A2 (A ⊕ B) and

C = A1 tA1⊕A1 Q. 12 AGNIESZKA BODZENTA

i d Exercise 2.4. A short exact sequence 0 → A1 −→ A −→ A2 → 0 deﬁnes a trivial element 1 in Ext (A2,A1) if there exists τ : A → A1 such that τ ◦ i = Id.

i d Exercise 2.5. A short exact sequence 0 → A1 −→ A −→ A2 → 0 deﬁnes a trivial element 1 in Ext (A2,A1) if there exists σ : A2 → A such that d ◦ σ = Id.

Proposition 2.6. Let d1 d0 ϕ P2 −→ P1 −→ P0 −→ A2 → 0 be an exact sequence in an abelian category A with P0, P1, P2 projective. Then the group 1 1 Ext (A2,A1) is isomorphic to the ﬁrst cohomology H of the complex

(3) 0 → Hom(P0,A1) → Hom(P1,A1) → Hom(P2,A1)

1 1 1 1 Proof. We deﬁne maps f : Ext (A2,A1) → H and g : H → Ext (A2,A1) and check that they are mutually inverse. i d 1 Let ζ = {0 → A1 −→ B −→ A2 → 0} be an element of Ext (A2,A1). As sequence

0 → Hom(P0,A1) → Hom(P0,B) → Hom(P0,A2) → 0 is exact, there exists α: P0 → B such that d ◦ α = ϕ. The map α is unique up to P0 → A1:

i d 0 / A1 / B / A2 / 0 O ` O O β α Id

d1 d0 ϕ P2 / P1 / P0 / A2 / 0

The composite d ◦ α ◦ d0 = ϕ ◦ d0 is zero, hence there exists map β from P1 to the kernel

A1 of d. The composite i ◦ β ◦ d1 = α ◦ d0 ◦ d1 = 0. As i is a monomorphism, it follows that β ◦ d1 = 0. We deﬁne f(ζ) as the class of β in the ﬁrst cohomology of the complex (3).

Let now γ : P1 → A1 be a morphism such that γ ◦ d1 : P2 → A1 is zero. Map γ factors via the cokernel of d1, i.e. via the kernel I of d0. Let C = A1 tI P0:

i d (4) 0 / A1 / C / A2 / 0 O O O γ µ Id ϕ 0 / I / P0 / A2 / 0

1 We deﬁne g(γ) as the class of {0 → A1 → C → A2 → 0} in Ext (A2,A1). To show that g is well-deﬁned we need to check that if γ = θ ◦ d0 for some θ : P0 → A1 then g(γ) = 0.

Assume γ = θ ◦ d0. Maps θ : P0 → A1, Id: A1 → A1 yield τ : C → A1 such that

τ ◦ i = IdA1 . It follows that A1 is a direct summand of C. As the quotient C/A1 is isomorphic to A2, we conclude that C ' A1 ⊕ A2. As diagram (4) commutes, we have fg(γ) = γ. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 13

i d 1 Let ζ = {0 → A1 −→ B −→ A2 → 0} be a class in Ext (A2,A1). Element gf(ζ) is the ι δ extension 0 → A1 −→ C −→ A2 → 0 where C is the cofiber product A1 tI P0. Morphisms i: A1 → B, α: P0 → B define ξ : C → B. By definition of ξ, we have ξ ◦ ι = i. The composite d ◦ ξ ◦ µ equals d ◦ α = ϕ, hence the diagram

i d 0 / A1 / B / A2 / 0 O O O Id ξ Id ι δ 0 / A1 / C / A2 / 0 O O O β µ Id ϕ 0 / I / P0 / A2 / 0 commutes. It follows by the ﬁve lemma that ξ is an isomorphism, i.e. ζ = gf(ζ). One can also consider an exact sequence

0 → A1 → I0 → I1 → I2

1 with injective I0, I1, I2. Then Ext (A2,A1) is isomorphic to the ﬁrst cohomology of the complex

0 → Hom(A2,I0) → Hom(A2,I1) → Hom(A2,I2)

2.7. Exact categories.

An exact category is an additive category E together with a ﬁxed class S of conﬂations, i.e. pairs of composable morphisms

(5) X −→i Y −→d Z such that i is the kernel of d and d is the cokernel of i. We shall say that i is an inflation and d a deflation. The class S is closed under isomorphisms and the pair (E, S) is to satisfy the following axioms: (Ex 0)0 → X −−→IdX X is a conflation, (Ex 1) the composite of two deflations is a deflation, (Ex 2) the pullback of a deflation against an arbitrary morphism exists and is a deflation, (Ex 2’) the pushout of an inflation along an arbitrary morphism exists and is an inflation. Exercise 2.7. Let A be an abelian category and E ⊂ A a full subcategory closed under extensions, i.e. given a short exact sequence 0 → E1 → A → E2 → 0 in A with E1,

E2 ∈ E, object A also belongs to E. Prove that E with conﬂations deﬁned as exact sequences in A whose all terms lie in E is an exact category. 14 AGNIESZKA BODZENTA

2.8. Serre subcategory and quotient.

The reference for this section is [Sta13, Sections 12.8 and 12.9].

Lemma 2.8. Let C be a preadditive category and S a left or right multiplicative system. Then there exists a canonical additive structure on S−1C such that the localisation functor Q: C → S−1C is additive.

Proof. Let α, β : X → Y be morphisms in S−1C. Lemma 1.7 implies that there exists s ∈ S such that α is the equivalence class of a pair (f, s) and β is the equivalence class of a pair (g, s). Then α + β is deﬁned as the equivalence class of (f + g, s). Functor Q commutes with ﬁnite (co)limits, hence S−1C has a zero object and direct sums.

Lemma 2.9. Let C be an additive category and S a multiplicative system. Let X be an object of C. The following are equivalent (1) Q(X) = 0 in S−1C; (2) there exists Y ∈ Ob(C) such that 0: X → Y is an element of S; and (3) there exists Z ∈ Ob(C) such that 0: Z → X is an element of S.

Proof. If (2) holds then 0 = Q(0): Q(X) → Q(Y ) is an isomorphism. As S−1C is additive, Q(X) = 0, i.e. (2) ⇒ (1). Similarly, (3) ⇒ (1). Suppose that Q(X) = 0. Then f : X → 0 is transformed into an isomorphism in S−1C. −1 −1 −1 Let s g = ht be the inverse morphism. IdX = s gf means there exists a commutative diagram X0 > ` gf u s X f 0 / YXo s0 O Id ν Id ~ X Hence ugf = f 0 = ν = s0 ∈ S and ug : 0 → Y is a morphism such that X → 0 → Y is in S. It proves (1) ⇒ (2). The implication (1) ⇒ (3) is proved analogously.

Proposition 2.10. Let A be an abelian category. (1) If S is a left multiplicative system then the category S−1A has cokernels and the functor Q: A → S−1A commutes with them. (2) If S is a right multiplicative system then the category S−1A has kernels and the functor Q: A → S−1A commutes with them. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 15

(3) If S is a multiplicative system then S−1A is abelian and Q: A → S−1A is exact.

Proof. Assume S is a left multiplicative system and let a: X → Y be a morphism in S−1A. Then a = (f : X → Y 0, s: Y → Y 0). Since Q(s) is an isomorphism, existence of the cokernel of a is equivalent to the existence of the cokernel of Q(f). Since Q commutes with ﬁnite colimits, Q(coker(f)) is the sought cokernel. The proof of (2) is similar. By (1) and (2) we know that S−1A has kernels and cokernels. It remains to check that Coim ' Im. As both are calculated in A, the isomorphism follows from the isomorphism in A.

A full subcategory X of an abelian category A is a Serre subcategory if given a short exact sequence 0 → A1 → A2 → A3 → 0 in A object A2 lies in X if and only if A1 and

A3 do. We deﬁne S = {f ∈ Mor(A) | ker(f), coker(f) ∈ X }.

Lemma 2.11. S is a multiplicative system.

Proof. Clearly IdX ∈ S. Let now f, g be composable morphisms in S. Exact sequences

0 → ker(f) → ker(gf) → ker(g), coker(f) → coker(gf) → coker(g) → 0

prove that gf ∈ S. Consider a solid diagram g A / B

t s f C / C tA B with t ∈ S. Then ker(t) → ker(s) is surjective and coker(t) → coker(s) is an isomorphism. This proves LMS2 and the proof of RMS2 is dual. Finally, consider f, g : B → C and s: A → B such that fs = gs, i.e. (f − g)s = 0. Then I = Im(f − g) is the quotient of cokers. Hence, t: C → C/I is an element of S and we have t(f − g) = 0. The proof of RMS3 is dual.

Exercise 2.12. Prove that sequence coker(f) → coker(gf) → coker(g) → 0 is exact. 16 AGNIESZKA BODZENTA

Let X be a Serre subcategory in an abelian category A. We say that X is localising if the quotient functor Q: A → A/X has a right adjoint Q!. Then functor Q! is fully faithful and

! 1 Q A/X = {A ∈ A | ∀X ∈ X , HomA(X,A) = 0 = Ext (X,A)} is the category of X -closed objects in A, [Gab62].

3. Triangulated categories

We introduce derived categories following the exposition in [Kel96]. I’ve also used [Hap87] and https://sites.math.washington.edu/~julia/teaching/581D_ Fall2012/StableFrobIsTriang.pdf

3.1. Stable category of an exact category with enough injectives. Let A be an additive category. Let C(A) be the category of complexes

n ... → An −→d An+1 → ...

· · i i i i+1 i i+1 i with morphisms A → B given by families f ∈ HomA(A ,B ) such that d f = f d .

Exercise 3.1. Show that category C(A) with conﬂations deﬁned as (i·, p·) such that in, pn is a split short exact sequence over A (i.e. An → An ⊕ Bn → Bn) is an exact category.

Let E be an exact category. An object I ∈ E is injective if for any conﬂation E1 → E →

E2 the sequence 0 → Hom(E2,I) → Hom(E,I) → Hom(E1,I) → 0 is exact. Similarly,

P ∈ E is projective, if for any conlfation E1 → E → E2 the sequence 0 → Hom(P,E1) →

Hom(P,E) → Hom(P,E2) → 0 is exact. We say that an exact category E has enough injectives if any object E ∈ E ﬁts into a conﬂation E → I → E0 with I injective.

Proposition 3.2. Let A be an additive category. Complex A· in C(A) is injective if and only if it is homotopic to zero, i.e. if there exists (hi : Ai → Ai−1) such that di ◦ hi + hi+1 ◦ i+1 d = IdAi for all i. The class of injectives and projectives objects in C(A) coincides and C(A) has enough injectives and projectives.

Proof. Let A· be any complex in C(A) and consider (IA)· defined as (IA)n = An ⊕ An+1 ! 0 1 with differential . 0 0 A morphism B· → IA· is given by ϕn : Bn → An. Indeed, such a collection defines a · · n n+1 n morphism of complexes ψ : B → IA , ψ = (ϕn, ϕ d ). HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 17

We show that IA· is injective. Any inflation in C(A) is of the form ιn : Bn → Bn ⊕ Cn ! d α where the differential on Bn ⊕Cn → Bn+1 → Cn+1 is B (then ι· is a morphism 0 dC · · n n n of complexes). Given ψ : B → IA corresponding to ϕ : B → A , morphism ψe:(B ⊕ · · C) → IA corresponding to (ψ, 0) is such that ψe ◦ ι = ψ. The map A· → IA· corresponding to Id: An → An is an inflation, hence C(A) has enough injectives. · · · n n+1 n n+1 n If A is itself injective, A → IA splits, i.e. there exist (dAh , h ): A ⊕ A → A n+1 · · such that dAhn + h dA = IdA (any morphism IA → B is determined by a family βn+1 : An+1 → Bn). Then hn is a homotopy, i.e. A· is null-homotopic. Analogous argument shows that IA· is projective, and that and projective complex is null-homotopic.

Let E be an exact category with enough injectives. We write SE for the cokernel of the inﬂation E → I(E).

Exercise 3.3. Describe SA for a complex A ∈ C(A) of objects of an additive category A.

Let E be an exact category with enough injective objects. The stable category E has the same objects as E and morphisms in E are morphisms in E modulo the subgroup of morphisms factoring through an injective object of E. The composition in E is induced from the composition in E. The main example Let A be an additive category. The homotopy category H(A) of A is the stable category of C(A). Often, the category H(A) is denoted by K(A).

Exercise 3.4. Show that morphisms in H(A) are morphisms of complexes modulo the homotopy relation.

Lemma 3.5. Let E and E0 be objects of an exact category E with enough injectives. If E ⊕ I ' E0 ⊕ I0 for some injective objects I,I0 ∈ E then E ' E0 in E. ! ! α β α0 β0 Proof. Let be an isomorphism E ⊕I → E0 ⊕I0 with inverse . Then γ δ γ0 δ0 0 0 0 0 0 α α + β γ = IdE. As β γ factors via injective I , we have α α = IdE in E.

Proposition 3.6. Let E be an object of an exact category E with enough projectives. Then the cokernel of an inﬂation E → I, for I ∈ E injective, is unique up to isomorphism in E. 18 AGNIESZKA BODZENTA

Proof. Consider E → I1 → C1, E → I2 → C2 and the pushout

E / I1 / C1

I2 / D / C1

C2 / C2

As I1 and I2 are injective, the middle row and column split, hence C1 ⊕ I2 ' C2 ⊕ I1. We conclude by above Lemma.

For E ∈ E we denote T (E) the cokernel of an inﬂation E → I with injective I. By the above Proposition T (E) ∈ E is well-deﬁned. We extend E 7→ T (E) to a functor T : E → E by mapping f : E → E0 in E to a morphism T (E) → T (E0) which makes the following diagram commutative:

i0 p0 E0 / I0 / T (E0) O O O f g h i p E / I / T (E)

Morphism g exists because I0 is injective and it determines h uniquely. Assume that g0 : I → I0 is another morphism such that g0 ◦i = i0 ◦f and let h0 be such that p0 ◦g0 = h0 ◦p. Then (g − g0)i = i0f − i0f = 0, hence there exits s: T (E) → I0 such that g − g0 = s ◦ p. Then (h − h0)p = p0(g − g0) = p0 ◦ s ◦ p, hence h − h0 = p0 ◦ s, as p is an epimorphism. It follows that morphism h is uniquely defined in E. Let f : A → B be a morphism in E and A → I → T (A) a conflation with injective I. Consider push-out of f along A → I and complete it to a conflation:

g h (6) B / C / T (A) O O O f '

A / I / T (A)

f g Then A −→ B −→ C −→h T (A) is a standard triangle in E. We deﬁne distinguished triangles in E as sequences A → B → C → T (A) isomorphic to standard triangles.

Theorem 3.7. Let E be an exact category with enough injectives. Then E is a suspended category, i.e. it satisﬁes: HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 19

(ST1) Every triangle isomorphic to a distinguished triangle is distinguished. Every morphism f : A → B can be embedded into a distinguished triangle A → B → C → T (A). For any object A triangle A −→Id A → 0 → T (A) is distinguished. f g g T (f) (ST2) If A −→ B −→ C −→h T (A) is a distinguished triangle then B −→ C −→h T (A) −−→ T (B) is distinguished. (ST3) Every solid diagram

f 0 g0 h0 A0 / B0 / C0 / T (A) O O O O ϕ ψ θ T (ϕ) f g h A / B / C / T (A)

can be completed to a dashed diagram (ST4) Given f : A → B, g : B → C there exists

T (f 0) T (B) / T (D) O O g00 j00 Id E / E O O g0 j0 h=gf h0 h00 A / C / F / T (A) O O O O Id g j Id f f 0 f 00 A / B / D / T (A)

with distinguished middle columns and middle rows.

Proof. A triangle isomorphic to a distinguished triangle is isomorphic to a standard triangle, hence it is distinguished. We have constructed a standard triangle for any f : A → B. From the commutative diagram

A / I(A) / T (A) O O O Id

A / I(A) / T (A) and the fact that I(A) ' 0 in E we conclude that A −→Id A → 0 → T (A) is distinguished. We have thus proved (TR1). To prove (TR2) we first note that any morphism in E is a class of an inflation. Indeed, let f : A → B be any map. Then diagram (6) implies that A → B ⊕ I(A) → C is a conflation. (We can present E as an extension-closed subcategory of an abelian category 20 AGNIESZKA BODZENTA

A in such a way that conﬂations in E are short exact sequences in A. Then it it enough to show that 0 → A → B ⊕ I(A) → C → 0 is a short exact sequence.) Then the composite A → B → I(B) is an inﬂation and we have a commutative diagram

A / I(B) / T (A) O O O Id f A / B / C with conﬂations in rows. Then

T (f) C / T (A) / T (B) O O O Id

B / I(B) / T (B) O O

Id A / A has conﬂations in rows and columns. It follows that B → C → T (A) → T (B) is distinguished. In particular, the composition of two consecutive morphism in a distinguished triangle is zero: A → B → C factors via I(A); for B → C → T (A) we can ’rotate’ the triangle. Consider a solid diagram:

v0 w0 B0 / C0 / T (A0) F O E O D O u0 x0 Id p0 0 0 0 A i0 / I(A ) / T (A ) F E D g h T (f) v B / C w / T (A) O O O f I(f) T (f) u x Id i p A / I(A) / T (A) which commutes in E, i.e. u0f − gu factors via an injective object. We can assume that there exists a: I(A) → B0 (not in the picture) such that gu − u0f = ai. Morphism f induces I(f) (non-unique) and T (f) (unique in E). Object C is a pushout, hence x0I(f)+v0a: I(A) → C0 and v0g : B → C0 ( they agree on A: (x0I(f) + v0a)i = x0i0f + v0gu − v0u0f = v0u0f − v0gu − v0u0f = v0gu) define h: C → C0. To check that it defines a morphism of triangles we need to check that w0h = T (f)w. As C is a pushout, it suffices to check that w0hv = T (f)wv and w0hx = T (f)wx in E. The first HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 21 equality is clear as w0hv = w0v0g = 0 = T (f)wv (composition of 2 consecutive morphisms in a distinguished triangle is zero). Finally w0hx = w0x0I(f) = p0I(f) = T (f)p = T (f)wx which finishes the proof of (TR3). To prove (TR4) we consider standard triangles:

f 0 f 00 h0 h00 B / D / T (A) C / F / T (A) O O O O O O f α Id gf γ Id

iA pA iA pA A / I(A) / T (A) A / I(A) / I(A)

g0 g00 C / E / T (B) O O O g β Id 0 0 −1 iDf T (f ) pD B / I(D) / T (B)

0 0 We can choose f to be an inflation and consider iD ◦ f and an injective envelope of B. We need a morphism D → F . As D is a pushout a pair h0g : B → F , γ : I(A) → F define unique j : D → E. Then jf 0 = h0g and jα = γ. 0 Next, we want F → E. Again a pair g : C → E and βiDα: I(A) → E define unique 0 0 0 0 0 j : F → E such that j γ = βiDα and j h = g . We have a diagram

h0 j0 C / F / E O O O g j β 0 f iD B / D / I(D) O O f α

iA A / I(A)

The bottom left square and the big left rectangle are pushouts, hence so is the top left square. The top rectangle is a pushout, too, hence the top right square is a pushout. We 00 00 thus get j : E → T (D) such that j β = pD : I(D) → D. 22 AGNIESZKA BODZENTA

To get a commutative diagram

T (f 0) T (B) / T (D) O O g00 j00 Id E / E O O g0 j0 h=gf h0 h00 A / C / F / T (A) O O O O Id g j Id f f 0 f 00 A / B / D / T (A) it remains to check that T (f 0)g00 = j00 and h00j = f 00. It is a straightforward veriﬁcation using the fact that E and D are pushouts. 3.2. Triangulated categories.

Deﬁnition 3.8. A suspended category is triangulated if the functor T is an equivalence.

An exact category E with enough injectives and projectives is Frobenius if the class of projective objects coincides with the class of injective objects.

Exercise 3.9. Show that stable category of a Frobenius category is triangulated.

We shall usually denote the shift functor in a triangulated category by [1]. P. May in http://www.math.uchicago.edu/~may/MISC/Triangulate.pdf proved that the octahedron axiom is equivalent to (ST4) A commutative square C / D O O

A / B can be completed to a diagram

A[1] / B[1] / E[1] / A[2] O O O O

F / G / H / G[1] O O O O

C / D / I / C[1] O O O O

A / B / E / A[1] HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 23

with distinguished rows and columns in which all squares except to upper right one commute and the upper right square anti-commutes.

Lemma 3.10. Let D be a triangulated category. The composition of two consecutive morphisms in a distinguished triangle is zero.

f g Proof. Let A −→ B −→ C −→h A[1] be a distinguished triangle. By TR1 triangle A −→Id A −→0 0 −→0 A[1] is distinguished. By TR2 we have morphisms of triangles

f g h A / B / C / A[1] O O O O Id f Id Id A / A / 0 / A[1]

In particular, the second square commutes, i.e. gf = 0.

Lemma 3.11. Let A −→u B −→v C −→w A[1] be a distinguished triangle in a triangulated category D. Then, for any D ∈ D complexes

Hom(D,A) → Hom(D,B) → Hom(D,C) Hom(C,D) → Hom(B,D) → Hom(A, D) are exact.

Proof. We already know that the composition of two consecutive morphisms in a triangle is zero. It suﬃces to show that given ϕ: D → B such that vϕ = 0 there exists ψ : D → A such that uψ = ϕ. Consider a solid morphism of triangles

u v w A / B / Z / A[1] O O O O ψ ϕ

Id D / D / 0 / D[1]

By (ST3) morphism ψ such that uψ = ϕ exists. Analogously one proves exactness of the second sequence. Let A be an abelian category and D a triangulated category. An additive functor F : D → A is cohomological if for any distinguished triangle A → B → C → A[1] the sequence F (A) → F (B) → F (C) is exact. Let F be a cohomological functor and F k := F ◦[k]. Then for any distinguished triangle A → B → C → A[1] the sequence

... → F i(A) → F i(B) → F i(C) → F i+1(A) → ... 24 AGNIESZKA BODZENTA is exact.

Lemma 3.12. Let D be a triangulated category. If in a morphism of triangles two arrows are isomorphisms, then so is the third.

Proof. Let A0 / B0 / C0 / A0[1] O O O O a b c a[1]

A / B / C / A[1] be a morphism of triangles. Assume that a and c are isomorphism. Take and D ∈ D. Applying Hom(D, −) to the above diagram gives a commutative 5 × 2 diagram of abelian groups. By the 5-lemma, the morphism Hom(D,B) → Hom(D,B0) given by the composition with b is an isomorphism. By Yoneda, b is an isomorphism.

