Notes on Homological Algebra

Home , Coimage, Cokernel, Epimorphism, Homological algebra, Image (category theory), Monomorphism

Mariusz Wodzicki

December 1, 2016

1 Foundations

1.1 Preliminaries 1.1.1 A tacit assumption is that A, B, ... , are abelian categories, i.e., additive categories with kernels, cokernels, every epimorphism being a kernel, and every monomorphism being a cokernel.

1.1.2 In an abelian category every morphism has an image-coimage factorization

• ﬂ [^ ι ^ θ  [ (1) ﬂ ••u α in which ι is an image of α, i.e., a kernel of a cokernel of α, and θ is a coimage of α, i.e., a cokernel of a kernel of α, cf. Notes on Category Theory.

1.1.3 Exact composable pairs A composable pair of arrows

α β •••u u (2) is said to be exact if β factorizes

κ ◦ β0 with κ being a kernel of α and β0 being an epimorphism.

1.1.4 If λ is a cokernel of β, then λ is a cokernel of κ , i.e., a cokernel of β and a kernel of α form an extension

λ κ . •••u u u (3) x Unlike the original condition this last condition is self-dual.

3 1.1.5 Note that a cokernel of β and a kernel of α form an extension precisely when κ is an image of β and λ is a coimage of α • fl [^^ α [λ fl β •••u α u [^ [  κ ^ flfl β¯ • in which case coimage factorization factorizes α through a cokernel of β with α being a monomorphism while image factorization factorizes β through a kernel of α with β¯ being an epimorphism. Let us record this fact for future use.

Lemma 1.1 For any composable pair of arrows, the following three conditions are equivalent (a) β factorizes through a kernel of α β ••u [^ [ ff κ ^ ffflfl β0 •

(b) α factorizes being a cokernel of β • 0 fl [^^ α ff [λ fffl ••u α

1.1.6 Condition α ◦ β = 0 A weaker condition, α ◦ β = 0, is equivalent to existence of a factorization of β β ••u u κ λ (4) v u ••u µ x

4 with µ being a monomorphism. Note that µ satisfying identity

α = κ ◦ µ ◦ λ is unique in view of κ being a monomorphism and λ being an epimorphism.

1.1.7 Homology A cokernel χ of µ is called a homology of composable pair (2) and it forms an extension χ µ •••u u u .(5) x

1.1.8 The target of χ is usually referred to as the homology (group) of the composable pair. This terminology originally was applied to homomorphisms of abelian groups but the tradition is so well established that it continues to be used also in the context of general abelian categories.

1.1.9 Note that a homology morphism χ is zero or, equivalently, in view of the fact that χ is an epimorphism, its target is zero, precisely when µ is an isomorphism, i.e., when (2) is exact.

1.1.10 Composable pairs (2) satisfying the condition α ◦ β = 0 form a full subcategory of the category of composable pairs A2,0 .

Exercise 1 Show that any assignment of a morphism χ on the class of such composable pairs admits a unique extension to a functor

χ : A2,0 −→ Arr A.

1.1.11 The compositions

Z ˜ s ◦ χ and H ˜ t ◦ χ

5 with the source and, respectively, the target functors

s : Arr A −→ A and t : Arr A −→ A are referred to as a cycles and, respectively, homology functors.

Exercise 2 Express the condition α ◦ β = 0 in terms of a certain factorization of α.

1.1.12 A dual approach to homology Based on the corresponding factorization of α instead of β one can develop a dual approach to homology.

1.1.13 Chain complices We shall use the term chain complex in two ways: either in a loose sense, to denote any sequence of composable morphisms of any length such that the composition of any two composable arrows is zero or, in a strict sense, as a Z-graded object C• = (Cl)l∈Z equipped with an endomorphism ∂ of degree −1 such that ∂ ◦ ∂ = 0. The graded endomorphism ∂ is referred to as the boundary morphism or operator.

1.1.14 Notation

We may use notation (C•, ∂•), (C, ∂), C• or, simply, C, to denote a chain complex (Cl, ∂l)l∈Z .

1.2 Projective and injective objects 1.2.1 Projective objects in an abelian category An object P is said to be projective if it possesses the Lifting Property for epimorphisms, more precisely, if for any diagram

u ••u u e

6 with e being an epimorphism, there exists an arrow

P \ \] • such that the diagram P \ \ \\] u ••u u e commutes.

1.2.2 The above deﬁnition makes sense in any category, in general categories, however, there may be “too many” epimorphisms making the Lifting Property with respect to the class of all epimorphisms very restrictive.

1.2.3 An example: the category of topological spaces In the category of topological spaces and continuous mappings, epimorphisms are mappings with dense image. For any nonsurjective epimorphism f : Y −→ Z and any nonempty space X the maping that sends all points in X to a point in Z \ f (Y) cannot be lifted to Y. Thus, the only topological space that has Lifting Property for epimorphisms is the empty space.

1.2.4 Projective objects in a general category For the above reasons, projective objects in general categories are deﬁned by the Lifting Property with respect to a