Lemma 3.13. Let f : A → B be a morphism in a triangulated category D. The following are equivalent: (1) f is an isomorphism, f (2) A −→ B → 0 → A[1] is a distinguished triangle, f (3) For any distinguished triangle A −→ B → C → A[1], C is isomorphic to 0.

f Proof. We know that A −→Id A → 0 → A[1] is distinguished. Triangle A −→ B → 0 → A[1] is isomorphic with it, hence it is also distinguished. It proves (1) ⇒ (2). Lemma 3.12 implies that (2) ⇒ (3), triangle A → B → C → A[1] has a map to A → B → 0 → A[1] which is an isomorphism on A and B, hence it is on C. Finally,

f 0 / A / B / 0 O O O O = Id f Id 0 / A / A / 0 is a morphism of triangles in which two morphisms are isomorphisms. Hence, by Lemma 3.12, f is an isomorphism, i.e. (3) ⇒ (1).

Proposition 3.14. Let A be an abelian category and H(A) the stable category of the category of complexes over A. Then functor H0(−): C(A) → A induces a cohomological functor H(A) → A.

Proof. Any distinguished triangle is isomorphic to a standard triangle and to a rotation of a standard triangle, hence it is enough to show that given a morphism f : A• → B• in 0 • 0 • 0 • C(A) the sequence H (B ) → H (Cf ) → H (A [1]) is exact. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 25

n+1 ! n n n+1 dB f The cone Cf is of the form Cf = B ⊕ A with differential . Hence 0 −dA • • • B → Cf → A [1] is a conflation in C(A). Hence, we prove that given a conflation A• → B• → C• in C(A) the sequence H0(A•) → H0(B•) → H0(C•) is exact. Snake lemma for the diagram

0 / An−1 / Bn−1 / Cn−1 / 0 O O O n n n dA dB dC

0 / An / Bn / Cn / 0

n n n n−1 n−1 proves that 0 → ker dA → ker dB → ker dC is exact. Similarly, coker dA → coker dB → n−1 coker dC → 0 is exact. Then the snake lemma for

n n n 0 / ker dA / ker dB / ker dC O O O

n−1 n−1 n−1 coker dA / coker dB // coker dC / 0

n · n · n · proves that H (A ) → H (B ) → H (C ) is exact.

Let D1, D2 be triangulated categories. An exact functor F : D1 → D2 is an additive functor which commutes with the shift functor and maps distinguished triangles to distinguished triangles.

Exercise 3.15. Let E1, E2 be Frobenius exact categories. A functor F : E1 → E2 which maps injective objects in E1 to injective objects in E2 induces an exact functor F : E 1 → E 2.

3.3. Localization of triangulated categories.

Let A be an abelian category. While studying derived functors we have discovered that sometimes it is convenient to replace objects of A by their resolutions. We want to consider the derived category D(A) of A. It shall be a triangulated category whose objects will be complexes of objects in A.We want to identify any two resolutions of an object of A. More generally, we would like to say that complexes are isomorphic if there is a morphism which induces an isomorphism of all cohomology objects (such a morphism will be called a quasi-isomorphism). To construct D(A) we want to consider the stable category of complexes H(A) and invert some morphisms in it (precisely the quasi-isomorphisms). We have already seen this kind of construction, it is a quotient of a category by a multiplicative system S. To ensure that the quotient category is still triangulated, we put extra conditions on S. 26 AGNIESZKA BODZENTA

Let D be a triangulated category. We say that a multiplicative system S of arrows in D is compatible with the triangulated structure if the following conditions hold:

MS5 For s ∈ S and n ∈ Z, s[n] ∈ S, MS6 Given a solid commutative square

X0 / Y 0 / Z / X0[1] O O O O s s0 s00 s[1]

X / Y / Z / X[1]

whose rows are distinguished and s, s0 ∈ S, there exists s00 ∈ S such that the diagram is a morphism of triangles.

Lemma 3.16. Let F : D → D0 be an exact functor of triangulated categories. Let

S = {f ∈ Mor(D) | F (f) is an isomorphism}.

Then S is a saturated multiplicative system compatible with the triangulated structure.

Proof. Identity morphisms lie in S and a composition of two morphisms in S lie in S, so MS1 is satisﬁed. Isomorphisms satisfy 2-out-of-3 property, so MS4 is satisﬁed. Clearly MS5 also holds. Property MS6 follows from Lemma 3.12. Next, we check that MS2 holds. Let f : A → B be a morphism and t: A → C a morphism in S. Then, we have a morphism of triangles

C / B0 / D / C[1] O O O O t Id t[1] f A / B / D / A[1]

Since S satisﬁes MS6, morphism B → B0 lies in S, i.e. LMS2 holds. The proof of RMS2 is dual. It remains to show MS3. Let f, g : A → B be morphism, such that, for t: C → A in S, ft = gt. In other words, at = 0, for a = g − t. Consider diagram

D O j a A / B O O Id b t q C / A / E HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 27 in which the bottom row is distinguished, morphism b is any morphism such that bq = a (it exists by Lemma 3.11) and the right column is distinguished. Then ja = jbq = 0. If we apply F to this diagram, Lemma 3.13 implies that F (E) ' 0, hence F (j) is an isomorphism. By deﬁnition, j ∈ S, which ﬁnishes the proof of LMS3. The proof of RMS3 is analogous.

Remark 3.17. In the proof of Lemma 3.16 we have in fact proved that if a multiplicative system satisﬁes MS1, MS5 and MS6, then it satisﬁes MS2.

Similarly, we have

Lemma 3.18. Let H : D → A be a cohomological functor between a triangulated category and an abelian category. Let

i S = {f ∈ Mor(D) | H (f) is an isomorphism for all i ∈ Z}. Then S is a saturated multiplicative system compatible with the triangulated structure.

Exercise 3.19. Prove the Lemma.

Proposition 3.20. Let D be a triangulated category and S a multiplicative system compatible with the triangulate structure. Then there exists unique structure of triangulated category on S−1D such that Q: D → S−1D is exact.

Proof. We already know that S−1D is additive. We deﬁne Q(D)[1] = Q(D[1]) and we say that a triangle in S−1D is distinguished if it is an image of a distinguished triangle in D.

Exercise 3.21. Check that axioms TR1-TR4 hold, cf. [Sta13, Proposition 13.5.5].

The quotient S−1D has universal property with respect to exact functors D → D0 and homological functors D → A. Now, we are ready to deﬁne the derived category D(A) of an abelian category. Let C(A) be the category of unbounded complexes of objects of A. We consider the functor

H0(−): C(A) → A.

By Proposition 3.14, it is cohomological. Hence, by Lemma 3.18

i S = {f ∈ Mor(H(A)) | H (f) is an isomorphism for all i ∈ Z} is a multiplicative system compatible with triangulated structure. We deﬁne

D(A) = S−1H(A). 28 AGNIESZKA BODZENTA

We could also start from the stable category Hb(A) of bounded complexes, the stable category H−(A) of bounded above complexes or the stable category H+(A) of bounded below complexes. Considering analogous quotients, we get

Db(A) = S−1Hb(A), D−(A) = S−1H−(A), D+(A) = S−1H+(A), the bounded, bounded above and bounded below derived category of A.

3.4. Derived category as a quotient by acyclic complexes.

The construction of derived category D(A) required the category A to be abelian, as we considered cohomology of complexes of objects of A. We shall now describe another deﬁnition of D(A) which admits an immediate generalisation to the case when A is only exact. Let D be a triangulated category. We say that full triangulated subcategory D0 ⊂ D is saturated if whenever X ⊕ Y is isomorphic to an object of D0 then X and Y are.

0 Lemma 3.22. Let F : D → D1 be an exact functor of triangulated categories. Let D be the full subcategory with objects

Ob(D0) = {D ∈ D | F (D) ' 0}.

Then D0 is a saturated triangulated subcategory of D.

Proof. Clear.

Let f : D1 → D2 be a morphism in a triangulated category. By TR1, there exists a f distinguished triangle D1 −→ D2 → D3 → D1[1]. We call D3 the cone of f. It is unique up to a non-unique isomorphism.

Proposition 3.23. Let D0 ⊂ D be a full triangulated subcategory of a triangulated category. Set

S = {f ∈ Mor(D) | the cone of f ∈ D0}.

Then the following are equivalent (1) S is a saturated multiplicative system. (2) D0 is a saturated subcategory.

Let D0 ⊂ D be a saturated full triangulated subcategory of D. The Verdier quotient D/D0 is S−1D, for the multiplicative system S deﬁned as in Proposition 3.23. From the universal property of S−1D (see Proposition 1.5) we conclude: HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 29

Proposition 3.24. Let D be a triangulated category and D0 ⊂ D a saturated full triangulated subcategory. Let Q: D → D/D0 be the quotient functor. (1) If H : D → A is a homological functor into an abelian category such that H(D0) ' 0, for all D0 ∈ D0, then there exists a unique factorization H : D/D0 → A such that H ◦ Q = H and H is a homological functor too. 0 (2) If F : D → D1 is an exact functor into a triangulated category such that F (D ) ' 0, 0 0 0 for all D ∈ D , then there exists a unique factorization F : D/D → D1 such that F ◦ Q = F and F is an exact functor too.

Let A be an abelian category and Ac(A) ⊂ H(A) be the category of acyclic complexes, i.e. complexes A· such that Hi(A·) = 0, for all i. Then

i S = {f ∈ Mor(H(A)) | H (f) is an isomorphism for all i ∈ Z} = {f ∈ Mor(H(A)) | the cone of f ∈ Ac(A)}.

It follows that the derived category D(A) is the quotient of H(A) by the subcategory of acyclic complexes. Similarly, Db(A) 'Hb(A)/Acb(A), D−(A) 'H−(A)/Ac−(A), D+(A) 'H+(A)/Ac+(A), where Ac∗(A), for ∗ ∈ {b, +, −} is deﬁned as Ac(A) ∩ H∗(A).

1 Lemma 3.25. Let A, B be objects of an abelian category A. Then ExtA(A, C) '

HomD(A)(A, C[1]).

Proof. For simplicity let us assume that A has enough projective objects. Let P • • be a projective resolution of A. Then HomD(A)(A, C[1]) ' HomD(A)(P ,C[1]) ' • 0 HomH(A)(P ,C[1]) is the 1’st cohomology group of the complex Hom(P ,C) → 1 • Hom(P ,C) → .... (We will show on tutorials that HomD(A)(P ,C[1]) ' • HomH(A)(P ,C[1]).) f g Proposition 3.26. Let 0 → A −→ B −→ C → 0 be a short exact sequence in an abelian f g ζ category A and ζ the corresponding class in Ext1(C,A). Then A −→ B −→ C −→ A[1] is a distinguished triangle in D(A).

Proof. Let us calculate the cone of A → B in H(A). The injective envelope of A is f complex A −→Id A concentrated in degrees −1 and 0. The pushout of A −→ B and A → IA f is the complex A −→ B. Indeed, a morphism (IA)• → D• is a map A → D−1 while a morphism B → D• is a map B → ker(D0 → D−1). If these agree on A, they deﬁne a f unique map of complexes {A −→ B} → D•. f By construction of triangulated structure A → B → {A −→ B} → A[1] is a distinguished f triangle in H(A). Morphism {A −→ B} → C induced by g is a quasi-isomorphism, hence f g ξ A −→ B −→ C −→ A[1] is distinguished in D(A). 30 AGNIESZKA BODZENTA

Let P • be a projective resolution of C. Then by Proposition 2.6 the element ζ is a f class of a map P −1 → A. The construction is such that P • → {A −→ B} is a morphism of complexes (we construct ϕ: P −1 → A by ﬁrst lifting P 0 → C to P 0 → B and then noticing that P −1 → P 0 → B → C is zero, so it factors via ϕ). We get a commuting triangle of quasi-isomorphisms

P • / C ; g $ f {A −→ B}

ξ f The map C −→ A[1] in the distinguished triangle is such that the composite {A −→ B} → ξ ξ C −→ A[1] is the projection to A[1]. Then P • → C −→ A[1] is the map ϕ. It follows that ξ = ζ.

Proposition 3.26 can be generalized, cf. [Kel96, Part 11]:

f g Proposition 3.27. Let A be an abelian category and let A· −→ B· −→ C· be a complex in C(A) such that An → Bn → Cn is a short exact sequence, for any n. Then there exists f g ζ ζ : C· → A·[1] such that A· −→ B· −→ C· −→ A·[1] is a distinguished triangle in D(A).

4. t-structures

4.1. The motivating example. Let A be an abelian category and D∗(A) its derived category. We consider the functor

(7) ι0 : A → C(A) → H(A) → D(A) which maps an object A ∈ A to the class of a complex {... → 0 → A → 0 → ...} with A in degree zero.

Exercise 4.1. Let A· be a complex in D(A) such that Hn(A·) = 0, for n ≥ N. Then A· is isomorphic to a complex concentrated in degrees ≤ N.

Proposition 4.2. Functor ι is fully faithful and induces an equivalence of A with the full subcategory of D(A) of complexes A· such that Hi(A) = 0, for i 6= 0.

Proof. Clearly, HomA(A, B) = HomC(A)(ι0(A), ι0(B)). As there cannot be a homotopy between two morphisms ι0(A) → ι0(B), we further have HomC(A)(ι0(A), ι0(B)) '

HomH(A)(ι0(A), ι0(B)). We check that the last space is isomorphic to the space of HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 31 morphisms in the derived category. A morphism i0(A) → i0(B) is

C = b f g '

ι0(A) ι0(B) where g is a quasi-isomorphism. As ι0(A) → C is a morphism of complexes, the image of A → C0 is contained in the kernel of d0. Hence, we can assume that Ci = 0, for i > 0. As g is a quasi-isomorphism, we know that coker d−1 ' B. The canonical map π : C0 → coker d−1 yields a commutative diagram

C : d f g π πf Id ι0(A) / ι0(B) o ι0(B) which proves that ι0 is full. Let now f, g : A → B be two morphisms whose images in D(A) are equal, i.e. we have a commutative diagram

ι0(B) f ; c Id α

h ι0(A) / Co ι0(B) O α α g # { Id ι0(B) As α is an isomorphism, as before, we can assume that Ci = 0, for i > 0. Then the composition B −→α C0 −→π coker d−1 is as isomorphism. Hence, αf = αg implies f =

παf = παg = g, i.e. ι0 is faithful. · It remains to show that ι0 is essentially surjective. Let A be a complex such that Hi(A·) = 0, for i 6= 0. Morphism of complexes

d−1 d0 ... / A−1 / A0 / A1 / ... O O O Id

... / A−1 / ker d0 / 0 / 0 is a quasi-isomorphism, hence we can assume that Ai = 0, for i > 0. Similarly,

0 / 0 / coker d−1 / 0 / 0 O O O

... / A−1 / A0 / 0 / 0 32 AGNIESZKA BODZENTA

· 0 · is a quasi-isomorphism, hence A is isomorphic (in D(A)) to a complex i0(H (A )).

Lemma 4.3. Let A be an abelian category. Consider functor ι0 : A → D(A) as in (7).

Then HomD(A)(i0(A), i0(B)[−1]) = 0, for all (A, B) ∈ A.

Proof. A morphism in D(A) is of the form

C· : f f g π 0 ι (A) / cokerd0 o ι (B)[−1] 0 ' 0 with a quasi-isomorphism g. We can assume that Ci = 0, for i > 1. Then the map B → C1 → coker d0 is an isomorphism, which proves that (f, C, g) is equivalent to

(0,B, Id), i.e. Hom(ι0(A), ι0(B)[−1]) = 0.

Proposition 4.4. Let A be an abelian category and consider functor ι0 : A → D(A) as above. Then a complex A → B → C of objects of A is a short exact sequence if and only if ι0(A) → ι0(B) → ι0(C) → ι0(A)[1] is a distinguished triangle.

Proof. With Proposition 3.26 we have proved one implication, that a short exact sequence yields a distinguished triangle. f g Let now A −→ B −→ C be a complex in A such that ι0(A) → ι0(B) → ι0(C) → ι0(A)[1] is a distinguished triangle. We show that f is the kernel of g and g the cokernel of f.

Let D ∈ A be any object and ϕ: D → B a morphism such that gϕ = 0. Then, as ι0 is fully faithful, ι0(g) ◦ ι0(ϕ) = 0. Exact sequence

Hom(ι0(D), ι0(C)[−1]) → Hom(ι0(D), ι0(A)) → Hom(ι0(D), ι0(B)) → Hom(ι0(D), ι0(C)) together with vanishing of Hom(ι0(D), ι0(C)[−1]) implies that there exists unique ψ such that f ◦ ψ = ϕ. It follows that f is the kernel of g. The proof that g is the cokernel of f is analogous. It follows that A ⊂ D(A) is a full subcategory and we can read oﬀ the short exact sequences in A from the triangulation of D(A).

Proposition 4.5. Any complex A· ∈ D(A) ﬁts into a distinguished triangle · · · · τ≤0A → A → τ≥1A → τ≤0A [1] with

( i ( i i · H (A) for i ≤ 0 i · H (A) for i ≥ 1 H (τ≤0A ) = H (τ≥1A ) = 0 for i ≥ 1. 0 for i ≤ 0. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 33 · Proof. We deﬁne τ≤0A as the bottom row of the diagram of a morphism of complexes

d−1 d0 ... / A−1 / A0 / A1 / ... O O O Id

... / A−1 / ker d0 / 0 / 0 · We deﬁne τ≥1A as the top row of the diagram of a morphism of complexes

0 / 0 / coker d0 / A2 / ... O O O

... / A0 / A1 / A2 / 0 The conditions on cohomology are clearly satisﬁed. Next we prove that these two maps ﬁt into a distinguished triangle. By Proposition · · 0 0 1 3.27 the cone of the map τ≤0A → A is a complex 0 → A / ker d → A → .... Then the cokernel of d0 is isomorphic to the cokernel of A0/ ker d0 → A1, hence the cone is · quasi-isomorphic to τ≥1A . Analogously as Lemma 4.3 one proves

Lemma 4.6. Let A be an abelian category. Consider complexes A·,B· ∈ D(A) such that i · j · · · H (A ) = 0 for i ≥ 1 and H (B ) = 0, for j ≤ 0. Then HomD(A)(A ,B ) = 0.

We deﬁne

(8) D(A)≤0 = {A· | Hi(A) = 0 for i ≥ 1} (9) D(A)≥1 = {A· | Hj(A) = 0 for j ≤ 0}.

Theorem 4.7. Consider an abelian category A. (t1) The subcategory D(A)≤0 is closed under the shift by one, D(A)≤0[1] ⊂ D(A)≤0. The subcategory D(A)≥1 is closed under the shift by minus one, D(A)≥1[−1] ⊂ D(A)≥1. · ≤0 · ≥1 · · (t2) For any A ∈ D(A) and B ∈ D(B) , HomD(A)(A ,B ) = 0. · · · · · (t3) Any A ﬁts into a distinguished triangle τ≤0A → A → τ≥1A → τ≤0A [1], with · ≤0 · ≥1 τ≤0A ∈ D(A) , τ≥1A ∈ D(A) .

Proof. As the shift is a shift to the left, Hi(A·[1]) = Hi+1(A·). If Hi(A·) = 0, for i ≥ 1, then Hi(A·[1]) = 0, for i ≥ 0. Property (t2) follows from Lemma 4.6. Property (t3) follows from Proposition 4.5. 34 AGNIESZKA BODZENTA

4.2. Deﬁnition and ﬁrst properties.

A t-structure on a triangulated category D is a pair of strictly full subcategories (D≤0, D≥1) which satisfy (t1)-(t3) above. The t-structure describe in Theorem 4.7 is the standard t-structure on D(A). For a t-structure (D≤0, D≥1), we deﬁne

D≤n = D≤0[−n], D≥n = D≥1[−n + 1], A = D≤0 ∩ D≥0.

The category A is the heart of the t-structure.

Lemma 4.8. Let (D≤0, D≥1) be a t-structure on a triangulated category D. Then (D≤n, D≥n+1) is a t-structure on D, for any n ∈ Z.

Proof. Follows immediately from the fact that [−n] is an equivalence of D. Lemma 4.9. Let (D≤0, D≥1) be a t-structure on a triangulated category D. Then the ≤0 triangle in (t3) is unique in A ∈ D. Moreover, τ≤0 deﬁnes a functor D → D , left ≤0 ≥1 adjoint to the inclusion D → D and τ≥1 deﬁnes a functor D → D right adjoint to the inclusion D≥1 → D.

Exercise 4.10. Prove the Lemma.

≤n ≥n Using appropriate shifts we deﬁne functors τ≤n : D → D and τ≥n : D → D . Example: Let A be the category of right modules over a quiver

1 / 2

f Objects of A are vector spaces V2, V1 and a morphism V2 → V1. We denote it by (V2 −→ V0). Let

0 0 T = {(0 −→ V1)}, F = {(V2 −→ 0)} be full subcategories of A with modules supported on one of the vertex. Then f Hom(T , F) = 0 and any (V2 −→ V0) ∈ A fits into a short exact sequence 0 f 0 0 → (0 −→ V1) → (V2 −→ V1) → (V2 −→ 0) → 0. The subcategories (T , F) yield an example of a torsion pair, i.e. a pair of subcategories T , F of an abelian category A such that Hom(T , F) = 0 and any A ∈ A fits into a short exact sequence 0 → T → A → F → 0 with T ∈ T and F ∈ F. Put D = D(A) and define

(10) pD≤0 = {A· ∈ D | H0(A·) ∈ T ,Hi(A·) = 0 for i ≥ 1}, (11) pD≥1 = {A· ∈ D | H0(A·) ∈ F,Hj(A·) = 0 for j ≤ −1}. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 35

Proposition 4.11. Let (T , F) be a torsion pair on an abelian category A. Then (pD≤0, pD≥1) is a torsion pair on the category D. It is the tilt of the standard t-structure in the torsion pair (T , F).

Proof. Condition (t1) is clearly satisﬁed. Let now A ∈ pD≤0 and B ∈ pD≥1. We can decompose A with respect to an appropriate shift of the standard t-structure:

τ≤−1A → A → τ≥0A → τ≤−1A[1].

Then in the long exact sequence

Hom(τ≤−1A[1],B) → Hom(τ≥0A, B) → Hom(A, B) → Hom(τ≤−1A, B)

≤−1 the ﬁrst an the last groups vanish (because τ≤−1A[1] ∈ D ) so Hom(A, B) '

Hom(τ≥0A, B). Similarly, we have the decomposition of B:

τ≤0B → B → τ≥1B → τ≤0B[1].

≤0 Again, τ≥0A ∈ D so in the exact sequence

Hom(τ≥0A, τ≤0B[1]) → Hom(τ≥0A, τ≤0B) → Hom(τ≥0A, B) → Hom(τ≥0A, τ≥1B) the ﬁrst and the last group vanish. It follows that Hom(A, B) ' Hom(τ≥0A, τ≤0A) = Hom(H0(A),H0(B)) = 0 (we use Propositions 4.2 and 4.5 and the fact that to know that cohomology objects of the truncations of A and B and to know that we can calculate morphisms in the category A). It remains to prove (t3). Let A· be a complex in D. We have a distinguished triangle

0 τ≤−1A → τ≤0A → H (A) and a decomposition of H0(A) with respect to the torsion pair 0 → T → H0(A) → F → 0. By Proposition 3.26 we can view the latter as a distinguished triangle too. Consider diagrams with distinguished rows and columns:

p T / τ≤−1A[1] / τ≤0A[1] O O O Id

T / H0(A) / F O O O

0 / τ≤0A / τ≤0A 36 AGNIESZKA BODZENTA

0 / τ≥1A / τ≥1A O O O

p p τ≤0A / A / τ≥1A O O O Id

p τ≤0A / τ≤0A / F We claim that the middle row of the second diagram is the triangle we were looking for. p We use the top row of the ﬁrst diagram to calculate cohomology of τ≤0A. We get that 0 p H ( τ≤0A) = T and higher cohomology vanish. The right column of the right diagram 0 p implies that H ( τ≥1A) = F and all lower cohomology vanish. The heart A] of the new t-structure is

A] = {A· ∈ D | H−1(A·) ∈ F,H0(A·) ∈ T ,Hi(A·) = 0 for i 6= −1, 0}.

Let B be an object of A]. The decomposition of B with respect to the standard t-structure is H−1(B)[1] → B → H0(B) → H−1(B)[2], hence B is determined by a class in Ext2(H0(B),H−1B). 2 ] In the example we considered Ext ((0 → V1), (V2 → 0)) = 0, hence any B ∈ A is isomorphic to H−1(B)[1] ⊕ H0(B). It follows that A] ' mod–k ⊕ k. In fact, the heart of a t-structure is always an abelian category. However, the derived category of the heart is not necessarily equivalent to the original triangulated category. (To see it on the example we note that any distinguished triangle in D(k ⊕ k) splits while in the original category it is not the case). Before we prove this fact we discuss the cohomology functors associated with a t-structure. In the following I use the presentation in [HTT08, Chapter 8].

Lemma 4.12. The following conditions on D ∈ D are equivalent (1) D ∈ D≤n (resp. D ∈ D≥n)

(2) The canonical morphism τ≤nD → D, (resp. D → τ≥nD) is an isomorphism

(3) τ>nD = 0 (resp. τ

Lemma 4.13. Let D0 → D → D00 be a distinguished triangle in D. If D0,D00 ∈ D≤0 then D ∈ D≤0. If D0,D00 ∈ D≥0 then D ∈ D≥0.

>0 Proof. We prove that τ>0D = 0. To do it, we show that Hom(D,E) = 0 for any E ∈ D . 00 0 It follows from the exact sequence Hom(D ,E) → Hom(D,E) → Hom(D ,E). Proposition 4.14. Let a, b be two integers. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 37

(1) If b ≥ a then τ≤b ◦ τ≤a ' τ≤a ◦ τ≤b ' τ≤a, τ≥b ◦ τ≥a ' τ≥a ◦ τ≥b ' τ≥b.

(2) If a > b then τ≤b ◦ τ≥a ' τ≥a ◦ τ≤b ' 0.

(3) τ≥a ◦ τ≤b ' τ≤b ◦ τ≥a.

Proof. For b ≥ a canonical morphism τ≤bτ≤a → τ≤a is an isomorphism as τ>bτ≤a = 0.

τ≤a◦τ≤b ' τ≤a as the composition of two adjoint functors is the adjoint to the composition. The remaining isomorphisms are proved similarly. For a > b part (2) is clear. In part (3) we might assume that b ≥ a. Let D ∈ D. We ﬁrst construct morphism

ϕ: τ≥aτ≤bD → τ≤bτ≥aD. We have a distinguished triangle

τ≤bτ≥aD → τ≥aD → τ>bτ≥aD ' τ>bD

≥a It follows that τ≤bτ≥aD ∈ D (see the previous lemma). Then we get isomorphisms

Hom(τ≤bD, τ≥aD) ' Hom(τ≤bD, τ≤bτ≥aD) ' Hom(τ≥aτ≤bD, τ≤bτ≥aD). We consider ϕ which corresponds to τ≤bD → D → τ≥aD. We have a distinguished triangle

≤b which implies that τ≥aτ≤bD ∈ D . Consider octahedron

0 / τ>bD / τ>bD O O O

In the distinguished triangle

τ≥aτ≤bD → τ≥aD → τ>bD the ﬁrst term lies in D≤b and the last in D>b. Hence, it is the canonical decomposition of

τ≤aD, i.e.τ≥aτ≤bD ' τ≤bτ≥aD.

Proposition 4.15. Let (D≤0, D≥1) be a t-structure on a triangulated category D. The heart A = D≤0 ∩ D≥0 is an abelian category.

Proof. Let A, B ∈ A. Then A ⊕ B ∈ A as τ<0 and τ>0 vanish. Let now f : A → B be a morphism in A. It ﬁts into a distinguished triangle A → B → 1 C. We shall show that the kernel of f is H (C) = τ≤−1C[1] and the cokernel is τ≥0C. For 38 AGNIESZKA BODZENTA any D in A we have exact sequences

0 ' Hom(A[1],D) → Hom(C,D) → Hom(B,D) → Hom(A, D) 0 ' Hom(D,B[−1]) → Hom(D,C[−1]) → Hom(D,A) → Hom(D,B) where the isomorphisms with zero follow from Hom(D≤0, D≥1) = 0. Moreover,

Hom(C,D) ' Hom(τ≥0C,D) and Hom(D,C[−1]) ' Hom(D[1],C) ' Hom(D[1], τ≤−1C), as functors τ≥0 and τ≤−1 are adjoint to appropriate inclusions. It remains to show that the Coimf → =f is an isomorphism. Octahedron

0 / H−1C[2] / H−1(C)[2] O O O

B / H0(C) / I[1] O O O

B / C / A gives distinguished triangles I → B → H0(C) and H−1(C) → A → I which imply that I ∈ A. Hence, I is the kernel of B → H0(C) = coker f and I is the cokernel of −1 ker f = H (C) → A. Let (D≤0, D≥1) be a t-structure on a triangulated category D with heart A. Deﬁne

0 HA = τ≥0τ≤0 : D → A.

Proposition 4.16. Functor H0 is cohomological.

Proof. Consider a distinguished triangle A → B → C → A[1]. We shall show that H0(A) → H0(B) → H0(C) is exact. We proceed in few steps. • When A, B, C ∈ D≤0 then H0(A) → H0(B) → H0(C) → 0 is exact. Let D be an object of A. We check the universal property of the cokernel 0 and we use the fact that Hom(H (A),D) ' Hom(A, ι0(D)) (because there are no

morphisms τ<0A → D). We have an exact sequence

Hom(A[1], ι0D) → Hom(C, ι0D) → Hom(B, ι0D) → Hom(A, ι0D).

As A[1] ∈ D<0 the ﬁrst space is isomorphic to zero, hence H0(C) is the cokernel of H0(A) → H0(B). • When A ∈ D≤0 then H0(A) → H0(B) → H0(C) → 0 is exact. >0 First we claim that τ>0B ' τ>0C. Let D ∈ D . Then Hom(A, D) = 0 and

Hom(A[1],D) = 0, so Hom(C,D) → Hom(B,D) are isomorphic. As τ>0 is left >0 adjoint to the inclusion of D , we conclude that τ>0B ' τ>0C. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 39

The map A → B factors via A → τ≤0B → B. From the octahedron

0 / τ>0B / τ>0C O O O

A / B / C O O O

A / τ≤0B / τ≤0C

we conclude that A → τ≤0B → τ≤0C is a distinguished triangle, hence by the previous point H0(A) → H0(B) → H0(C) → 0 is exact. • For C ∈ D≥0 sequence 0 → H0(A) → H0(B) → H0(C) is exact. It is proved analogously or by considering the opposite categories. • The general case. Octahedrons

0 / C / C τ>0A / 0 / τ>0A[1] O O O O O O

τ≤0A / B / Q A / B / C O O O O O O

τ≤0A / A / τ>0A τ≤0A / B / Q

give triangles τ≤0A → B → Q and Q → C → τ>0A[1]. By previous points

H0(A) → H0(B) → H0(Q) → 0, 0 → H0(Q) → H0(C) → H1(A)

are exact, hence H0(A) → H0(B) → H0(C) is exact.

T ≤n T ≥n We say that a t-structure is bounded if it is non-degenerate, i.e. n D = 0 = n D and for any D ∈ D all but ﬁnitely many cohomology objects H(D) vanish.

Theorem 4.17. Let A be the heart of a bounded t-structure on a triangulated category D, F : Db(A) → D a t-exact functor. The functor F is an equivalence of categories if ∗ 1 n and only if ExtD is generated by ExtD, i.e. for any X,Y ∈ A any class in Ext (X,Y ) = 1 Hom(X,Y [n]) is a linear combination of monomials β1 . . . βn with βj ∈ ExtD(Xj,Xj+1), for someq Xj ∈ A.

≤0 ≥1 ≤0 ≥1 Let (D1 , D1 ), (D2 , D2 ) be t-structures on triangulated categories D1, D2. An exact functor F : D1 → D2 (i.e. F (−[1]) ' F (−)[1], F maps distinguished triangles to 40 AGNIESZKA BODZENTA

≥1 ≥1 ≤0 distinguished triangles) is left t-exact if F (D1 ) ⊂ D2 . F is right t-exact, if F (D1 ) ⊂ ≥0 D2 and F is t-exact if it is both left and right t-exact.

Exercise 4.18. Let F : D1 → D2 be left adjoint to G: D2 → D1. Assume that G is right t-exact, for some t-structures on D1 and D2. Prove that F is left t-exact.

4.3. Semi-orthogonal decompositions and recollements.

A semi-orthogonal decomposition hD1, D2i of a triangulated category D is a pair of full triangulated subcategories of D such that (D2, D1) is a t-structure on D. ≤n Let us write down explicitly what it means. Category D2 is D , for all n, while ≥n D1 = D , for all n. In particular, Hom(D2,D1) = 0 and, any object D ∈ D ﬁts into a distinguished triangle

D2 → D → D1 → D2[1]

∗ with D1 ∈ D1, D2 ∈ D2. Moreover, j∗ : D1 → D has left adjoint j : D → D1 and ! i∗ : D2 → D has right adjoint i : D → D2. A full subcategory D0 of a triangulated category D is right admissible if the inclusion 0 ! 0 functor i∗ : D → D has right adjoint i : D → D . A full subcategory D0 of a triangulated category D is left admissible if the inclusion 0 ∗ 0 functor j∗ : D → D has left adjoint j : D → D .

0 Proposition 4.19. [Bon89] Let i∗ : D → D be the inclusion of a full triangulated subcategory. The following conditions are equivalent: (1) Category D0 is right admissible 0 (2) Category D admits a semi-orthogonal decomposition D = hD1, D i where D1 = {D ∈ D | Hom(D0,D) = 0}.

Proof. Assume that D0 is right admissible. We check that conditions (t1)-(t3) are satisﬁed.

The ﬁrst two are clear. For D ∈ D let D1 be such that

∗ D1 → D → i∗i D → D1[1]

∗ is distinguished, where D → i∗i D is the adjunction unit. Then D1 ∈ D1, hence (t3) holds. 0 0 In the opposite direction, if D = hD1, D i then D is right admissible.

We say that a full triangulated subcategory D0 ⊂ D is admissible if it is left and right admissible. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 41

Example 4.20. Let E ∈ D be an exceptional object in a k-linear Ext-ﬁnite (i.e. L dimk i Hom(D1,D2[i]) < ∞ for any pair D1,D2 of objects of D) triangulated category D, i.e. assume that Hom(E,E) = k and Hom(E,E[i]) = 0, for all i 6= 0. Then functor b i∗ : D (k) → D such that i∗(k) = E is fully faithful and has left and right adjionts: M M i!(D) = Hom(E,D[i])[−i] i∗(D) = Hom(D,E[i])∨[i]. i i

Example 4.21. Let X be a regular projective scheme. Then Db(Coh(X)) is a k-linear i Ext-ﬁnite triangulated category. Assume that H (OX ) = 0, for i > 0. Let L be a line bundle on X. Then L ∈ Db(Coh(X)) is an exceptional object.

An admissible subcategory i∗ : D2 → D yields two semi-orthogonal decompositions

D = hD1, D2i = hD2, D3i.

! Exercise 4.22. Let k∗ : D3 → D be the inclusion functor and k its right adjoint. Prove ! ∗ that the mutation functor k j∗ : D1 → D3 is an equivalence with quasi-inverse j k∗ : D3 →

D1.

Exercise 4.23. For semi-orthogonal decompositions D = hD1, D2i = hD2, D3i show that ∗ functor i1 : D → D1 left adjoint to the inclusion, is right adjoint to the inlusion of D3 → D.

Classically, a admissible subcategory is known as recollement, i.e. triangulated categories D1, D, D2 together with 6 functors:

∗ o i o j! ∗ (12) D1 i∗ / D j / D2 o i! o j∗ such that

∗ ! ∗ (1) Functors i a i∗ a i , j! a j a j∗ are adjoint and i∗, j!, j∗ are fully faithful, ∗ (2) i∗D1 is the kernel of j (3) Every object D ∈ D ﬁts into distinguished triangles

! ∗ ! ∗ ∗ ∗ i∗i D → D → j∗j D → i∗i D[1], j!j D → D → i∗i D → j!j D[1].

Exercise 4.24. Prove that the data of a recollement is equivalent to the data of an admissible subcategory i∗D1.

Exercise 4.25. Given a recollement (12), prove that D2 is the quotient of D by D1. 42 AGNIESZKA BODZENTA

4.4. Gluing of t-structures.

≤0 ≥1 ≤0 ≥1 Consider a recollement (12) and t-structures (D1 , D1 ), (D2 , D2 ).

∗ Theorem 4.26. There exists a unique t-structure on D such that functors i∗, j are t-exact. The t-structure is deﬁned as

≤0 ∗ ≤0 ∗ ≤0 D = {D ∈ D | j D ∈ D2 , i D ∈ D1 }, ≥1 ∗ ≥1 ! ≥1 D = {D ∈ D | j D ∈ D2 , i D ∈ D1 }.

Proof. Condition (t1) is clearly satisﬁed. Let now X ∈ D≤0 and Y ∈ D≥1. We consider the following diagram with exact rows and columns

∗ ! ∗ ∗ ∗ Hom(j!j X, i∗i Y ) / Hom(j!j X,Y ) / Hom(j!j X, j∗j Y ) O O O

! ∗ Hom(X, i∗i Y ) / Hom(X,Y ) / Hom(X, j∗j Y ) O O O

∗ ! ∗ ∗ ∗ Hom(i∗i X, i∗i Y ) / Hom(i∗i X,Y ) / Hom(i∗i X, j∗j Y )

∗ ∗ ∗ ! ∗ given by distinguished triangles j!j X → X → i∗i X → j!j X[1], i∗i Y → Y → j∗j Y . ∗ ≤0 ! ≥1 The bottom left space in the diagram vanishes because i X ∈ D1 , i Y ∈ D1 and i∗ is fully faithful. ∗ The top left space is zero because j i∗ = 0. For the same reason the bottom right space ∗ ∗ ∗ vanishes. Finally, the top right is isomorphic to Hom(j X, j Y ), as j j∗ ' Id, hence it is ∗ ≤0 ∗ ≥1 also zero because j X ∈ D2 , j Y ∈ D2 . ! ∗ It follows that Hom(X, i∗i Y ) = 0 = Hom(X, j∗j Y ), hence Hom(X,Y ) = 0 and (t2) holds. Now, let X be any object of D. Consider octahedrons:

∗ Y / X / j∗τ≥1j X O O O Id

! ∗ i∗i X / X / j∗j X O O O

∗ ∗ j∗τ≤0j X[−1] / 0 / j∗τ≤0j D HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 43

∗ A / Y / i∗τ≥1i Y O O O Id

∗ ∗ j!j Y / Y / i∗i Y O O O

∗ ∗ i∗τ≤0i Y [−1] / 0 / i∗τ≤0i Y

∗ ∗ i∗τ≥1i Y / B / j∗τ≥1j X O O O

∗ Y / X / j∗τ≥1j X O O O

Id A / A / 0

We claim that A → X → B is a distinguished triangle with A ∈ D≤0 and B ∈ D≥1. ∗ ∗ ≥1 From the top row of the third diagram we conclude that j B ' τ≥1j X ∈ D2 (because ∗ j i∗ = 0). The top row of the second diagram implies that j∗A ' j∗Y , which, by the left column ∗ of the ﬁrst diagram, is isomorphic to τ≤0j X. ∗ ! ∗ ∗ As j i∗ = 0 so are its right i j∗ and left i j! adjoints. The vanishing of i j! and the left ∗ ∗ ! column of the second diagram implies that i A ' τ≤0i Y . Finally, as i j∗ = 0, the top ! ∗ row of the third diagram implies that i B ' τ≥1i Y .

4.5. Intermediate extension.

≤0 ≥1 ≤0 ≥1 Consider a recollement (12) and non-degenerate t-structures (D1 , D1 ), (D2 , D2 ) with hearts A1, respectively A2. Let A ⊂ D be the heart of the glued t-structure (D≤0, D≥1).

The six functors in (12) induce functors between abelian categories: i∗ : A1 → A and ∗ p ∗ p ! j : A → A2 are the restrictions to the hearts of the t-exact functors. i and i are 0 ∗ 0 ! ∗ ! p ∗ deﬁned as H ◦ i |A and H ◦ i |A. Since i is right t-exact and i is left t-exact, functor i p ! p p is right exact while i is left exact. Functors j!, j∗ : A2 → A are deﬁned analogously, as 0 0 p p H ◦ j!|A2 and H ◦ j∗|A2 . As before, j! is right t-exact and j∗ is left t-exact. ∗ As functor j! is fully faithful, the adjunction unit Id → j j! is an isomorphism. Its ∗ inverse corresponds under the j a j∗ adjunction to a morphism

j! → j∗. 44 AGNIESZKA BODZENTA

By considering the 0’th cohomology we obtain a morphism

p p j! → j∗.

Its image deﬁnes the intermediate extension

j!∗ : A2 → A.

5. Perverse sheaves

Let M be an n-dimensional oriented closed manifold. Then Poincare duality states that the k’th cohomology group of M is isomorphic to the n − k’th homology group of M:

k H (M, Z) ' Hn−k(M, Z).

If, however, M is singular, Poincare duality does not hold. One needs to replace the cohomology with the intersection cohomology groups of a variety with coefficients in a local system. Originally, the intersection cohomology was introduced to study the failure, due to presence of singularities, of Poincare duality. Later, it has been noticed that the intersection complexes of irreducible subvarieties of M are the simple perverse sheaves. The category of perverse sheaves will be introduced as a heart of an appropriate t-structure. We shall consider a topological space X and a sheaf R of rings with unity on X. We look at the category ShR(X) of sheaves of R-modules and its derived category D(ShR(X)). Under some additional conditions, given a closed subset Z ⊂ X with open complement U, we shall construct a recollement o o D(ShR(Z)) / D(ShR(X)) / D(ShR(U)) o o To define functors in the recollement, we first discuss derived functors.

5.1. Derived functors.

Let F : A → B be a left (resp. right) exact functor between abelian categories. We deﬁne its extension RF : D+(A) → D+(B) (resp. LF : D−(A) → D−(B)) which is the right (resp. left) derived functor of F . The functors RF , LF are exact, i.e. they map distinguished triangles to distinguished triangles. If functor F is exact, construction of the derived functor is easy: HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 45

Proposition 5.1. Assume that F : A → B is exact. Then, for ∗ ∈ {, −, +, b}, functor H∗(F ): H∗(A) → H∗(B) maps acyclic complexes to acyclic complexes, hence it induces an exact functor D∗(F ): D∗(A) → D∗(B).

The main idea of the construction of derived functors is to apply F componentwise to some selected representatives in equivalence classes of quasi-isomorphic complexes. A class R ⊂ A is adapted to a left (resp. right) exact functor F if it is stable under direct sums and satisﬁes (1) F maps any acyclic (no cohomology in A) complex from Kom+(R) (resp. Kom−(R)) into an acyclic complex. (2) Any object of A is a subobject (resp. a quotient object) of an object of R.

Proposition 5.2. Let R be a class of objects adapted to a left exact functor F : A → B + and SR the class of quasi-isomorphisms in K (R). Then SR is a localising class of morphisms in K+(R) and the canonical functor

+ −1 + K (R)[SR ] → D (A) is an equivalence.

+ −1 We deﬁne the right derived functor RF on objects of K (R)[SR ] termwise. Fixing the inverse of the equivalence in the above proposition, we get a functor

RF : D+(A) → D+(B).

To prove the independence of the choice of R we need to deﬁne the derived functor using a universal property.

Deﬁnition 5.3. The derived functor of an additive left exact functor F : A → B is a pair consisting of an exact functor D+(F ): D+(A) → D+(B) and a morphism of functors + + εF : QB ◦ K (F ) → D (F ) ◦ QA:

D+(F ) D+(A) / D+(B) O O QA QB K+(F ) K+(A) / K+(B) satisfying the following universal property: for any exact functor G: D+(A) → D+(B) + and a morphism of functors ε: QB ◦ K (F ) → G ◦ QA there exists a unique morphism of + functors η : D (F ) → G such that (η ◦ QA) ◦ εF = ε. The derived functor of a right exact functor is a pair consisting of a functor − − − − + D (F ): D (A) → D (B) and a morphism εF : D (F ) ◦ QA → QB ◦ K (F ) satisfying the universal property with a morphism η : G → D−(F ). 46 AGNIESZKA BODZENTA

The universal property implies that the derived functor is unique. One checks that the functor deﬁned using an adapted class satisﬁes the universal property.

Theorem 5.4. If A contains suﬃciently many injective (resp. projective) objects then the class of all these objects is adapted to any left (resp. right) exact functor F .

Proof. Let I· be a bounded below acyclic complex of injective objects. Then the identity morphism is homotopic to the zero morphism (we checked on tutorials that there are no morphisms in the homotopy category from a complex with projective terms to an acyclic complex. Here we use the dual argument). The image under F of the homotopy shows · · that the identity on F (I ) is homotopic to the zero morphism, hence F (I ) is acyclic. To define derived functors on unbounded derived categories, one uses K-projective complexes. A complex P · is K-projective if for any acyclic complex A· the complex Hom(P ·,A·) is acyclic (we consider all morphisms f i : P i → Ad+i, differential is given by composition with differentials of P · and A·. Then the morphism in the category C(A) of complexes are the closed degree-zero morphisms, while morphisms in H(A) are zero’th cohomology of the complex Hom(P ·,A·)). A K-projective left resolution of a complex A· is a quasi-isomorphism P · → A· with K-projective P ·. We can use K-projective or K-injective resolutions to calculate the right and left derived functors. In [Spa88] Spaltenstein showed that: (1) Let R be an associative ring with unit and A the category of right R modules. Then any complex of objects of A admits a K-projective and K-injective resolution. (2) Let O be a sheaf of rings on a topological space X and A the category of sheaves of O-modules. Then any complex of objects of A admits a K-injective resolution.

5.2. The six functors formalism.

Let (X, RX ) be a ringed space and F1, F2 sheaves on X. Then F1⊗F2 and Hom(F1, F2) are sheaves of abelian groups on X.

Proposition 5.5. Let X be a topological space, R a sheaf of rings with unit on X. Then any sheaf of R-modules can be embedded into an injective sheaf of R-modules.

Proof. The proof is based on Godement’s method. Let F be a sheaf of R-modules. For any x ∈ X we can construct a monomorphism FX → I(x) where I(x) is injective over

Rx. We deﬁne sheaf Y I(U) := I(x). x∈U HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 47

It is injective and F is its subsheaf.

A sheaf F1 is flat if functor F1 ⊗ (−) is exact. F1 is flat if and only if the stalk of F1 at any point of X is flat. As any sheaf on X is a quotient of a flat sheaf, we have functor

L − − F1 ⊗ (−): D (ShRX ) → D (ShZX ). As any sheaf on X is a subsheaf of an injective sheaf, we also get

+ + R Hom(F1, −): D (ShRX ) → D (ShZX ). Let f : X → Y be a morphism of topological spaces and let F be a sheaf on X. The direct image of F, i.e the sheaf f•(F) on Y is deﬁned as

−1 f•(F)(V ) = F(f (V )).

Let (f, ϕ):(X, RX ) → (Y, RY ) be a morphism of ringed spaces, so that ϕ: RY → f•(RX ) is a morphism of sheaves of modules. Then for any F ∈ ShRX the sheaf f•(F) inherits via ϕ a structure of RY -module so we have a functor f• : ShRX → ShRY . It is left exact. To construct its right derived functor we use Proposition 5.5 and get

+ + Rf• : D (ShRX ) → D (ShRY ).

When Y is a point and RY = Z, we get the derived functor

+ + RΓ: D (ShRX ) → D (Ab).

Theorem 5.6. Let Φ: ShRX → ShZ be the forgetful functor (of the structure of RX - S module). Then the functors RΓ and R(Γ ◦ Φ) are naturally isomorphic. If X = Ui is q i an open covering with H (Ui1 ∩ ... ∩ Uip , F) = 0 for all q > 0, p ≥ 1 then H (X, F) can be computed via the Cechˇ complex.

• Functor f• : ShRX → ShRY has left adjoint, the inverse image functor f . For a sheaf G on Y f •(G) is a sheaf associated to the presheaf

U 7→ lim G(V ) f(U)⊂V where the limit is taken over all open subsets V ⊂ Y containing f(U). If G was a sheaf of modules, not just abelian groups, then f •(G) has a structure of • f (RY )-module. The morphism RY → f•RX given by the map (f, ϕ) of ringed spaces • yields f (RY ) → RX . It allows us to deﬁne

∗ • • f (G) = RX ⊗f (RY ) f (G). 48 AGNIESZKA BODZENTA

Functor f • is exact (the stalk of f •F at y is the stalk of F at f(y)), while f ∗ is right exact. Let now f : X → Y be a morphism of locally compact topological spaces and F a sheaf on X. For V ⊂ Y open put

−1 f!(F)(V ) = {s ∈ F(f (U)) | supp (s) → V is proper}.

Recall, that morphism is proper if the preimage of any compact set is compact. Then f!F is a sheaf on Y , the direct image with compact support of F. It is a subsheaf of f·F and functor

f! : ShRX → ShRY is left exact. Indeed, let 0 → F1 → F2 → F3 be an exact sequence of sheaves on X. We need to show that f!F1 is the kernel of f!F2 → f!F3. It is clear, as any section of f·F2 which becomes zero as a section of f·F3 is a section of f·F1. Considering injective resolutions, we obtain functor

+ + Rf! : D (ShRX ) → D (ShRY ).

When Y is a point, we recover Γc(X, F), global sections with compact support. Let us now assume that h: W → X is an embedding of a locally closed subset. For a sheaf E on W and U ⊂ X open, we have

h!E(U) = {s ∈ E(U ∩ W ) | supp (s) ⊂ U is closed }.

Lemma 5.7. Let x ∈ X be a point which does not lie in W . Then the stalk h!Ex is zero.

Proof. Let U ⊂ X be an open neighbourhood of x and let s ∈ h!E(U) be a section. Then s is a section of E(W ∩ U) and supp s ⊂ U is closed. Hence, there exists U 0 ⊂ U such 0 0 that x ∈ U and U ∩ supp s = ∅. Hence, any germ is zero at x.

As the stalks of h!E at x ∈ W are equal to the stalks of E, h! is an exact functor. Functor h induces an equivalence of Sh with the full subcategory ShW of Sh ! RW RX RX consisting of sheaves whose stalks outside W are zero. The inverse of this equivalence is given by h∗. For a sheaf F on X we deﬁne a sheaf F W which, for any U ⊂ X is

F W (U) = {s ∈ F(U) | supp (s) ⊂ W }.

Then F W ∈ ShW and we put RX h!F := h∗F W .

Lemma 5.8. Functor h! is right adjoint to h!. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 49

! Proof. We ﬁx a map h!h F → F and show that the induced morphism

! ! (13) HomW (E, h F) → HomX (h!E, h!h F) → HomX (h!E, F)

is an isomorphism for any E ∈ ShRW , F ∈ ShRX . By deﬁnition ! h!h F(U) = {s ∈ F(U) | supp (s) ⊂ W }

! so we have a canonical map h!h F → F. Moreover, it follows that for any sheaf G whose ! stalks outside W are zero, any morphism G → F factors uniquely via h!h F. In particular, the second map in (13) is an isomorphism.

The ﬁrst map in (13) is an isomorphism because h! is an equivalence of categories Sh → ShW . RW RX

As f! is in general not right exact, it cannot have right adjoint on the abelian level. However, there exists ! + + f : D (ShRY ) → D (ShRX ) such that ! Hom(Rf!F, G) ' Hom(F, f G). Functor f ! is the inverse image with compact support. For f !G we should have

! ! RΓ(U, f G) = R Hom(RU , f G) = R Hom(f!RU , G).

Unfortunately, U 7→ R Hom(f!RU , G) is not a sheaf on X. We need to replace the constant sheaf RX by its ﬂat resolution:

0 → Ln → ... → L0 → 0.

· ! · · Then, for an injective complex G , f (G ) is the total complex of Hom(f!L, G ).

For a ﬁnite-dimensional, locally compact topological space X with a constant sheaf ZX

(or kX for some ﬁeld k) let

· ! (14) DX = f (Z), · where f : X → pt. Object DX is the dualizing complex on X. By adjunction, we have · Hom(Γc(X, F), Z) ' Hom(F,DX ). · Exercise 5.9. If X is a smooth manifold with the constant sheaf kX then DX = ! i kX [dim X]. Show that the f! a f adjunction implies Poincare duality for X, Hc(X) ' Hn−i(X)∨. 50 AGNIESZKA BODZENTA

b For F ∈ D (ShZX ) we put · DX (F) := R Hom(F,DX ).

Object DX (F) is the Verdier dual of F. As we always have F 7→ Hom(Hom(F, G)), we have a natural transformation

αX : IdDb(Sh ) → DX DX . ZX α is generally not an isomorphism. F However, let us assume that X is a stratified space X = s∈S S whose strata are non- singular topological spaces. A sheaf F on X is constructible with respect to S if for any · S ∈ S and its embedding iS : S → X, iS(F) is locally constant on S. (Recall that a constant sheaf is a sheafification of a constant presheaf whose all restriction morphisms are the identity morphisms. A sheaf is locally constant if every point has an open neighbourhood such that restriction to this neighbourhood is a constant sheaf). Let b b DS (ShZX ) ⊂ D (ShZX ) be the full subcategory consisting of complexes with constructible · b b cohomology. Then DX ∈ DS (ShZX ) and α is an isomorphism when restricted to DS (ShZX ).

DX |Db (Sh ) is the Verdier duality. S ZX Let f : X → Y be a continuous morphism of locally compact and ﬁnite-dimensional spaces. For F ∈ D(ShRX ) and G ∈ D(ShRY ) we have

· L · L · f (G1 ⊗ G2) ' f (G1) ⊗ f (G2),

R Hom(F1 ⊗ F2, F3) ' R Hom(F1,R Hom(F2, F3)), · Rf·R Hom(f G, F) ' R Hom(G, Rf·F), L · L Rf!(F ⊗ f G) ' Rf!F ⊗ G, ! · ! f R Hom(G1, G2) ' R Hom(f G1, f G2), ! ! · · · · f DY (G) = f R Hom(G,DY ) ' R Hom(f G,DX ) ' DX (f G),

The proofs consist of checking equalities for sheaves and then considering appropriate resolutions.

5.3. Recollement for a closed subset.

Let X be a locally compact ﬁnite-dimensional space and i: F → X an embedding of a closed subset. Then i• = i!, so the functor

i• : ShZF → ShZX HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 51 has left and right adjoints

• ! i , i : ShZX → ShZF .

Let now j : U → X be an embedding of an open subset. Then, for any sheaf F ∈ ShZX U • • ! F is isomorphic to j•j F, hence j = j . It follows that functor

• j : ShZX → ShZU has left and right adjoints

j!, j• : ShZU → ShZX .

• • ! Functors i•, i , j and j! are exact while i and j• are left exact.

The sheaf j!E is a sheaﬁﬁcation of the presheaf ( E(V ) if V ⊂ U V 7→ 0 otherwise.

• Let us now assume that U = X \F is the complementary open set to F . Then j i• = 0, • ! hence also i j! = 0 and i j• = 0. Given a sheaf F on X sequence

• • 0 → j!j F → F → i•i F → 0

• is exact, as the stalks of j!j F are either zero or equal to the stalks of F. Precisely where • the stalks are zero, the stalks of i•i F are equal to the stalks of F. • On the other hand, j•j F(V ) = F(U ∩ V ), for any V ⊂ U closed. Hence, the kernel of • the morphism of (pre)sheaves F → j•j F on V consists of s ∈ F(V ) such that supp s ⊂ F . It follows that sequence

! • 0 → i•i F → F → j•j F

is exact for any F ∈ ShZX . • An injective sheaf I is in particular ﬂabby, i.e. the map I → j•j I is epi and we have

! • 0 → i•i I → I → j•j I → 0.

Considering bounded below complexes of injective objects we get distinguished triangles

• · · • · • · j!j I → I → i•i I → j!j I [1], (15) ! · · • · ! · i•i I → I → j•j I → i•i I [1].

The fact that the triangles are distinguished follows from the fact that they are term-wise exact sequences, see Proposition 3.27. 52 AGNIESZKA BODZENTA

+ · Let now F be any object of D (ShZX ) and I its injective resolution. Then triangles (15) can be read as

• • • j!j F → F → i•i F → j!j F[1], ! • ! i•Ri F → F → Rj•j F → i•Ri F[1].

In the ﬁrst triangle we all four functors are exact, so we can just apply them to any complex termwise and get derived functor. In the second triangle i! should be applied to an injective resolution (as it is) while j• to a ﬂabby one (which again is the case).

Functors i•, Rj• and j! considered as functors between derived categories are fully • • • faithful, because the compositions with the adjoints i i•, j Rj•, j j! are isomorphic to identity, which can be checked term-wise (on complexes of injective objects, if necessary) cf. [Huy06, Remark 1.23]. + + Now, for any A ∈ D (ShZU ) and B ∈ D (ShZF )

Hom(j!A, i•B) ' 0 ' Hom(i•B, Rj•A).

Indeed, both i• and j• map injective objects to injective objects hence the above Hom- spaces can be computed term-wise using injective resolution of B, resp. A.

Exercise 5.10. A functor F : A → B of abelian categories with exact left adjoint maps injective objects to injective objects.

Thus, we have proved

Proposition 5.11. Let F ⊂ X be a closed subset of a topological space with the + complementary open set U. Then category D (ShZX ) admits a recollement o o + + + D (ShZF ) / D (ShZX ) / D (ShZU ) o o

5.4. Perverse sheaves.

Let now X be a topological space with a ﬁxed sheaf of rings OX . We suppose that X is partitioned by a ﬁnite family S = {S} of non-empty locally closed subspaces. We assume that • The closure S of any stratum is a union of strata.

•O X is the constant sheaf R on X, where R is a left Noetherian ring. • The components of a given strata are topological manifolds, all of the same dimension; if S is contained in the closure T of a stratum T , then dim S < dim T . HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 53

• Let iS : S → X be the inclusion map of a stratum S into X. The direct image

functor j∗ from OS-modules to OX -modules has finite homological dimension l (there exists NS such that R iS∗F = 0 for l > NS and any F ∈ ShOS ). An example of such stratification is Whitney stratification: if S is of dimension i and

T of dimension j and sequences of points si ∈ S, ti ∈ T converging to t ∈ T are such that secant lines between si and ti converge to a line L and the sequence of tangent i-planes to si ∈ S converge to an i-plane then L is contained in T . In general, we shall consider X to be a complex analytic or algebraic variety stratiﬁed by (real or complex) subvarieties. We consider [ Xi = S. dim S≤i

Then Xi ⊂ X is a closed variety. The assumptions on the stratification imply that the recollement given by Proposition 5.11 restricts to the bounded and bounded below derived category of complexes of sheaves with S -constructible cohomology, which we shall denote b + by DS (X) and DS (X). It follows that we have recollements o o + + + (16) DS (Xi−1) / DS (Xi) / Dc (Si) o o b where Si is the union of strata of dimension i. Analogously for DS (X). + Category Dc (Si) of complexes with locally constant cohomology has the standard + ≤p + ≥p+1 t-structure and its shifts (Dc (Si) , Dc (Si) ) for p ∈ Z. We fix perversity function p: S → Z + and define the category of perverse sheaves MS (p, X) as the t-structure on DS (X) glued + along recollements (16) from the shifts by p(S) of the standard t-structure on Dc (S). For p = 0 we get the standard t-structure

ShOX 'MS (0,X). Proposition 5.12. The aisles of the perverse t-structure are:

p + ≤0 + ∗ + ≤p(S) DS (X) = {K ∈ DS (X) | iSK ∈ Dc (S) ∀S ∈ S }, p + ≥0 + ! + ≥p(S) DS (X) = {K ∈ DS (X) | iSK ∈ Dc (S) ∀S ∈ S }. Proof. We proceed by induction on the dimension of X that The case dim X = 0 is clear. Let us now assume that the above description holds for X of dimension < n and let X be a stratiﬁed topological space of dimension n.

The embedding in−1 : Xn−1 → X is closed and Sn ⊂ X is open. Hence, for any S ∗ ! of dimension n, iS ' iS. For T ∈ S with dim T < n let kT : T → Xn−1 denote the 54 AGNIESZKA BODZENTA embedding. Then, the deﬁnition of the glued t-structure and the inductive hypothesis imply:

p + ≤0 + ∗ + ≤p(S) ∗ p + ≤0 DS (X) ={K ∈ DS (X) | iS(K) ∈ Dc (S) ∀S dim S = n, in−1K ∈ DS (Xn−1) } = + ∗ + ≤p(S) ={K ∈ DS (X) | iS(K) ∈ Dc (S) ∀S dim S = n, ∗ ∗ + ≤p(S) kT in−1K ∈ Dc (T ) ∀T dim T < n}, p + ≥0 + ∗ + ≥p(S) ! p + ≥0 DS (X) ={K ∈ DS (X) | iS(K) ∈ Dc (S) ∀S dim S = n, in−1K ∈ DS (Xn−1) } = + ! + ≤p(S) ={K ∈ DS (X) | iS(K) ∈ Dc (S) ∀S dim S = n, ! ! + ≥p(T ) kT in−1K ∈ Dc (T ) ∀T dim T < n}.

For S ∈ S with dim S < n the composition of the embedding kS : S → Xn−1 with in−1 is ∗ ∗ ∗ ! ! ! the embedding iS. Hence iS = kS ◦ in−1 and iS = kS ◦ in−1. The statement follows.

Let us now assume that OX = kX is the constant sheaf with value k, for some ﬁeld k. + op ' + Recall that the dualising complex (14) yields Verdier duality DX : DS (X) −→DS (X). · For a smooth manifold S we have DS = kS[dimR S].

Exercise 5.13. Let F : C → D be an equivalence of triangulated categories. Show that a t-structure (C≤0, C≥1) on the category C induces a t-structure on D.

Exercise 5.14. Show that a t-structure on category C induces a t-structure on the category Cop.

+ Proposition 5.15. Verdier duality DS on Dc (S), for a smooth manifold S + ≤p + ≥p+1 with dimR(S) = n, maps the t-structure (Dc (S) , Dc (S) ) to the t-structure + ≤−p−n + ≥−p−n (Dc (S) , Dc (S) ).

+ ≥p+1 + ≤p + Proof. We know that (DS(Dc (S) ), DS(Dc (S) ) is a t-structure on Dc (S). We check + ≤p + ≥−p−n + ≥q + ≤−q−n that DS(Dc (S) ) ∈ Dc (S) and DS(Dc (S) ) ∈ Dc (S) . + p,q q − Let A ∈ Dc (S). Spectral sequence with E2 = Ext (H p(A), kS[n]) converges to i i p,q H (DS(A)) ' Ext (A, kS[n]). As E2 are Ext-groups of vector spaces they are non-zero only when they are Hom’s. It follows that the spectral sequence degenerates and

i i i+n −i−n H (DS(A)) ' Ext (A, kS[n]) ' Ext (A, kS) ' Hom(H (A),S).

i j i Then, if H (A) = 0 for i > p then H (DS(A)) = 0 for j < −p − n. Similarly, if H (A) = 0 j for i < q then H (DS(A)) = 0 for j > −q − n.

∗ Proposition 5.16. For perversity p: S → Z deﬁne p (S) = −p(S) − dimR(S). Then the image under Verdier duality of the t-structure for perversity p is the t-structure for perversity p∗. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 55

p + ≥0 p + ≤−1 + Proof. We know that (DX ( DS (X) ), DX ( DS (X) ) is a t-structure on DS (X). It p + ≤0 p∗ + ≥0 thus suﬃces to check that F ∈ DS (X) if and only if DX (F) ∈ DS (X) . ∗ ! By Proposition 5.12 and the fact that Verdier duality exchanges iS with iS, we have

p + ≤0 F ∈ DS (X) ⇔ ∗ + ≤p(S) ⇔ iS(F ) ∈ Dc (S) ∀ S ∈ S ⇔

∗ + ≥−p(S)−dimR(S) ⇔ DS(iS(F )) ∈ Dc (S) ∀ S ∈ S ⇔

! + ≥−p(S)−dimR(S) ⇔ iSDX F ∈ Dc (S) ∀ S ∈ S ⇔ p∗ + ≥0 ⇔ DX (F) ∈ DS (S) .

Corollary 5.17. Assume that X is a topological space with a stratiﬁcation S whose all strata have even dimension over R. Let 1 p (S) = − dim (S). 1/2 2 R

Then Verdier duality restricts to a duality on the category MS (p1/2,X).

∗ 1 Proof. It suﬃces to check that p (S) = −p(S) − dimR(S) = 2 dimR(S) − dimR(S) = 1 − 2 dimR(S) = p(S).

The perversity p1/2 is often called the middle perversity. It exists whenever X is a complex manifold stratiﬁed by complex submanifolds.

An open set U ⊂ X has stratification induced from S Let Un ⊂ U be the intersection of U with the open strata. The intersection complex IC(U) is defined as the intermediate extension j!∗ of kUn [−1/2 dimR(Un)]. These were first described as explicit complexes by Goresky and MacPherson [GM80].

5.4.1. Subdivision of stratification. Let a stratification T be a subdivision of a stratification S (each stratum of S is the union of several strata of T ). Then we have an + + embedding of categories DS (X) → DT (X). Let p: S → Z, q : T → Z be perversities such that p(S) ≤ q(T ) ≤ p(S) + dim S − dim T whenever T ⊂ S.

+ Proposition 5.18. Under the above assumptions the t-structure of perversity q on DT (X) + + + induces the t-structure of perversity p on DS (X) (under the embedding DS (X) ⊂ DT (X)). In particular, any p-perverse sheaf is also q-perverse. 56 AGNIESZKA BODZENTA

Consider now the case when X is a complex variety, S is a stratiﬁcation of X by non- singular subvarieties and the perversity p depend only on the dimension of S ∈ S. We also assume that

(17) 0 ≤ p(n) − p(m) ≤ m − n for n ≤ m.

Under these assumptions the subdivision of stratification is compatible with t-structures b and we can define a t-structure on the triangulated category Dc(X) of complexes with cohomology constructible with respect to some stratification. The following proposition describes perverse sheaves.

b Proposition 5.19. For F ∈ Dc(X) the following conditions are equivalent: (1) F is a perverse sheaf, (2) Any irreducible submanifold S ⊂ X contains a Zariski open U ⊂ S such that, for i · i ! j : U → X, H (j F) = 0 for i > p(dimR(S)), H (Rj F) = 0 for i < p(dimR(S)).

5.4.2. Simple objects. In case the perversity depends only on the dimension of strata and satisﬁes (17) the category MS (p, X) admits the following description

Proposition 5.20. The category MS (p, X) is Artinian. Its simple objects are of the p form L(S, E) = (iS)!∗E[p(S)] where S ∈ S and E is an irreducible locally constant sheaf of vector spaces on S. In particular, if all strata are simply connected, simple objects on

MS (p, X) are in one-to-one correspondence with strata S ∈ S.

If we do not ﬁx a stratiﬁcation then the category of perverse shaves is Noetherian.

However, if p = p1/2 the Verdier duality implies that the category is also Artinian. In particular, if X is a complex variety we obtain the following proposition

Proposition 5.21. The category M(p1/2,X) of perverse sheaves with the cohomology that are constructible with respect to some stratiﬁcation with non-singular varieties is Artinian. Its simple objects are of the form L(S, E) where S ⊂ X is a non-singular irreducible variety and E is an irreducible locally constant sheaf on S. L(S, E) ' L(S0, E 0) 0 0 0 if and only if S ∩ S is dense in S and S and E|S∩S0 = E |S∩S0 .

5.5. Gluing of perverse sheaves.

Finally, we consider a smooth complex variety X, its closed subvariety Z deﬁned by f = 0 and the complement U = X \ Z. We consider the self-dual perversity p1/2 and discuss how to ’glue’ a perverse sheaf on Z and on U to get a perverse sheaf on X. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 57

We denote B = C1, B∗ = C1 \{0} and Be∗ the universal covering of B∗, p: Be∗ → B∗ the covering map. We have the following commutative diagram:

i j Z / X o U

f f f {0} / BBo ∗

Denote by Xe∗ the ﬁber product π Xe∗ / X

f Be∗ / B + For F ∈ Dc (X) deﬁne · · ψf (F) = i Rπ·π F.

+ Proposition 5.22 (SGA VII). ψf extends to the nearby cycles functor ψf : Dc (X) → + Dc (Z). If F ∈ M(p1/2,X) then ψf (F) ∈ M(p1/2,Z).

∗ · Since π(Xe ) ⊂ U, ψf F depends only on j F. Hence

ψf : Dc(U) → Dc(Z). · The unit Id → Rπ·π gives a morphism · θ : i → ψf .

The vanishing cycles functor is a functor ϕf : Dc(X) → Dc(Z) such that for any F ∈

Dc(X) · θF canF · i F −→ ψf F −−−→ ϕf F → i F[1] is distinguished.

Proposition 5.23. If F ∈ M(p1/2,X) then ϕf (F) ∈ M(p1/2,Z).

We have the complete turn t: Be∗ → Be∗ and p ◦ t = p. It determines τ : Xe∗ → Xe∗ with · · · π ◦ τ = τ. The unit Id → Rτ·τ gives a morphism λ: Rπ·π → Rπ·π which induces

T : ψf → ψf .

Morphism T , called monodromy satisﬁes T ◦ θ = θ, hence it induces the monodromy action on vanishing cycles

T : ϕf → ϕf . Since t is an isomorphism, T is an automorphism of functors. 58 AGNIESZKA BODZENTA

The deﬁnition of ϕf gave us morphism can: ψf → ϕf . We also have var: ϕf → ψf deﬁned as a lift to ϕf of T − Id: ψf → ψf (its composition with θ is zero, as T ◦ θ = θ). Then, we have

var ◦ can = T − Id.

We deﬁne the category Glue(T,U). Objects are quadruples (G, H, a, b) where G ∈

M(U, p1/2), H ∈ M(Z, p1/2), a: ψf (G) → H, b: H → ψf (G) are such that b ◦ a = TG − Id.

· Theorem 5.24 (Beilinson). Functor M(X, p1/2) → Glue(T,U), F 7→ (j F, ϕf F, canF , varF ) is an equivalence of categories.

Consider the simplest example X = B, Z = {0}, U = B∗, f(z) = z and the category of perverse sheaves with respect to the fixed stratification by Y and U. Then M(U) is the category of locally constant sheaves on U, i.e. its is equivalent to the category of pairs (V,A) where V is a finite dimensional vector space and A: V → V is an invertible linear operator (monodromy). The category M(Z) is the category of vector spaces. Functor ψf maps (V,A) to V . The monodromy operator is A − Id. It follows that the category of perverse sheaves is equivalent to the category whose objects are diagrams of vector spaces

F (18) Φ / Ψ o E such that Id +FE is invertible. n Another example is X = C stratiﬁed by XI = {(x1, . . . , xn) | xi = 0 for i ∈ I, xi 6= 0 for i∈ / I}, where I ⊂ {1, . . . , n} is any subset. The category of perverse sheaves is equivalent to the category whose objects are ﬁnite-dimensional vector spaces VI and linear maps:

EI,i : VI → VI∪{i},FI,i : VI∪{i} → VI , satisfying

EI∪{j},iEI,j = EI∪{i},jEI,i,

FI,jFI∪{j},i = FI,iFI∪{i},j,

EI\{j},iFI\{j},j = FI∪{i}\{j},jEI,i,

FI,iEI,i + Id is invertible. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 59

5.6. Perverse sheaves on hyperplane arrangements.

M. Kapranov and V. Schechtman in [KS16] considered the case X = Cn. The stratiﬁcation they considered came from an arrangement H of linear hyperplanes in Rn.

Let H = {fH = 0}, for all H ∈ H. L = LH is the poset of ﬂats of H, i.e. linear subspaces T H∈I H for various subsets I ⊂ H. We assume that L contains {0}. The stratiﬁcation of Cn is given by

o LC = LC \H+L HC. We consider the category M(X, H) of perverse sheaves with middle perversity. n n Each x ∈ R gives a sign vector (sgnfH (x))H∈H. Level sets of this vector subdivide R 0 into locally closed subsets, faces. We denote by C = CH the poset of faces with C C if C0 ⊂ C. An ordered triple (A, B, C) of faces is collinear if there are point a ∈ A, b ∈ B, c ∈ C such that b lies in the straight line segment [a, c].

A perverse sheaf gives a vector space EC for every C ∈ C. If A B then we have the restriction map γA,B : EA → EB, as every open set which contains a point of A has an open subset which is contained in B. Then, γ yield a representation of a quiver of C with arrows A → B if A B. On the other hand, we have the Verdier duality which preserves M(X, H). The restriction morphisms for the Verdier dual give maps δB,A : EB → EA which yield a representation of the opposite quiver.

Theorem 5.25. The category M(X, H) is equivalent to the category of representations of the double quiver of the poset C satisfying the following conditions

• For any A B we have γA,BδB,A = IdEB . It allows us to deﬁne for any A, B ∈ C

the transition map ϕA,B = γC,BδA,C : EA → EB where C is any cell less than A and B.

• If (A, B, C) is a collinear triple of faces then ϕA,C = ϕB,C fA,B.

• If C1, C2 are faces of the same dimension d lying in the same d-dimensional

subspace, on the opposite sides of a (d − 1)-dimensional face D then ϕC1,C2 is an isomorphism.

Let us again consider X = C. Then we have 3 faces in C: −, 0 and +. The category M(X, H) is equivalent to the category whose objects are diagrams of vector spaces

δ− γ+ (19) E / E / E − o 0 o + γ− δ+ 60 AGNIESZKA BODZENTA such that

γ−δ− = IdE− , γ+δ+ = IdE+ ,

γ+δ− is an isomorphism, γ−δ+ is an isomorphism.

Exercise 5.26. Show that the two descriptions of the category of perverse sheaves on C1 stratiﬁed by a point and its complement are equivalent. (1) Show that the category with objects (19) is equivalent to the category B whose

objects are (E0,P+,P−) where P+,P− : E0 → E0 are idempotents such that

P− : Im(P+) → Im(P−) and P+ : Im(P−) → Im(P+) are isomorphisms.

(2) For (E0,P+,P−) ∈ B put Φ = ker(P−), Ψ = Im(P+), F = P+, E = P− − Id. Show that it deﬁnes a functor Υ from the category B to the category with objects (18). ! ! 0 0 0 E (3) Given an object (18) let E0 = Φ ⊕ Ψ, P+ = and P− = . F 1 0 1 Show that this way we get a functor quasi-inverse to Υ.

6. Derived categories of coherent sheaves

This section is mainly based on [Huy06].

6.1. Crash course on spectral sequences.

A spectral sequence in an abelian category A is a collection of objects p,q n p,q p,q p+r,q−r+1 (Er ,E ), n, p, q, r ∈ Z, r ≥ 1 and morphisms dr : Er → Er such that p+r,q−r+1 p,q (i) dr ◦ dr = 0, r,q 0 p+·r,q−·r+1 (ii) Ep+1 is isomorphic to H (Er ), the isomorphisms are part of the data p,q p−r,q+r−1 (iii) For any (p, q) there exists r0 such that dr = dr = 0 for r ≥ r0. In p,q p,q p,q particular, Er ' Er0 for all r ≥ r0. This object is called E∞ . (iv) There is a decreasing ﬁltration ... ⊂ F p+1En ⊂ F pEn ⊂ ... ⊂ En such that T p n S p n n p,q p p+q p+1 p+q F E = 0, F E = E and E∞ ' F E /F E . We write it as

p,q p+q Er ⇒ E . We shall mostly encounter spectral sequences given by double complexes K·,·. They have i,j i,j i+1,j i,j i,j i,j+1 dI : K → K and dII : K → K satisfying

2 2 dI = 0, dII = 0, dI dII + dII dI = 0.

· n L i,j The total complex K is K = i+j=n K with d = dI + dII . HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 61

l n L n−j,j n−l,l The total complex has a ﬁltration F K = j≥l K . If K = 0 for |l| >> 0 then there is a spectral sequence

p,q p q ·,· p+q · E2 = HII HI (K ) ⇒ H (K ).

+ + + + Proposition 6.1. Let F1 : K (A) → K (B) and F2 : K (B) → K (C) be two exact functors. Suppose that A and B contain enough injectives and that the image under F1 of · a complex I of injective objects in A is cntained in an F2-adapted triangulated subcategory · + KF2 . Then for any A ∈ D (A) there exists a spectral sequence

p,q p q · p+q · E2 = R F2(R F1(A )) ⇒ R (F2 ◦ F1)(A ).

Proof. We take the Cartan-Eilenberg resolution of A·, i.e. we resolve by injective objects · the cohomology and every term of A . Our double complex is F1 of this resolution. Then p q ·,· p q · HII HI (K ) ' R F (H (A )). Let A·, B· be complexes in D(A) with B· bounded below. If A has enough injectives then there exists a spectral sequence

p,q · q · · · E2 = Hom(A ,H (B )[p]) ⇒ Hom(A ,B [p + q]).

The ﬁrst functor we use is the identity, the second Hom(A·, −). Looking directly at a double complex induced by an injective resolution of B· we also get p,q −q · · · · E2 = Hom(H (A ),B [p]) ⇒ Hom(A ,B [p + q]).

6.2. Preliminaries.

A scheme X is quasi-separated if the diagonal map ∆: X → X × X is quasi-compact, i.e. the inverse image of any quasi-compact open set is quasi-compact.

By a sheaf on X we mean a sheaf of OX -modules. We denote by Sh(X) the category of sheaves of OX modules and by QCoh(X) the category of quasi-coherent sheaves.

Let D(Sh(X)) denote the unbounded derived category and DQCoh(X)(Sh(X)) its full subcategory of objects with quasi-coherent cohomology sheaves. By Perf(X) we denote the full subcategory of D(Sh(X)) whose objects are perfect, i.e. locally isomorphic to a bounded complex of vector bundles. If X is quasi-compact and quasi-separated then

Perf(X) consists precisely of compact objects of DQCoh(X)(Sh(X)), i.e. objects such that morphisms from them commute with arbitrary direct sums. If X is quasi-compact and separated then the functor D(QCoh(X)) → D(Sh(X)) ' induces an equivalence D(QCoh(X)) −→DQCoh(X)(Sh(X)). 62 AGNIESZKA BODZENTA

Lemma 6.2. If G F is an OX -module homomorphism from a quasi-coherent sheaf G onto a coherent sheaf F on a noetherian scheme X, then there exists a coherent subsheaf G0 ⊂ G such that the composition G0 → G → F is still surjective.

Proof. The statement is clear for modules. We cover X by finitely many affine open, find 0 a coherent subsheaf HU for any U and take as G the coherent subsheaf of G such that 0 HU ⊂ G |U .

Proposition 6.3. If X is noetherian we have an equivalence D−(Coh(X)) −→' − DCoh(X)(QCoh(X)). This remains true if we replace ’−’ by ’b’.

Proof. Let ... Gn → ... → Gm → 0 be a bounded above complex of quasi-coherent sheaves. Assume that Gi are coherent for i > j. We have surjective morphisms Gj → Imdj and ker(dj) → Hj(G·). Imdj is a subsheaf of Gj+1, hence it is coherent and, by assumption, j · j j j j j so is H (G ). Hence, there exist coherent subsheaves G1 ⊂ G and G2 ⊂ ker d ⊂ G . Let G0j be the coherent subsheaf of Gj generated by them. Let also G0j−1 be the ﬁber j−1 0j j j−1 0j 0j−1 product G ×Gj G . Then complex with G and G exchanged by G and G is quasi-isomorphic to the original complex and has coherent terms for i ≥ j.

Proposition 6.4. Let X be a Noeatherian separated scheme. Then Perf(X) = Db(Coh(X)) implies that X is regular. If X is of ﬁnite dimension the converse is also true.

We shall be mostly interested in

Db(X) = Db(QCoh(X)).

Category Coh(X) does not contain enough injective objects so in order to compute derived functors we pass to the category QCoh(X). If X is projective over a field k then cohomology Hi(X,F ) of any coherent sheaf F is finite dimensional. This can be used to show that Exti(E,F ) is finite dimensional for any two coherent sheaves E,F . Then the spectral sequence

p,q p · q · p+q · · E2 = Ext (E , H (F )) ⇒ Ext (E ,F ) implies that Exti(E·,F ·) are ﬁnite dimensional for any E·,F · ∈ Db(X).

Deﬁnition 6.5. The support of a complex F · in Db(X) is the union of the support of all its cohomology sheaves.

· b · Lemma 6.6. Suppose F ∈ D (X) and supp F = Z1 t Z2, where Z1,Z2 ⊂ X are disjoint · · · closed subsets. Then F ' F1 ⊕ F2 with supp Fi ⊂ Zi, for i = 1, 2. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 63

Proof. We proceed by induction on the length n of the complex. If n = 1 the case is clear. Let F · be a complex of length at least two. Suppose m is minimal such that Hm(F ·) 6= 0. m · Then H = H (F ) can be decomposed as H = H1 ⊕ H2 with supp Hi ⊂ Zi. Consider distinguished triangle

H[−m] → F · → G· → H[−m + 1] · · · given by the truncation in the standard t-structure. By inductive hypothesis G = G1 ⊕G2 · with supp Gi ⊂ Zi. We use spectral sequence

p,q −q · · E2 = Hom(H (G1),H2[p]) ⇒ Hom(G1,H2[p + q])

i · −q · to conclude that Ext (G1,H2) = 0, for all i ∈ Z. Indeed, H (G1) and H2 are coherent sheaves with disjoint support so there are no morphisms between them (we can take injective resolution of push-forward of H2 to X\Z2. As the embedding j of the complement of Z2 to X is open, functor j∗ maps injective objects to injective objects. This way we get an injective resolution of H2 whose support is disjoint with the support of cohomology of · G2, so all Ext-groups vanish.) · Similarly we show that there are no Ext-groups between G2 and H1. So the map · · · · G1 ⊕ G2 ' G → H[−m + 1] ' H1[−m + 1] ⊕ H2[−m + 1] is diagonal. It follows that F · · · is isomorphic to F1 ⊕ F2 , for Fi deﬁned as the ﬁber of Gi → Hi[1 − m]: · · Hi[−m] → Fi → Gi → Hi[−m + 1].

A triangulated category D is decomposed into triangulated subcategories D1, D2 if D1 and

D2 are both non-trivial and D admits semi-orthogonal decompositions D = hD1, D2i = hD2, D1i. Category D is indecomposable if it cannot be decomposed.

Proposition 6.7. Let X be a noetherian scheme. Then Db(X) is an indecomposable triangulated category if and only if X is connected.

b b Proof. If X = X1 t X2 we take D1 = D (X1) and D2 = D (X2). b Assume now that X is connected and assume that D (X) is decomposed into D1 and

D2. Let OX = F1 ⊕ F2 be a decomposition of the structure sheaf. Both F1 and F2 are concentrated in zero’th cohomology, hence we can assume they are pure sheaves. As Fi is a subsheaf of OX it is a coherent ideal sheaf of some closed subscheme Xi ⊂ X. Moreover,

OX = IX1 + IX2 ⊂ IX1∩X2 and IX1∪X2 = IX1 ∩ IX2 = 0. Therefore, X = X1 ∪ X2 and

X1 ∩ X2 = ∅. As X is connected, we can assume that X = X1. Therefore OX ∈ D1. 64 AGNIESZKA BODZENTA

If x ∈ X is a closed point then sheaf Ox is indecomposable. Therefore, Ox ∈ D1 or

Ox ∈ D2. As Hom(OX , Ox) 6= 0, we know that Ox ∈ D1. · m · Let now F be an object of D2. Let m be maximal such that H (F ) 6= 0. Pick a m · closed point x in the support of H = H (F ). There exists surjection H → Ox. Applying

Hom(−, Ox[−m]) to the distinguished triangle · · · τ≤m−1F → F → H[−m] → τ≤m−1F [1] yields an isomorphism · Hom(H[−m], Ox[−m]) ' Hom(F , Ox[−m]) · · as Hom(τ≤m−1F [1], Ox[−m]) ' 0 ' Hom(τ≤m−1F , Ox[−m]) for degree reasons · b ≤m−1 · b ≤m−2 b ≥m (τ≤m−1F ∈ D (X) , τ≤m−1F [1] ∈ D (X) and Ox[m] ∈ D (X) ). This contradicts the assumption that Hom(D1, D2) = 0.

6.3. Hom and Hom.

Let A be an abelian category and A· a complex of objects of A. We have

Hom·(A·, −): K+(A) → K(Ab) which to a complex B· assigns the complex Hom·(A·,B·) with Homi(A·,B·) = L k i+k i Hom(A ,B ) and d(f) = dB ◦ f − (−1) f ◦ dA. Assume A has enough injective objects. The category of complexes of injective objects is adapted to this functor and we may deﬁne

R Hom·(A·, −): D+(A) → D(Ab).

Then Exti(A·,B·) := Hi(R Hom·(A·,B·)).

i · · · · One checks that Ext (A ,B ) ' HomD(A)(A ,B [i]). The Ext-groups depend only on the quasi-isomorphism class of A·, so in fact we have

R Hom·(=, −): D(A)op ⊗ D+(A) → D(Ab).

If A has enough projectives, we get

R Hom·(=, −): D−(A)op ⊗ D(A) → D(Ab).

If A = QCoh(X) a sheaf F ∈ QCoh(X) deﬁnes a left exact functor

Hom(F, −): QCoh(X) → QCoh(X). HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 65

As QCoh(X) has enough injective objects, we have the derived functor

R Hom(F, −): D+(QCoh(X)) → D+(QCoh(X)).

By deﬁnition Exti(F,E) = Ri Hom(F,E). If F is coherent, the deﬁnition is local:

Exti(F,E) ' Exti (F ,E ). x OX,x x x Restricting to coherent sheaves yields the functor

R Hom(F, −): D+(X) → D+(X).

If X is regular, it can be further restricted to

R Hom(F, −): Db(X) → Db(X).

Let now F · ∈ D−(X). We have

Hom·(F ·, −): K+(QCoh(X)) → K+(QCoh(X)),

i · · Y p i+p i Hom (F ,E ) := Hom(F ,E ), d = dE − (−1) dF .

The class of complexes of injective sheaves is adapted to the functor Hom·(F ·, −). As in the global case we get

R Hom(=, −): D−(QCoh(X))op ⊗ D+(QCoh(X)) → D+(QCoh(X)).

If F · is a complex of locally free sheaves then R Hom(F ·, −) can be computed as Hom(F ·, −). This is the consequence of the fact that we can calculate cohomology of Hom locally and free modules are projective. The theory of spectral sequences work here and we have:

p,q p · q · p+q · · E2 = Ext (F , H (E )) ⇒ Ext (F ,E ), p,q p −q · · p+q · · E2 = Ext (H (F ),E ) ⇒ Ext (F ,E ). We also have the trace map · · R Hom(E ,E ) → OX .

We assume that X is regular and replace E· with a complex of locally free sheaves. Then R Hom(E·,E·) 'Hom(E·,E·) and Hom0(E·,E·) ' L Hom(Ei,Ei) On each of the i i component we have the trace map Hom(E ,E ) → OX and

M i trE· = (−1) trEi . 66 AGNIESZKA BODZENTA

The dual of F · ∈ D−(QCoh(X)) is

·∨ · + F := R Hom(F , OX ) ∈ D (QCoh(X)).

· ·∨ i If F is a complex of locally free sheaves, F is the complex with terms Hom(F , OX ).

6.4. Serre duality.

Deﬁnition 6.8. Let X be a smooth projective variety of dimension n. Then one deﬁned the Serre functor SX as the composition:

ω ⊗(−) [n] D∗(X) −−−−−→DX ∗(X) −→D∗(X) where ∗ = b, +, −.

Theorem 6.9. Let X be a smooth projective variety. Then for any E·,F · there exists functorial isomorphism

· · ' · · ∨ Hom(E ,F ) −→ Hom(F , SX (E )) .

Exercise 6.10. Suppose that E and F are coherent sheaves on a smooth projective variety X of dimension n. Prove that Exti(E,F ) = 0, for i > n.

Exercise 6.11. Let C be a smooth projective curve. Prove that any object of Db(C) is L isomorphic to a direct sum Ei[i], where Ei ∈ Coh(C).

6.5. Derived functors in algebraic geometry.

Cohomology Let X be a noetherian k-scheme. Global section functor Γ: QCoh(X) → mod-k is left exact. As QCoh(X) has enough injective objects, we can form the derived functor RΓ: D+(QCoh(X)) → D+(mod-k).

Theorem 6.12 (Grothendieck). For any quasi-coherent sheaf F on a noetherian scheme X, Hi(X,F ) = 0 for i > dim(X).

Therefore, we can restrict RΓ to Db(QCoh(X)), the functor will take values in Db(mod-k).

Theorem 6.13 (Serre). If F is a coherent sheaf on a projective scheme X over a ﬁeld, then all cohomology groups Hi(X,F ) are of ﬁnite dimension. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 67

Direct image Let f : X → Y be a morphism of noetherian schemes. The direct image is a left exact functor

f∗ : QCoh(X) → QCoh(Y ). As QCoh(X) has enough injective objects, we can derive to get

+ + Rf∗ : D (QCoh(X)) → D (QCoh(Y )). · The cohomology sheaves of Rf∗(F ) are called higher direct images.

Theorem 6.14. For a quasi-coherent sheaf F on X and a morphism f : X → Y of i noetherian schemes the higher direct images R f∗F are trivial for i > dim X.

Therefore, we can restrict to bounded derived categories of quasi-coherent sheaves. Finally

Theorem 6.15. If f : X → Y is a projective (or proper) morphism of noetherian schemes i then the higher direct images R f∗(F ) of a coherent sheaf on X are again coherent.

Flabby sheaves, i.e. sheaves such that for any U ⊂ X open the restriction map F (X) → F (U) is surjective, form an adapted class for the direct image functor.

Lemma 6.16. Any injective OX -sheaf is flabby. Any flabby sheaf F on X is f∗-acyclic for any morphism f : X → Y and, moreover, f∗F is again flabby.

f g For a composition X −→ Y −→ Z we can conclude that R(g ◦f)∗ = Rg∗ ◦Rf∗. Moreover, we get Leray spectral sequence:

p,q p q · p+q · E2 = R g∗(R f∗(F )) ⇒ R (g ◦ f)∗(F ) for any F · ∈ Db(QCoh(X)). Tensor product Let F be a coherent sheaf on a projective k-scheme X. It defines right exact functor F ⊗ (−): Coh(X) → Coh(X). Any coherent sheaf admits a resolution by locally free sheaves (we use the fact that X is projective here. More generally, the resolution property holds for quasi-projective schemes over affine schemes but there are some normal 3-dimensional toric examples for which the resolution property is not known). As locally free sheaves form an adapted class for tensor product (at every point they are projective, hence flat), we get

F ⊗ (−): D−(X) → D−(X).

By deﬁnition −i L T ori(F,E) = H (F ⊗ E). 68 AGNIESZKA BODZENTA

If X is smooth of dimension n any coherent sheaf admists a locally free resolution of length n. Hence, then the functor F ⊗L (−) restricts to Db(X). In more general situation let F · be a bounded above complex of coherent sheaves on X. Deﬁne

F · ⊗ (−): K−(Coh(X)) → K−(Coh(X)),

· · i M p q i (F ⊗ E ) = F ⊗ E , d = dF ⊗ 1 + (−1) 1 ⊗ dE. p+q=i The subcategory of complexes of locally free sheaves is adapted to F · ⊗ (−) (we use two spectral sequences of the double complex and the fact that locally free sheaves are adapted to tensor product with a sheaf to show that if E· is an acyclic complex of locally free sheaves then F · ⊗ E· is also acyclic). So we can consider the derived functor

F · ⊗L (−): D−(X) → D−(X).

For a complex of locally free sheaves E· and an acyclic complex F · the complex F · ⊗ E· is acyclic, therefore the functor

(−) ⊗L (−): K−(Coh(X)) × D−(X) → D−(X) need not be derived in the ﬁrst factor and descends to the bifunctor (−) ⊗L (−) for the derived categories. If X is smooth, the functor restricts to the bounded derived categories. Computing the derived tensor product as the ordinary tensor product of complexes of locally free sheaves yields functorial isomorphisms:

F · ⊗L E· ' E· ⊗L F ·, F · ⊗L (E· ⊗L G·) ' (E· ⊗L F ·) ⊗L G·.

We also have spectral sequence

p,q q · · · · E2 = T or−p(H (F ),E ) ⇒ T or−p−q(F ,E ).

· q · · We may assume that E is a complex of locally free sheaves. Then T or−p(H (F ),E ) can be computed via the p-th cohomology of the complex Hq(F ·) ⊗ E·. Similarly, · · · · T or−p−q(F ,E ) is the p + q-th cohomology of the complex F ⊗ E which is the double complex of a double complex. The spectral sequence is then the standard spectral sequence of a double complex.

Inverse image Let f :(X, OX ) → (Y, OY ) be a morphism of ringed spaces. Then

∗ f : ShOY (Y ) → ShOX (X) HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 69

−1 −1 is a right exact functor, the composition of an exact functor f with OX ⊗f OY (−). Then ∗ L −1 − − Lf := (O ⊗ −1 (−)) ◦ f : D (Y ) → D (X) X f OY and we have a spectral sequence

p,q p ∗ q · p+q ∗ · E2 = L f (H (E )) ⇒ L f (E ). If f : X → Y is ﬂat, functor f ∗ is exact and need not to be derived.

Consider a morphism S → X. For a closed point x ∈ X denote ix : Sx → S the closed embedding of the ﬁbre over x.

Lemma 6.17. Suppose Q ∈ Db(S) and assume that for all closed points x ∈ X the ∗ b derived pull-back LixQ ∈ D (Sx) is a complex concentrated in degree zero, i.e. a sheaf. Then Q is isomorphic to a sheaf which is ﬂat over X.

Proof. We look at the spectral sequence

p,q p ∗ q p+q ∗ E2 = H (LixH (Q)) ⇒ H (LixQ). The right hand side is trivial except possibly for p + q = 0. Choose m maximal with m 0,m 0 ∗ m H (Q) 6= 0. Then there exists a closed point x ∈ X with E2 = H (LixH (Q)) 6= 0. This non-triviality survives passing to the limit of the spectral sequence, so m = 0. 0,−1 −1 ∗ 0 Also E2 = H (LixH (Q)) with x ∈ X arbitrary also survives and must, therefore, 0 p,0 p ∗ 0 be trivial. This shows that H (Q) is ﬂat over X. Hence, E2 = H (LixH (Q)) = 0. It remains to show that Q has no non-trivial negative cohomology. Again, choose m maximal among all non-trivial negative cohomology and a point x in the support of m −p,q −p ∗ q H (Q). Since all E2 = H (LixH (Q)) = 0 for q > m and p < 0, this would again m m ∗ 0 ∗ m yield the contradiction E = H (LixQ) = H (LixH (Q)) 6= 0 in the limit.

As higher derived functors of f∗ can be calculated by integrating along the ﬁbers of f i we know that R f∗(F) = 0 for any coherent sheaf F on X and i ≥ max dimXf . The following proposition tells us when higher derived functors of f ∗ vanish.

Proposition 6.18. For f : X → Y , a projective morphism between smooth projective k ∗ varieties L f (F) = 0 for any coherent sheaf F on Y and k > dimY −dimX+max dimXf .

Proof. As f is projective we have a factorization

i X / X × Y

p f # Y 70 AGNIESZKA BODZENTA where i is given by the graph of f and p is ﬂat. Let Z ⊂ Y be a closed subscheme of Y . Then · ∗ · ∗ ∗ · ∗ L f (OZ ) = L i p (OZ ) = L i Op−1(Z).

The functor i∗ is exact and hence understanding cohomology sheaves of the complex · ∗ · ∗ L i (Op−1(Z)) is the same as understanding the cohomology sheaves of i∗L i (Op−1(Z)) = L Op−1(Z) ⊗ i∗OX . The image of X in X × Y is locally a complete intersection and hence there exists a vector bundle E of rank r = dimY and s ∈ H0(X × Y,E∗) such that X = Z(s) and we have a Koszul complex

2 ... → Λ E → E → OX×Y which we can also restrict to p−1(Z). The set of zeroes of s restricted to p−1(Z) is −1 −1 Z(s|p−1(Z)) = i(f (Z)). We can estimate dim f (Z) ≤ dim Z + max dim Xf . We need one fact about Koszul complex of a not necessarily regular section. Namely, let E be a vector bundle on a smooth scheme W and s ∈ H0(W, E∗) let be a section. Put t = codimW ({s = 0}). Then

max{n | H−n(λ·E) 6= 0} = rk(E) − t.

Indeed, locally E = O⊕rk(E) and s is given by rk(E) functions. If the functions do not form a regular sequence then after permutation we can assume that the first t functions do. The Koszul resolution for s is then a tensor product of the exact Koszul complex for the first t functions and the Koszul sequence for the remaining rk(E) − t functions which is not exact. When E· is an exact complex of flat modules such that Hi(E·) = 0 for i 6= 0 and F· is any complex then the second page of a spectral sequence

pq p · q p+q · · E2 = H (E ⊗ F ) ⇒ H (E ⊗ F ) degenerates and hence E· ⊗ F· has nonzero cohomology groups in the same places as F· does. Now, let our W be p−1(Z). Then dim W = dim Z + dim X. We also know that dim Z(s|p−1(Z)) ≤ dim Z + max dim Xf . Hence,

codimW Z(s|p−1(Z)) = dim W − dim Z(s|p−1(Z)) ≥ dim X − max dim Xf .

Then, the bound for the nonzero cohomology of the Koszul complex of i∗OX when restricted to p−1(Z) is

k = rk(E) − codimW Z(s|p−1(Z)) ≤ dim Y − dim X + max dim Xf .

k ∗ We have hence proved that for structure sheaves of closed subschemes L f (OZ ) = 0 for k > dim Y − dim X + max dim Xf . The question about vanishing of higher derived HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 71 functors of f ∗ is local on Y and so we can assume that Y = SpecR for some noetherian ring R and that we have a finitely generated R-module M. Then (by theorem 7.E from Matsumura, for example) M admits a finite filtration with quotients of the form R/P for some prime ideal P ⊂ R. Let M 0 ,→ M → R/P be a short exact sequence. Then M 0 has a shorter filtration than M and we can assume by induction that the higher Tor- groups vanish both for M 0 and R/P . The long exact sequence then proves that higher Tor-functors vanish also for M. Compatibilities (1) Let f : X → Y be a proper morphism of projective schemes over a field k. For any F · ∈ Db(X), E· ∈ Db(Y ) there exists natural isomorphism, projection formula

· l · · ∗ · Rf∗(F ) ⊗ E ' Rf∗(F ⊗ Lf E ).

To prove it we present E· as a complex of locally free sheaves. (2) Let f : X → Y be a morphism of projective schemes and let E·,F · ∈ Db(Y ). Then Lf ∗(F ·) ⊗L Lf ∗(E·) ' Lf ∗(F · ⊗ E·).

∗ (3) Let f : X → Y be a projective morphism. Then Lf a Rf∗ because we can present the complex on Y as a complex of locally free sheaves and the complex on X as a complex of injective sheaves. (4) Suppose X is smooth and projective and take E·,F ·,G· ∈ Db(X). Then

R Hom(F ·,E·) ⊗L G· = R Hom(F ·,E· ⊗L G·), R Hom(F ·,R Hom(E·,G·)) ' R Hom(F · ⊗L E·,G·), R Hom(F ·,E· ⊗L G·) ' R Hom(R Hom(E·,F ·),G·).

because we can assume that all three complexes are complexes of locally free sheaves. (5) Let F · ∈ D−(X). Then Γ ◦ Hom·(F ·, −) = Hom·(F ·, −). Hence

RΓ ◦ R Hom(F ·, −) = R Hom(F ·, −)

and we get a spectral sequence

p.q p q · · p+q · · E2 = H (X, Ext (F ,E )) ⇒ Ext (F ,E ).

(6) Let f : X → Y be a morphism of projective schemes, F · ∈ D−(Y ), E· ∈ Db(Y ). Then ∗ · · ∗ · ∗ · Lf R HomY (F ,E ) ' R HomX (Lf F , Lf E ). 72 AGNIESZKA BODZENTA

(7) Consider a ﬁbre product diagram

v X ×Z Y / Y

g f u X / Z with u: X → Z flat and f : Y → Z proper. Then flat base change asserts, for any F · ∈ D(QCoh(Y )), ∗ · ' ∗ · u Rf∗F −→ Rg∗v F . Let us consider the special case of the product X×Y , q : X×Y → X, p: X×Y → Y . For F · ∈ Db(Y ) the flat base change yields

∗ · · q∗p F ' RΓ(Y,F ) ⊗ OX .

As a consequence we get K¨unnethformula, for F ∈ Db(Y ), E ∈ Db(X),

∗ · L ∗ · ∗ · L ∗ · RΓ(X × Y, q E ⊗ p F ) ' RΓ(X, Rq∗(q E ⊗ p F )) ' · L ∗ · · · ' RΓ(X,E ⊗ Rq∗p F ) ' RΓ(X,E ⊗ RΓ(Y,F ) ⊗ OX ) ' ' RΓ(X,E·) ⊗ RΓ(Y,F ·).

6.6. Grothendieck-Verdier duality.

This part is based on [Nee10] and [Sta13, Section 46.3]. Let f : X → Y be a morphism of quasi-separated, quasi-compact schemes. The pushforward functor Rf∗ : D(QCoh(X)) → D(QCoh(Y )) (deﬁned via h-injective ! complexes) has right adjoint f . The proof is ’abstract nonsense’, functor Rf∗ maps direct sums to direct sums, hence it has a right adjoint. In order to formulate some properties of the functor f ! we need to introduce the following

Conjecture 6.19. Let X be a quasi-compact separated scheme and let U ⊂ X be a quasi- compact open subset. Let j : U → X be the inclusion. Then there exists a compact object

E ∈ D(QCoh(X)) and an integer n ≥ 1 such that Rj∗OU ∈ hEin, the smallest subcategory obtained from E, its shifts and direct sums of copies of those, by taking n cones.

The conjecture holds under the assumptions that (1) X is noetherian, finite dimensional and smooth over a finite dimensional noetherian ring, (2) X is a locally closed subscheme of Y and Conjecture holds for Y , (3) There is an affine morphism X → Y and the Conjecture holds for Y . HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 73

Let f : X → Y be such that Rf∗ maps perfect complexes to perfect complexes and X satisﬁes the Conjecture. Then f ! restricts to a functor Db(Y ) → Db(X). · · Grothendieck-Verdier duality states that for F ∈ DQCoh(X)(OX ) and E ∈ ! DQCoh(Y )(OY ) functors Rf∗ and f are locally adjoint:

· ! · · · Rf∗R HomX (F , f (E )) ' R HomY (Rf∗(F ),E ). · Let now X be a noetherian separated scheme. There exists dualizing complex ωX ∈ Db(X) so that the natural functor

· b op b R Hom(−, ωX ): D (X) → D (X) is an equivalence. The dualizing complex is unique up to shift and twist by a line bundle. The variety X · · is Gorenstein if and only if ωX is a line bundle. X is Cohen-Macaulay if and only if ωX is a sheaf (up to shift, of course), [Har66].

If f : X → Y is such that Rf∗ maps perfect complexes to perfect complexes and X ! · · satisfies the Conjecture, then f ωY ' ωX . As a result, we can define the dualizing complex for a proper noetherian scheme X over a field k for which the Conjecture holds. Let p: X → Spec k be the structure morphism. Then

· ! ωX ' p (OSpec k).

6.7. Spanning classes in the derived category.

Deﬁnition 6.20. A collection Ω of objects in a triangulated category D is a spanning class of D if for any D ∈ D

(i) if Hom(A, D[i]) = 0 for all A ∈ Ω, all i ∈ Z, then D ' 0. (ii) If Hom(D[i],A) = 0 for all A ∈ Ω, all i ∈ Z, then D ' 0.

Exercise 6.21. If D is endowed with a Serre functor conditions (i) and (ii) of the above deﬁnition are equivalent.

Proposition 6.22. Let X be a smooth projective variety. Then the objects Ox, for x ∈ X closed, span Db(X).

Proof. As Db(X) has Serre functor, it suﬃces to check that for any non-zero F · ∈ Db(X) · there exists x ∈ X such that Hom(F , Ox[i]) 6= 0, for some i. 74 AGNIESZKA BODZENTA

Let m be maximal such that Hm(F ·) is non-trivial. Let x be a closed point in the m · m · support of H (F ). Then Hom(H (F ), Ox) 6= 0. Distinguished triangle

· · m · τ≤m−1F → F → H (F )[−m]

· m · implies an isomorphism Hom(F , Ox[−m]) ' Hom(H (F )[−m], OX [−m]) which ﬁnishes the proof. Spanning classes prove useful when proving fully faithfulness:

Exercise 6.23. Let F : D → D0 be an exact functor between triangulated categories with left and right adjoints G a F a H. Suppose that Ω is a spanning class for D and that for all A, B ∈ Ω the natural homomorphisms

F : Hom(A, B[i]) → Hom(F (A),F (B)[i]) are bijective for all i. Show that F is fully faithful.

Deﬁnition 6.24. A sequence of objects Li ∈ A, i ∈ Z in a k-linear abelian category A is ample if for any A ∈ A there exists an integer i0(A) such that for i < i0(A) the following conditions are satisﬁed:

(i) The natural morphism Hom(Li,A) ⊗ Li → A is surjective.

(ii) If j 6= 0 then Hom(Li,A[j]) = 0.

(iii) Hom(A, Li) = 0.

An abelian category A has ﬁnite cohomological dimension if there exists N such that for i > N and any A, B ∈ A, HomDb(A)(A, B[i]) = 0. Spectral sequence

p,q −q · · E2 = Hom(H (A ),B[p]) ⇒ Hom(A ,B[p + q]) implies that for any A· ∈ Db(A) and B ∈ A there exists N such that Hom(A·,B[i]) = 0 for i > N.

Proposition 6.25. Let Li, i ∈ Z be an ample sequence in a k-linear abelian category A b of ﬁnite cohomological dimension. Then Li span the derived category D (A).

· b · Proof. Let A ∈ D (A) be such that Hom(Li,A [j]) = 0 for all i, j. Let n be minimal such n · · n · that H (A ) 6= 0. Then, as Hom(Li[−n], τ>nA [−1]) = 0, Hom(Li[−n], H (A )[−n]) ⊂ · n · Hom(Li[−n],A ). By assumption, the latter space is zero, hence Hom(Li, H (A )) = 0 n · n · for all i. On the other hand, there exists i0 such that Hom(Li, H (A )) ⊗ Li → H (A ) is surjective. The contradiction implies that A· ' 0. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 75 · Let us now assume that Hom(A ,Li[j]) = 0 for all i, j. Let n be maximal such that n · n · n · H (A ) 6= 0. Let i be such that Hom(Li,H (A )) ⊗ Li → H (A ) is surjective and · · n · let B1 denote its kernel. As Hom(A ,Li[j]) = 0 for all j, 0 6= Hom(A ,H (A )[−n]) ⊂ · Hom(A ,B1[1 − n]). n · We continue exchanging H (A ) with B1; there exists i1 < i such that Hom(Li1 ,B1) ⊗ · · Li1 → B1 is surjective. If B2 is its kernel then 0 6= Hom(A ,B1[1 − n]) ⊂ Hom(A ,B2[2 − · n]). This way we obtain a sequence of objects Bj ∈ A such that Hom(A ,Bj[j − n]) 6= 0 which contradicts the assumption that A is of ﬁnite homological dimension. Hence n · H (A ) = 0 for all n.

Proposition 6.26. Let X be a projective variety over a ﬁeld. If L is an ample line bundle on X then powers of L, Li, i ∈ Z form an ample sequence in the abelian category Coh(X).

Proof. By deﬁnition, an ample line bundle L has the property that for any coherent sheaf n F on X there exists n0 such that for any n ≥ n0 the sheaf F ⊗ L is globally generated. This means that 0 n n H (X,F ⊗ L ) ⊗ OX → F ⊗ L is surjective. Tensoring with L−n we get a surjective map

Ln ⊗ Hom(L−n,F ) → F.

Condition (ii) follows from the fundamental theorem of Serre that Hi(X,F ⊗ Ln) = 0 for any i > 0 and any n > n0. It remains to check that Hom(F,Li) = 0. We can assume taht L is very ample (by passing to some power Lk and proving the vanishing for F ⊗ Li, i = 1, . . . , k). 0 As L is very ample for any x ∈ X there exists a section 0 6= sx ∈ H (X,L) with 0 = i sx(x) ∈ L(x). If 0 6= ϕ: F → L then there exists a closed point x such that ϕ(x): F (x) → Li(x) is non-trivial. Hence, ϕ is not in the image of the inclusion Hom(F,Li−1) −→sx Hom(F,Li). It follows that dim Hom(F,Li−1) < dim Hom(F,Li). As Hom(F,L) is ﬁnitely i dimensional, it can happen only ﬁnitely many times before Hom(F,L ) become trivial.

Theorem 6.27. Let F : Db(A) → Db(A) be an exact autoequivalence. Suppose ' f : Id{Li} −→ F |{Li} is an isomorphism of functors on the full subcategory {Li} given by an ample sequence Li in A. Then there exists a unique extension to an isomorphism fe: IdDb(A) → F .

Proof. Step 1 We characterize objects in A in terms of the ample sequence: An object A· ∈ Db(A) is isomorphic to an object in A if and only if · Hom(Li,A [j]) = 0 76 AGNIESZKA BODZENTA for all j 6= 0 and i << 0. One direction is immediate from the deﬁnition of ample sequence and the other can be veriﬁed using the spectral sequence:

p,q q · · E2 = HomA(Li,H (A )[p]) ⇒ Hom(Li,A [p + q]).

(We assume that A has enough injectives, for Coh(X) we consider the embedding into QCoh(X).) Since A· is bounded, its cohomology is concentrated in degree [−k, k]. Hence, p,q q E2 = 0 for |q| > k. We can ﬁnd i0 such that Hom(Li,H (A)[p]) = 0 for i ≤ i0 and p 6= 0. Then the spectral sequence is supported in vertical axis, hence Hom(Li,Hq(A·)) = Hom(Li,A·[q]). By deﬁnition, Hom(Li,Hq(A·)) 6= 0 for i << 0, as soon as Hq(A) 6= 0. · q As Hom(Li,A [q]) = 0 for q 6= 0, we conclude that H (A) = 0, for q 6= 0. Step 2 We show that for any A ∈ A also F (A) ∈ A. It follows from the isomorphisms

Hom(Li,F (A)[j]) ' Hom(F (Li),F (A)[j]) ' Hom(Li,A[j])

(we assumed that F is an equivalence, hence it is fully faithful) and the previous step.

Step 3 We construct for any A ∈ A an isomorphism feA : A → F (A) which is functorial in A and extends f. We have short exact sequence:

⊕k 0 → B → Li → A → 0 with i << 0. Its image under F is again a short exact sequence in A (by the previous step):

⊕ (20) F (B) / F (Li k) / F (A) O O f

⊕k B / Li / A

We want to show that the dashed arrow exists and is unique. To show its existence we ⊕k ⊕k shall check that g : B → Li → F (Li ) → F (A) is zero. If the map exists, then it is ⊕k unique as Hom(A, F (A)) ⊂ Hom(Li ,F (A)). ⊕l We choose a surjective map Lj → B and get a commutative diagram

⊕l ⊕k F (Lj ) / F (Li ) O O f

⊕l ⊕k Lj / Li HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 77

⊕l ⊕k ⊕l as f|{Li} is a natural transformation. As Lj → Li → A is zero, so is F (Lj ) → ⊕k ⊕l g F (Li ) → F (A). So, the map Lj → B −→ F (A) is zero. As the ﬁrst map is a surjection, it implies that g = 0. We have constructed a map A → F (A). Now we check that it does not depend on the choice of Li. Any two surjections can be dominated by a third one, hence it is enough to consider ⊕l ⊕k F (Lj ) / F (Li ) / F (A) O O O f f fe

⊕l ⊕k Lj / Li / A If th smaller square commutes then the bigger one does too. We have seen that there is unique morphism A → F (A) with this property, hence fe is equal to the morphism ⊕l constructed via the surjection Lj → A.

We check that A → F (A) is functorial. Morphism ϕ: A1 → A2 can be lifted to the 1 ⊕k morphisms of kernels, because Ext (Li ,B) = 0 for i << 0. We get

⊕k F (Li ) / F (A1) c <

⊕k Li / A1

⊕l Lj / A2

{ ! ⊕l F (Lj ) / F (A2) and we want to show commutativity of the right part. As everything else commutes, ⊕k ⊕k Li → A1 composed with the upper path A1− > F (A2) equals Li → A1 composed with ⊕k the lower path. As Li → A1 is surjective, we get that fe is functorial. It remains to check that it’s an isomorphism. In (20) we have not only map A → F (A) but also B− > F (B). Snake Lemma implies that A → F (A) is surjective, for any A. Hence, also B → F (B) is surjective and Snake Lemma again implies that A → F (A) is injective. b Step 4 We deﬁne feA· for any A ∈ D (A) recursively on the length of the complex. We · · will assume that we have constructed an isomorphism feA· : A → F (A ) for any complex A· with

· q1 · q2 · length(A ) := max{q1 − q2 | H (A ) 6==6= H (A )} + 1 < N 78 AGNIESZKA BODZENTA functorial in A·. Suppose length of A· is N. We can assume that A· looks like ...Am−1 → Am → 0 and m · m · H (A ) 6= 0. There exists i such that Hom(H (A ),Li) = 0 and there exists surjection ⊕k m Li → A . This surjection gives us a distinguished triangle

⊕k · · ⊕k Li [−m] → A → B → Li [−m + 1].

The map L⊕k → Am → Hm(A·) is surjective, hence Hm(B·) = 0. It follows from the long exact sequence of cohomology that Hi(A·) ' Hi(B·) for i < m − 1. Hence, B· is of length less than N. We have

⊕k · ⊕k F (Li )[−m] / F (A ) / F (B) / F (Li )[−m + 1] O O O O fe fe fe fe ⊕k · ⊕k Li [−m] / A / B / Li [−m + 1]

· · · ⊕k Morphism fe: A → F (A ) exists because of TR3. It is unique as Hom(A ,F (Li )[−m]) ' · ⊕k m · ⊕k Hom(A ,Li [−m]) ' Hom(H (A ),Li ) ' 0. Also it is an isomorphism as the remaining two morphisms are.

We need to check that feA· is independent of the choices and functorial. For ⊕l ⊕k m independence of the choices we again consider Lj → Li → A . We have a diagram: · · F (A ) / F (B1) = ;

⊕l · · Lj [−m] / A / B1

= ⊕k · · Li [−m] / A / B2

" $ · · F (A ) / F (B2) · · which implies that two ways of getting from A to F (B2) are identical. In the diagram, · · existence of B1 → B2 is ensured by TR3 and the right part commutes because of inductive · · · · hypothesis. We have already seen that the map Hom(A ,F (A )) → Hom(A ,F (B2)) is · · injective (the proof of uniqueness of fe), so also the two ways of getting A → F (A ) are equal. We are left with functoriality. Let ϕ: A· → C· be a morphism of complexes of length less than or equal to N. Assume A· is quasi-isomorphic to ... → An−1 → An → 0 and C· is quasi-isomorphic to ... → Cm−1 → Cm → 0. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 79

⊕k n · ⊕k First assume that m < n. Let Li → A be a surjection and B the cone of Li [−n] → A· → B·. Then Hom(B·,C·) → Hom(A·,C·) is surjective. Indeed, we can assume that k · there are no higher Ext-groups from Li to the cohomology of C . Then the usual spectral k · · · sequence implies that Hom(Li [−n],C ) is zero. Then ϕ admits a lift to ϕ1 : B → C . By · · · · inductive hypothesis fe is functorial with respect to A → B and B → C , as the length of B· is less than the length of A·:

ϕ / A· / B· / C·

fe fe fe F (A·) / F (B·) / F (C·) / F (ϕ)

⊕l m ⊕l · · If n ≤ m we choose Lj → C and consider the triangle Lj [−m] → C → D . As · · the length of D is less than the length of C , morphism fe is functorial with respect · · · · · · · to ϕ1 : A → C → D and ψ : C → D . It follows the two maps A → F (C ) are equal when composed with F (ψ). We just need to check that Hom(A·,F (C·)) → · · · ⊕l Hom(A ,F (D )) is injective. Its kernel is a quotient of Hom(A ,Lj [−m]). For degree · ⊕l m · ⊕l reasons, Hom(A ,Lj [−m]) ' Hom(H (A ),Lj ). By deﬁnition of an ample sequence, the latter space vanishes for j << 0.

7. Full exceptional collections

We shall discuss non-standard t-structures on Db(X) induced by (strong) full exceptional collections. A full exceptional collection in a k-linear triangulated category D is a semi-orthogonal decomposition of the form D = hDb(k),..., Db(k)i.

Equivalently, it is a collection hE1,...,Eni of objects of D such that · (1) R Hom (Ei,Ei) ' k, · (2) R Hom (Ei,Ej) ' 0 for i ≥ j,

(3) The smallest triangulated subcategory of D containing E1,...,En is equivalent to D. Moreover, an exceptional collection is strong if

Hom(Ei,Ej[l]) = 0, for l 6= 0. 80 AGNIESZKA BODZENTA

7.1. Beilinson’s result.

The ﬁrst example of a full strong exceptional collection (though it was not called this way) was constructed by A. Beilinson on projective space [Be˘ı78]. The structure sheaf of the diagonal on Pn × Pn admits resolution

∗ n ∗ ∗ 1 ∗ 0 → p1(Ω (n)) ⊗ p2(O(−n)) → ... → p1(Ω (1)) ⊗ p2(O(−1)) → OPn×Pn → O∆ → 0.

· b n ∗ · L · Let now E be an object of D (P ). Then Rp2∗(p1E ⊗ O∆) ' E . ∗ · L Object p1E ⊗ OD is quasi-isomorphic to

∗ · n ∗ ∗ · 1 ∗ ∗ · p1(E ⊗ Ω (n)) ⊗ p2(O(−n)) → ... → p1(E ⊗ Ω (1)) ⊗ p2(O(−1)) → p1(E ).

We get a spectral sequence ( · r,s s n −r · E if r + s = 0 E1 = H (P , Ω (−r) ⊗ E ) ⊗ O(r) ⇒ 0 otherwise.

· b n It follows that E belongs to the smallest triangulated subcategory of D (P ) containing · i b n RΓ(E ⊗ Ω (i)) ⊗ O(−i). In other words, the smallest triangulated subcategory of D (P ) b n n containing O(−n),..., O is equivalent to D (P ). Similarly, for Ω (n),..., Ω1(1), O. Standard computation shows that hO(−n),..., Oi, hΩn(n),..., Ω1(1), Oi are strong exceptional collections in Db(Pn).

7.2. Equivalence of categories.

Theorem 7.1. [Bon89] Let hE1,...,Eni be a full exceptional collection in a k-linear Hom-ﬁnite triangulated category D. Then functor

· M b M R Hom ( Ei, −): D → D (End( Ei)) is an equivalence of D with the bounded derived category of the category of ﬁnite L dimensional right modules over End( Ei).

Let us come back to the example of Pn ' P(V ∨). Then

Hom(O(−i), O(−j)) ' Si−j(V )

L0 · and the composition in End( i=−n O(−i)) agrees with the composition in S (V ). In other words, Theorem 7.1 implies

b ∨ b · n−1 D (P(V )) 'D (S (V )/S V ). HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 81

Exterior powers of the Euler sequence 0 → Ω1 → O(−1) ⊗ V → O → 0 yield short exact sequences 0 → Ωi(i) → Λi(V ) ⊗ O → Ωi−1(i) → 0. By induction one proves that

Hom(Ωi(i), Ωj(j)) ' Λi−j(V ∨).

The multiplication in the exterior algebra again coincides with the composition of morphisms in the collection Ω(i), hence

b ∨ b · ∨ D (P(V )) 'D (Λ (V )).

7.3. Braid group action.

The subcategory hEi generated by an exceptional object E is always admissible, functor left adjoint to the embedding i: Db(k) → D, i(k) = E is i∗(−) = R Hom·(−,E)∨, the right adjoint is i!(−) = R Hom·(E, −). It follows that an exceptional object induces two semi-orthogonal decompositions

b b D = hD0, D (k)i = hD (k), D1i

Functors

∗ ! i0i1 : D1 → D0, i1i0 : D0 → D1 are quasi-inverse equivalences called mutation functors.

In the case of an exceptional collection hE1,...,Eni we can consider the subcategory hEi,Ei+1i and perform the mutation there. The left mutation LEi Ei+1 of Ei+1 over Ei ﬁts into a distinguished triangle · · Ei ⊗ R Hom (Ei,Ei+1) → Ei+1 → LEi Ei+1 → Ei ⊗ R Hom (Ei,Ei+1)[1]

(we use the triangle associated to the semi-orthogonal decomposition hLEi Ei+1,Eii). Note that then · · R Hom (Ei+1,LEi Ei+1) ' k, R Hom (Ei,LEi Ei+1) ' 0, · · R Hom (Ei+1,Ei) ' 0,R Hom (Ei,Ei) ' k.

The right mutation REi+1 Ei of Ei over Ei+1 ﬁts into a distinguished triangle

· ∨ REi+1 Ei → Ei → Ei+1 ⊗ R Hom (Ei,Ei+1) → REi+1 Ei[1]. 82 AGNIESZKA BODZENTA

Again, we have

R Hom(REi+1 Ei,Ei) ' k, R Hom(Ei+1,Ei) ' 0,

R Hom(REi+1 Ei,Ei+1) ' 0,R Hom(Ei+1,Ei+1) ' k.

Mutations LEi and REi+1 deﬁne the action of the braind group with n strands on the set of exceptional collections in a triangulated category D.

In particular, the half twist gives the collection hFn,...,F1i left dual to hE1,...,Eni:

Fi = LE1 ...LEi−1 Ei. One checks that ( · k if i = j R Hom (Ei,Fj) ' 0 otherwise.

If the collection hE1,...,Eni is full, the left dual collection hFn,...,F1i is determined by the Kronecker-type condition on R Hom·. One checks that on Pn collection hΩn(n)[n],..., Ω1(1)[1], Oi is left dual to hO,..., O(n)i.

7.4. Glued t-structure.

A full exceptional collection hE1,...,Eni yields two admissible ﬁltrations

hE1i ⊂ hE1,E2i ⊂ ... hE1 ...Eni

hEni ⊂ hEn−1,Eni ⊂ ... hE1,...Eni. In both cases the ’graded components’ are equivalent to Db(k). We can use recollements o o b hE1,...Ei−1i / hE1,...Eii / D (k) o o and the standard t-structure on Db(k) to glue t-structure on D.

Proposition 7.2. Let hE1,...,Eni be a full exceptional collection in a triangulated category D. Denote by hFn,...,F1i the left dual exceptional collection. The t-structure glued on D along the ﬁltration

hE1i ⊂ hE1,E2i ⊂ ... hE1 ...Eni from the standard t-structures on Db(k) is

≤0 · b ≥0 D = {D ∈ D | R Hom (D,Fi) ∈ D (k) , ∀i}, ≥0 · b ≥0 D = {D ∈ D | R Hom (Ei,D) ∈ D (k) , ∀i}. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 83

7.5. Full exceptional collections on P2 and Markov numbers.

A. Rudakov studied full exceptional collection on P2 obtained by mutations from the collection hO, O(1), O(2)i. He found that these collections contain (up to shift) line bundles. If a, b, c are ranks of the bundles then (a, b, c) is a solution to the Markov equation:

a2 + b2 + c2 = 3abc.

All solutions of the Markov equation are obtained by ’mutations’ from the solution (1, 1, 1). If (a, b, c) is a solution to Markov equation then putting b0 = 3ac − b we get a new solution (a, b0, c). If we do this procedure again, we recover the original solution (a, b, c). However, we can change a or c. Solutions to the Markov equation can be organised in a tree with vertices (a, b, c) and arrows joining two solutions related by a mutation. Generally, from every solution there are three arrows:

(a, b, c) → (a, 3ac − b, c), (a, b, c) → (a, 3ab − c, b), (a, b, c) → (b, 3bc − a, c).

The only singular solutions are (1, 1, 1) and (1, 2, 1) where some of the triples coincide.

(1, 1, 1)

(1, 2, 1)

(1, 5, 2)

y % (1, 13, 5) (5, 29, 2) 7.6. Full exceptional collections on homogeneous spaces and toric varieties.

Further examples of full exceptional collections were constructed by M. Kapranov on Grassmannians and ﬂag varieties. They were also constructed via a ’nice’ resolution of the structure sheaf of the diagonal. Consider G(k, V ), the Grassmannian of k-dimensional subspaces in V . We denote by S the tautological k-dimensional bundle on G(k, V ) and by U the tautological n − k- dimensional quotient bundle. α b α The resolution of O∆ implied that Σ U generate D (G(k, V )) where Σ is the Schur functor assigned to a Young tableau α and α ﬁts into a k × (n − k) rectangle. 84 AGNIESZKA BODZENTA

(Recall, that given a vector space W and a Young tableau α with l entries ΣαW is the projection of the subspace of W ⊗l consisting of tensors which are unchanged by any permutation of boxes in the same row to the subspace of vectors that change sign under any permutation that changes boxes in the same column. This operation is functorial, so it can be applied to vector bundles.) Full exceptional collections have been also constructed by Yu. Kawamata on projective toric varieties. (Quotient singularities are allowed but then one needs to consider derived category of a relevant Deligne-Mumford stack.)

7.7. Derived category under blow-up and projective bundles.

Beilinson’s result was generalised in a diﬀerent direction by D. Orlov [Orl92]. He constructed semi-orthogonal decompositions for projective bundles and blow-ups. Let X be a smooth variety and let E be a vector bundle of rank r on X. Let p: P(E) →

X be the projective bundle determined by E. Let OE(1) be the relative ample line bundle. Morphism p is ﬂat, moreover the derived functor p∗ : Db(X) → Db(P(E)) is fully faithful:

∗ · ∗ · · ∗ · · · · · Hom(p E , p F ) ' Hom(E , Rp∗(p F )) ' Hom(E ,F ⊗ Rp∗(O)) ' Hom(E ,F ).

∗ b b We denote by D(X)0 the full-subcategory p D (X) ⊂ D (P(E)). By D(X)l we denote ∗ · · the full subcategory with objects p (F ) ⊗ OE(k), for F ∈ D(X).

Lemma 7.3. For 0 ≤ l ≤ r − 1, Hom(D(X)l, D(X)0) = 0.

Proof.

∗ · ∗ · · ∗ · Hom(p (F ) ⊗ Ol, p (G )) ' Hom(F , Rp∗(p (G ) ⊗ O(−l)) ' · · ' Hom(F ,G ⊗ Rp∗O(−l)) ' 0.

Beilinson’s resolution of the diagonal works also in this case and we get

Theorem 7.4. [Orl92] Let E be a rank r vector bundle on a smooth projective variety X. Then Db(P(E)) admits a semi-orthogonal decomposition

b D (P(E)) = hD(X)0,..., D(X)r−1i.

Let Y be a smooth variety, Z ⊂ Y a closed subvariety (which is locally a complete intersection) and f : X → Y the blow-up of Y along Z. Let E ⊂ X be the exceptional HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 85 divisor of f. Then E is a Cartier divisor [Har77, Proposition II.7.3] and we have a diagram

i E / X

π f j Z / Y

Moreover, since Z is a locally complete intersection, OE(E) 'Oπ(−1).

b ! Lemma 7.5. For C ∈ D (E), we have i∗i i∗C ' (i∗C ⊕ i∗C ⊗ OX (E)[−1]).

! Proof. Local duality i∗R HomE(−, i (=)) ' R HomX (i∗(−), =) implies isomorphisms:

! ! i∗i i∗C ' i∗R HomE(OE, i i∗C) ' R HomX (i∗OE, i∗C)

' R HomX ([OX (−E) → OX ], i∗C) ' i∗C ⊕ i∗C(E)[−1].

The direct sum decomposition follows from the fact that the adjunction unit i∗C → ! i∗i i∗C splits the distinguished triangle obtained by applying R Hom(−, i∗C) to the exact sequence

0 → OX (−E) → OX → OE → 0.

∗ b b Lemma 7.6. Functor Ri∗ ◦ Lπ : D (Z) → D (X) is fully faithful.

b Proof. For A1,A2 ∈ D (Z), we have

∗ ∗ ! ∗ HomX (Ri∗Lπ A1, Ri∗Lπ A2) ' HomZ (A1, Rπ∗i Ri∗Lπ A2).

∗ ! ∗ We shall show that Rπ∗ of the adjunction unit Lπ A2 → i Ri∗Lπ A2 is an isomorphism, i.e. that for a distinguished triangle

∗ ! ∗ ∗ (21) Lπ A2 → i Ri∗Lπ A2 → B → Lπ A2[1] we have Rπ∗B = 0. Since j is a closed embedding, it suﬃces to show that Rj∗Rπ∗B =

Rf∗Ri∗B = 0. In view of Lemma 7.5, we have

∗ ∗ ∗ ∗ Ri∗B ' Ri∗Lπ A2(E)[−1] ' Ri∗(Lπ A2 ⊗ Li OX (E))[−1] ' Ri∗(Lπ A2 ⊗ Oπ(−1))[−1].

∗ Since Ri∗ is conservative, B ' Lπ A2 ⊗ Oπ(−1)[1]. Then,

∗ Rf∗Ri∗B ' Rj∗Rπ∗B ' Rj∗Rπ∗(Lπ A2⊗Oπ(−1))[−1] ' Rj∗(A2⊗Rπ∗Oπ(−1))[−1] ' 0.

For k ∈ Z, we put

∗ b b De(Z)k := Ri∗Lπ D (Z) ⊗ OX (kE) ⊂ D (X).

Let d = codimY Z. 86 AGNIESZKA BODZENTA

b ∗ Lemma 7.7. For any k ∈ {1, . . . , d} and any A ∈ D (Z), object Ri∗Lπ A ⊗ OX (kE) lies b b in the kernel of Rf∗ : D (X) → D (Y ).

Proof. It follows immediately from the projection formula:

∗ ∗ ∗ Rf∗(Ri∗Lπ A ⊗ OX (kE)) ' Rf∗Ri∗(Lπ A ⊗ Li OX (kE)) ∗ ' Rj∗Rπ∗(Lπ A ⊗ Oπ(−k)) ' Rj∗(A ⊗ Rπ∗(Oπ(−k))) ' 0,

Lemma 7.8. For any k ∈ {1, . . . , d − 1} and A ∈ Db(Z), we have

! ∗ Rf∗Ri∗i (Ri∗Lπ A ⊗ OX (kE)) = 0.

Proof. Local duality gives

! ∗ ! ∗ Rf∗Ri∗i (Ri∗Lπ A ⊗ OX (kE)) ' Rf∗Ri∗R HomE(OE, i (Ri∗Lπ A ⊗ OX (kE))) ∗ ∗ ' Rf∗R HomX (OE, Ri∗Lπ A ⊗ OX (kE)) ' Rf∗(R HomX (OE, Ri∗Lπ A) ⊗ OX (kE)) ! ∗ ' Rf∗(Ri∗i Ri∗Lπ A ⊗ OX (kE)).

In view of Lemma 7.5, we have

! ∗ ∗ ∗ Rf∗(Ri∗i Ri∗Lπ A ⊗ OX (kE)) ' Rf∗((Ri∗Lπ A ⊕ Ri∗Lπ A ⊗ OX (E)[−1]) ⊗ OX (kE)) ∗ ∗ ' Rf∗(Ri∗Lπ A ⊗ OX (kE)) ⊕ Rf∗(Ri∗Lπ A ⊗ OX ((k + 1)E)[−1]).

The statement follows from Lemma 7.7.

Lemma 7.9. For 0 ≤ k < l ≤ d − 1, we have HomX (De(Z)k, De(Z)l) = 0.

b Proof. For A1,A2 ∈ D (Z), we have

∗ ∗ HomX (Ri∗Lπ A1 ⊗ OX (kE), Ri∗Lπ A2 ⊗ OX (lE)) ! ! ' HomZ (A1 ⊗ Rπ∗i (Ri∗Lπ A2 ⊗ OX ((l − k)E)).

! ∗ ! ∗ In view of Lemma 7.7, object Rj∗Rπ∗i (Ri∗Lπ A2⊗OX ((l−k)E)) ' Rf∗Ri∗i (Ri∗Lπ A2⊗ ! ∗ OX ((l − k)E)) is zero. Since j is a closed embedding, it follows that Rπ∗i (Ri∗Lπ A2 ⊗ OX ((l − k)E)) = 0 which ﬁnishes the proof.

b Proposition 7.10. Let F ∈ D (X) be such that Rf∗F = 0. Then F lies in the b b subcategory of D (X) generated by i∗D (E).

Proof. We shall show that all cohomology sheaves of F are (set-theoretically) supported on E. Then, we check that any Fe ∈ Db(X) whose cohomology are set-theoretically supported on E is an iterated extension of shifts of sheaves supported on E. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 87

p,q p q Since Rf∗F = 0, spectral sequence with the E2 = R f∗H F converges to zero. We i l note that R f∗H (F) together with their subobjects and quotients are supported on Z, p,q for i ≥ 1 and arbitrary l. It follows that Ew is a sheaf whose support is contained in Z, as soon as p ≥ 1 and w ≥ 2. On the other hand, if Hk(F) is not supported on E, k then, since f is an isomorphism outside of E, f∗H (F) is non-zero and its support is not 0,k p,q contained in Z. It follows that E∞ is non-zero which contradicts the fact that E ⇒ 0. For a sheaf G ∈ Coh(X) set-theoretically supported on E, let l(G) ∈ N be minimal b 0 such that G ∈ D (l(G)E). Then, for the kernel G of the restriction G → G|E, we have l(G0) = l(G) − 1. For a complex F ∈ Db(X) with all cohomology sheaves supported on E, we put l(F) := P l(Hi(F)). We prove by descending induction on l(F) that F lies in triangulated subcategory of Db(X) generated by Db(E). Let k be maximal such that Hk(F) 6= 0 and let F 0 ∈ Db(X) be a complex deﬁed by mattress:

0 k τ≤k−1F / F / H (F)|E[−k] O O O '

k τ≤k−1F / F / H (F)[−k] O O

' G0[−k] / G0[−k]

0 b Then l(F ) = l(F) − 1, hence, by inductive hypothesis, l(F) ∈ hi∗D (E)i. The statement k follows from the fact that H (F)|E is isomorphic to i∗G, for some G ∈ Coh(E).

Lemma 7.11. Let E be a locally free sheaf of rank d on Y and Z ⊂ Y be given by the i ∗ i zeros of a section of E. Then, for i = 1, . . . , d, the sheaf L f OZ = Ωπ(i) lies in the subcategory hDe(Z)d−1,..., De(Z)1i.

Proof. We use Koszul resolution

d ∗ ∗ 0 → Λ E → ... → E → OX → 0

i ∗ i of OZ to get L f OZ = Ωπ(i). i b To prove that Ωπ(i) ∈ hDe(Z)d−1,..., De(Z)1i ⊂ D (X) we use relative Euler sequence 0 → Ωp(p) → f ∗ΛpE ∗ → Ωp−1(p) → 0

p and its twists by OX (kE). We prove by decreasing induction on p that Ω (k) ∈ hDe(Z)d−1,..., De(Z)1i, for k = 1, . . . , p and p = 1, . . . , d − 1. d−1 d−1 Since Ωπ = OE(dE) and OE(E) 'Oπ(−1), we have Ωπ (k) = OE((d − k)E) ∈

De(Z)d−k and the case p = d − 1 is clear. Let us now assume that the statement holds for 88 AGNIESZKA BODZENTA p. The sheaf Ωp−1(p − k) belongs to any triangulated subcategory of Db(X) containing p ∗ p ∗ Ω (p − k) and f Λ E ⊗ OX (kE). The Lemma follows from the fact that the ﬁrst sheaf belongs to hDe(Z)d−1,..., De(Z)1i for k = 0, . . . , p − 1, the second for k = 1, . . . , d − 1.

Theorem 7.12. Category Db(X) admits a semi-orthogonal decomposition

b ∗ b (22) D (X) ' hDe(Z)d−1,..., De(Z)1, Lf D (Y )i

Proof. The fact that categories are semi-orthogonal follows from Lemmas 7.7 and 7.9. It remains to show that any object B ∈ Db(X) admits a “ﬁltration” with subquotients in ∗ b De(Z)k and Lf D (Y ). Let F ∈ Db(X). Then

∗ 0 ∗ Lf Rf∗F → F → F → Lf Rf∗F[1]

∗ ∗ b 0 0 is a decomposition of F into Lf Rf∗F ∈ Lf D (Y ) and F such that Rf∗F = 0, hence Hom(Lf ∗Db(Y ), F 0) = 0. b ∗ ! For any F ∈ D (X), the Lπ a Rπ∗ and i∗ a i adjunction counits give a morphism of ∗ ! functors Ri∗Lπ Rπ∗i ((−) ⊗ OX (−kE)) → (−) ⊗ OX (−kE), hence

∗ ! κk : Ri∗Lπ Rπ∗i ((−) ⊗ OX (−kE)) ⊗ OX (kE) → (−).

b The cone F of κk applied to F ∈ D (X) is such that HomX (De(Z)k, F) = 0. b ∗ Let now F be an arbitrary element of D (X). The cone F0 of the Lf a Rf∗ adjunction b ∗ b unit is an element of D (X) orthogonal to Lf D (Y ). Let F1 be the cone of κ1 applied to ∗ ! ∗ b F0. Since both F0 and Ri∗Lπ Rπ∗i (F0 ⊗OX (−E))⊗OX (E) are orthogonal to Lf D (Y ),

Lemma 7.8, so is F1. Moreover, F1 is orthogonal to De(Z)1. Continuing, we define Fk as the cone of κk applied to Fk−1. Then, Lemmas 7.8 and 7.9 imply that Fk is orthogonal to ∗ b hDe(Z)k,..., De(Z)1, Lf D (Y )i. It follows that (22) is a semi-orthogonal decomposition if and only if Fd−1 is zero, i.e. κd−1 applied to Fd−2 is an isomorphism. ∗ The Lf a Rf∗ adjunction counit and morphisms κk are defined locally over Y . Therefore, to prove that (22) is a semi-orthogonal decomposition, it suffices to choose an open cover of Y and prove that (22) holds for an arbitrary open set U ⊂ Y in the cover and the blow-up fU of U along Z ∩ U. Therefore, one can assume that Z ⊂ Y is the zero set of a section of a locally free sheaf of rank d. b 0 ∗ Let F be an arbitrary element of D (X) and let F be the cone of the Lf a Rf∗ 0 0 adjunction counit applied to F. Then Rf∗F = 0, hence by Proposition 7.10, F belongs b b to the triangulated subcategory of D (X) generated by i∗D (E). Thus, in order to prove 0 ∗ b b that F belongs to hDe(Z)d−1,..., De(Z)1, Lf D (Y )i ⊂ D (X) it suffices to check that any b b element of i∗D (E) belongs to this subcategory of D (X). HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 89

b ∗ b ∗ b By Theorem 7.4 we have D (E) = hπ D (Z) ⊗ OE((d − 1)E), . . . , π D (Z) ⊗ ∗ b b ∗ b OE(E), π D (Z)i. Thus, to show that i∗D (E) is contained in hDe(Z)d−1,..., De(Z)1, Lf D (Y )i, ∗ b it suffices to check that i∗π D (Z) is contained there. Since morphism π is flat and ∗ ∗ i is a closed embedding, functor i∗π is t-exact. Hence, to show that i∗π F belongs ∗ b b to hDe(Z)d−1,..., De(Z)1, Lf D (Y )i, for any F ∈ D (Z), it suffices to check it for F ∈ Coh(Z). Since Z ⊂ Y is given by a zero-section of a locally free sheaf E, morphism f fits into a commuting diagram k i q * E / X / P(E)

π f p j } Z / Y Since morphism p is ﬂat and E is the ﬁber product of P(E) and Z over Y , we have, ∗ ∗ k∗π F' p j∗F, for any F ∈ Coh(Y ). Further,

∗ ∗ ∗ ∗ ∗ M k ∗ ∗ (23) q∗Lf j∗F' q∗Lq p j∗F' q∗Lq q∗i∗π F' q∗(Λ N [k] ⊗ i∗π F) where N is the normal bundle to the embedding q. The last isomorphism follows from [Huy06, Proposition 11.1] and the fact that X ⊂ P(E) is the set of zeros of a section of ∗ 2 ∗ p Λ E ⊗ Og(1). ∗ ∗ In particular, it follows that i∗π F = f j∗F. In view of distinguished triangle

∗ ∗ ∗ ∗ (24) τ≤−1Lf j∗F → Lf j∗F → f j∗F → τ≤−1Lf j∗F[1],

∗ ∗ b in order to show that i∗π F lies in hDe(Z)d−1,..., De(Z)1, Lf D (Y )i, it suﬃces to check ∗ that τ≤−1Lf j∗F is an element of this category. Since q∗ is t-exact, (23) implies that k ∗ k ∗ ∗ k ∗ k L f j∗F' Λ N ⊗ i∗π F. It follows from Lemma 7.11 that Λ N |E ' Ωπ(k). Finally, ∗ k ∗ k ∗ as i∗π F is scheme-theoretically supported on E, we have L f j∗F' Ωπ(k) ⊗ i∗π F' k ∗ i∗(Ωπ(k) ⊗ π F). As in the proof of Lemma 7.11 we use the relative Euler sequence

p ∗ p p−1 0 → Ωπ(p) → f Λ E → Ω (p) → 0

∗ and its twist by Opi(k) and π F (note that all sheaves in the sequence are locally free, hence it remains exact after twist with an arbitrary sheaf on E) to prove by decreasing p ∗ induction on p that i∗(Ωπ(k) ⊗ π F) is an element of hDe(Z)d−1,..., De(Z)1i, for k = ∗ 1, . . . , p. It implies that τ≤−1Lf j∗F is an element of this category, hence, in view of ∗ ∗ ∗ b (24), f j∗F' i∗π F is an element of hDe(Z)d−1,..., De(Z)1, Lf D (Y )i, which ﬁnishes the proof. 90 AGNIESZKA BODZENTA

8. Modern approach 8.1. The stable category of spectra.

The ﬁrst example of a ’distinguished triangle’ is the Puppe sequence. Given a continuous morphism f : X → Y of pointed topological spaces, we have the mapping cone I Mf = {(x, ω) ∈ X × Y | ω(0) = y0, ω(1) = f(x)} which ﬁts into a sequence ΩY → Mf → X → Y.

Remark 8.1. Note that we calculated Mf as a homotopy ﬁber product

Mf / X

∗ / Y which means that we replaced the map ∗ → Y with a ﬁbration Y I → Y and calculated the usual ﬁber product: Mf / X

Y I / Y. The sequence can be extended to the left and for any topological space Z the sequence

[Z, ΩY ] → [Z, Mf] → [Z,X] → [Z,Y ] is exact, where [Z,X] denotes the (set of) homotopy classes of pointed continuous morphisms Z → X.

Puppe sequence almost makes Top∗ into a triangulated category; the problem is that not every space is a loop space, so Ω is not an equivalence.

We denote by HoTop∗ the homotopy category of pointed compactly generated weak Hausdorﬀ spaces with homotopy classes of pointed continuous morphisms. There is a functor ∞ Σ : HoTop∗ → HoSpectra∗ to the stable homotopy category. It is a category with suspension functor Σ: HoSpectra∗ →

HoSpectra∗ which agrees with the suspension on HoTop∗ and is an equivalence. The functor Σ∞ has a right adjoint

∞ Ω : HoSpectra∗ → HoTop∗. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 91

Moreover, as all objects of HoSpectra∗ are suspensions, [X,Y ] is an abelian group for any pair of objects X,Y ∈ HoSpectra∗. The sphere spectrum ∞ 0 S := Σ S allows us to deﬁne stable homotopy groups

n πn(X) = [S,X]n = [Σ S,X].

A spectrum is connective if πn(X) = 0 for n < 0.

The category HoSpectra∗ is a symmetric monoidal category with respect to X ∧Y . The unit object is S.

A prespectrum E is a sequence of based spaces E0,E1,... along with structure maps

ΣEn → En+1. A map of prespectra is a sequence fn : En → Fn which commute with the structure maps.

A prespectrum is a CW prespectrum if all Ei are CW complexes and all structure maps

ΣEn → En+1 are inclusions of subcomplexes. A cell of a CW prespectrum is a cell of one of the Ei’s together with all of its suspensions. A k-cell of En is a stable (k − n)-cell of E·.

CW perspectra are objects of HoSpectra∗. Morphisms are homotopy classes of ’eventually deﬁned’ f : E· → F·. f is a map on each m-cell of E· which is deﬁned on (m − n)-cell of En for n >> 0.

This is the original deﬁnition due to Adams. A modern point of view on HoSpectra∗ is as the homotopy category of a stable (∞, 1)-category.

8.2. DG categories and DG enhancements.

The first solution to the problem that cones are not functorial and that a triangulated category is a structure not a property was proposed by Bondal and Kapranov in [BK90]. They proposed, in some sense, not to make the last step and, in particular, view derived categories as the homotopy categories of complexes of injective objects. More precisely, a DG category is a (pre)additive category D together with a grading and a differential ∂ on HomD(D1,D2) for any pair D1,D2 of objects of D. We assume that the graded Leibniz rule is satisfied

∂(fg) = ∂(f)g + (−1)|f|f∂(g) and that for any D ∈ D, IdD is closed of degree zero. Examples of DG categories are categories of complexes over an additive category A with homogeneous morphisms; f· : A· → B· is of degree n if f i is a morphism Ai → Bi+n. 92 AGNIESZKA BODZENTA

The diﬀerential is

i i+1 n+1 i (∂f) = f dA + (−1) dBf .

A homotopy category D of a DG category D has the same objects as D and morphisms are the zero cohomology groups of morphisms in D. The DG category D is pretriangulated if for any D ∈ D and n ∈ Z functor Hom(−,D)[n] is representable and if for any closed degree zero morphism f : D1 → D2 the functor

Cone(Hom(−,D1) → Hom(−,D2)) is representable. If a DG category D is pretriangulated then D is triangulated. Let T be a triangulated category. A DG enhancement of T is a pretriangulated DG category D together with an equivalence D −→T' . Let A be an abelian category. If A has enough projectives then the DG category of bounded above complexes of projective objects in A is a DG enhancement for D−(A). If A has enough injectives then the DG category of bounded below complexes of injectives in A is a DG enhancement for D+(A). Note that the category of complexes of objects of A is not a DG enhancement for the derived category. V. Lunts and D. Orlov proved that the DG enhancement for Db(X), for a reasonable scheme X is unique, [LO10]. The result was further generalised by A. Canonaco and P. Stellari [CS18].

8.3. Inﬁnity categories.

The following is based mostly on [Lur09]. Nowadays, the language of infinity categories is believed to be the most useful. Naively speaking, an infinity category is a category with objects, morphisms, morphisms between morphisms and so on. These have to satisfy axioms subject to higher coherence conditions. The first approach to infinity categories was suggested by Dwyer and Kan. For them an infinity category was a category enriched over simplicial sets. Their construction turned out to be not enough to construct limits of infinity categories. The remedy is to define infinity categories as Kan complexes. Before we do it, we give THE example of an infinity category. Let X be a topological space. We can form an infinity category πX whose objects are points of X. Morphisms in πX are given by paths I → X, 2-morphisms by homotopies between paths, 3-morphisms as homotopies between homotopies and so on. The category πX is often called an infinity groupoid as all morphisms are invertible. In fact, every infinity groupoid is πX, for some topological space X. These are often referred to as homotopy types. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 93

We shall be mostly interested in (∞, 1) categories, i.e. inﬁnity categories in which all k-morphisms, for k > 1 are invertible. Therefore, it is a collection of objects and

MapC(X,Y ) is an infinity groupoid, i.e. a homotopy type. One way to define an (∞, 1) category is to say that it is a category enriched over the category of compactly generated and weakly Hausfdorff topological spaces. This is a ’strict’ version of an (∞, 1)-category, therefore it is not very convenient to work with. However, the ’strict’ version is equivalent to a weak version which is defined via simplicial sets. Following Lurie, we shall refer to (∞, 1)-categories as infinity categories. If X is a topological space then the simplicial set SingX is a Kan complex, i.e. it satisfies the property that for any 0 ≤ i ≤ n any diagram of solid arrows can be completed to a diagram of dotted arrows:

n Λi / SingX ;

∆n

n n Here, Λi denotes the i’th horn obtained from ∆ by deleting the i’th face and the interior. According to Quillen the functor Sing and geometric realisation provide mutually inverse equivalences of the category of topological spaces and Kan complexes.

We can also assign a simplicial set to a category A, the nerve of a category. N(A)n = Map(∆n, A) is the set of n composable arrows. In fact, a simplicial set K is equivalent to N(A), for some category A if and only if any 0 < i < n any diagram of solid arrows can be completed to a diagram of dotted arrows:

Λn / K i >

∆n

We can view an arbitrary simplicial set K is a generalised category whose objects are vertices of K, morphisms are the edges of K and a 2-simplex ∆2 → K should be thought of as

Y φ > ψ

θ X / Z together with a ’homotopy’ between θ and ψφ. However, this way we cannot always compose morphisms in K unless it satisﬁes the additional property: 94 AGNIESZKA BODZENTA

Deﬁnition 8.2. An (∞, 1)-category is a simplicial set K which has the following property, n n for any 0 < i < n any map f0 :Λi → K admits an extension to f : ∆ → K. The simplicial set K is often referred to as a weak Kan complex.

This approach to infinity categories was developed by Joyal. If C is a topological category it is easy to define its homotopy category C. Its objects are objects of C and morphisms are π0MapC(X,Y ). As every (∞, 1)-category C is equivalent to a topological category, we can consider a homotopy category of an infinity category. There is also a (nontrivial) way to write down explicitly the simplicial set MapC(X,Y ). The role of vector spaces in the world of infinity categories is played by the category S of spaces. This is the (appropriate ∞-enhancement of the) subcategory of the category of simplicial sets consisting of Kan complexes. An important source of ∞-categories come from localisation. For a category C and a set W of morphisms there is an infinity category L(C,W ) which is universal with respect to functors to infinite categories that map W to equivalences. Its homotopy category is equivalent to the usual localisation. The localisation procedure can be applied to the category of ∞-categories and strict ∞-functors for W being equivalences (F is an equivalence when it is fully faithful, Map(X,Y ) → Map(F (X),F (Y )) are weak homotopy equivalences, and essentially surjective, which can be checked on homotopy categories). This way we get an infinity category of infinity categories which allows us, in particular, to consider homotopy limits of infinity categories.

8.4. Stable inﬁnity categories.

We would like to enhance triangulated categories to infnity categories. Therefore, we need to know when the homotopy category of an infnity category is triangulated. This is precisely the condition of stability.

Definition 8.3. Let C be an infinity category. A zero object of C is an object which is both initial and final. We say that C is pointed if it contains a zero object.

An object 0 ∈ C is zero if for any object X spaces Map(0,X), Map(X, 0) are contractible. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 95

Deﬁnition 8.4. Let C be a pointed inﬁnity category. A triangle in C is a diagram ∆1 × ∆1 → C depicted as

f X / Y

g 0 / Z

A triangle is a ﬁber sequence if it is a pullback square and a coﬁber sequence if it is a pushout square.

A triangle is a pair of morphisms f, g, a 2-simplex corresponding to the diagram

Y f > g

h X / Z and a 2-simplex

0 ?

h X / Z

Definition 8.5. Let C be a pointed category and g : X → Y a morphism. A fiber of g is a fiber sequence

W / X

g 0 / Y

A coﬁber of g is a coﬁber sequence

g X / Y

0 / Z

Definition 8.6. An infinity category C is stable if it satisfies the following

(1) There exists a zero object 0 ∈ C, (2) Every morphism in C admits a fiber and a cofiber, (3) A triangle in C is a fiber sequence if and only if it is a cofiber sequence. 96 AGNIESZKA BODZENTA

Let C be a stable category and X ∈ C an object of C. The shift X[1] ∈ C is deﬁned via the pushout diagram

X / 0

0 / X[1]

Deﬁnition 8.7. Let C be a pointed category which admits coﬁbers. Suppose given a diagram f g X −→ Y −→ Z −→h X[1] in the homotopy category C. We will say that it is a distinguished triangle if there exists a diagram ∆1 × ∆2 → C

fe X / Y / 0

ge eh 00 / Z / W satisfying (1) Objects 0, 00 are zero, (2) Both square are pushout diagrams, (3) Morphisms fe and ge represent f and g, respectively, (4) The map h: Z → X[1] is the composition of the homotopy class of eh with the equivalence W ' X[1] determined by the outer rectangle.

Theorem 8.8. Let C be a stable inﬁnity category. Then the homotopy category C with the above distinguished triangles is a triangulated category.

The useful property of stable categories is that the categories of functors are also stable

Proposition 8.9. Let C be a stable category and K a simplicial set. Then the ∞-category Fun(K, C) is stable.

8.5. Examples of stable inﬁnity categories.

The first example of a triangulated category is the derived category of an abelian category A. We shall briefly discuss how to construct its (∞, 1)-enhancement. For simplicity, let us assume that has enough injective objects. In this case we already know the DG enhancement for D+(A). Therefore, it suffices to describe an (∞, 1)- enhancement of a DG category D. HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 97

Deﬁnition 8.10. Let D be a DG category. The diﬀerential graded nerve of D is a n simplicial set Ndg(D) with Hom(∆ ,Ndg(D)) = ({Xi}0≤i≤n, {fI }) where

(1) for 0 ≤ i ≤ n Xi is an object of D,

(2) For every subset I = {i− < im < im−1 < . . . < i1 < i+} ⊂ [n] with m ≥ 0 fI is an m element of Hom (Xi− ,Xi+ ) satisfying

X j ∂fI = (−1) (fI\{ij } − fij <...

If α:[m] → [n] is a nondecreasing function then the induced map Ndg(D)n → Ndg(D)m is given by

({Xi}0≤i≤n, fI ) 7→ ({Xα(j)}, {gJ }) where  f if α| is injective  α(J) J 0 0 gJ = IdXi if J = {j, j } with α(j) = α(j ) = i,   0 otherwise.

Proposition 8.11. Let D be a DG category. Then Ndg(D) is an ∞-category. The homotopy categories of D and Ndg(D) are equivalent.

If A does not have enough projectives or injectives, we can localise the ∞-category corresponding to the DG category of complexes over A. Why do we prefer inﬁnity categories to DG categories? Because for inﬁnity categories we have descent. Let X be a scheme. Denote by QCoh(^ X) the (∞, 1)-enhancement of D(QCoh(X)). Then

QCoh(^ X) = lim QCoh(^ U) U⊂X where the limit is taken in the category of (presentable) ∞-categories. Presentability is some ﬁniteness condition. Any pointed ∞-category C has the universal stable ∞-category; the category of spectrum objects. It is the homotopy inverse limit of the tower

... C −→CΩ −→CΩ .

If we take C to be the category S of spaces, the resulting stable ∞-category will be an ∞-enhancement for the stable homotopy theory. 98 AGNIESZKA BODZENTA

8.6. Derived algebraic geometry. This part is based on [To¨e14]. Derived algebraic geometry is based on 2 principles (1) The smooth algebraic varieties or more generally smooth schemes and smooth maps are good. A non-smooth variety, scheme or a map between schemes must be replaced by the best possible approximation by smooth objects. (2) Approximations of varieties, schemes and maps of schemes are expressed in terms of simplicial resolutions. The simplicial resolutions must only be considered up to the notion of weak equivalence, and are controlled by higher categorical or homotopical structures.

Deﬁnition 8.12. A derived scheme consists of a pair (X, AX ) where X is a topological space and AX is a sheaf of commutative simplicial rings on X such that

(1) The ringed space (X, π0(AX )) is a scheme,

(2) For all i > 0, πi(AX ) is quasi-coherent.

We would like derived schemes to form a nice category. To this end we consider the inﬁnity category of simplicial commutative rings sComm, or simply the category of derived rings, which is the localisation of the category of simplicial commutative rings with respect to weak homotopy equivalences. For a topological space X there is an ∞-category of sComm(X) of stacks on X with coeﬃcients in the ∞-category of derived rings. These are prestack satisfying descent condition. We can introduce the category of derived ringed spaces dRgSp whose objects are pairs

(X, OX ) of a topological space X and OX ∈ sComm(X). Any derived ringed space has a truncation to a ringed space (X, π0(OX )) and we can consider the full subcategory of dRgSp consisting of derived ringed spaces whose truncations are locally ringed spaces.

Deﬁnition 8.13. The ∞-category of derived schemes is the full subcategory of dRgSp consisting of (X, OX ) such that

(1) The ringed space (X, π0(OX )) is a scheme,

(2) For all i > 0, πi(OX ) is quasi-coherent.

A ring can be considered as a constant simplicial ring, which defines a functor Comm → sComm. This inclusion has right adjoint π0. This adjunction extends to an adjunction between the category of schemes and derived schemes. The inclusion is fully faithful. For a derived scheme we define the derived category of quasi-coherent sheaves as a limit of such categories for open affine subsets. An affine U ⊂ X corresponds to some simplicial commutative ring A, which can be normalised to a DG ring. The category of HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY 99 quasi-coherent sheaves is then the localisation of the category of DG modules in quasi- isomorphisms. In the world of derived schemes the base change holds without flatness assumption. There is also a better behaved version of a cotangent complex. In characteristic zero the category of simplicial commutative rings is equivalent to the category of non-positively graded DG algebras which simplifies the picture a bit.

8.7. Why do we care about derived algebraic geometry.

Except for descent, the derived point of view on algebraic geometry helps us to understand better the ’standard facts’. Let us mention the one regarding deformation theory. Let X be a scheme. Consider the deformation functor X∧ which to a local commutative Artinian algebra A assigns a deformation of X over A, i.e. X → Spec A such that the fiber over the closed point is isomorphic to X (in fact, we should fix this isomorphism). Deformations over dual numbers C[ε]/(ε2), the first order deformations, are known to 1 be equivalent to H (X,TX ). To every first order deformation η1 of X we can assign an 2 obstruction class θ ∈ H (X,TX ) which vanishes if and only if η extends to a deformation over C[ε]/(ε3). The proof of the last statement is usually an ad hoc argument. However, we can extend the deformation functor to the category of local commutative non-positively graded Artinian DG rings. We can then consider C[δ] := C ⊕ C[1], the square zero extension 2 ∧ with ε in degree -1. Then H (X,TX ) ' π0X (C[δ]) which gives (classically not known) geometric interpretation of this cohomology group. The ring C[ε]/(ε3) fits into a pull-back diagram

C[ε]/(ε3) / C[ε]/(ε2)

C / C[δ] This determines a pull-back square

X∧(C[ε]/(ε3)) / X∧(C[ε]/(ε2))

X∧(C) / X∧(C[δ])

As X∧(C) is a point, we get a ﬁber sequence of spaces

∧ 3 ∧ 2 ∧ X (C[ε]/(ε )) → X (C[ε]/(ε )) → X (C[δ]). 100 AGNIESZKA BODZENTA

∧ 2 In particular, a ﬁrst order deformation η ∈ π0X (C[ε]/(ε )) determines an element in ∧ 2 π0(X (C[δ]) ' H (X,TX ) which vanishes if and only if η can be lifted.

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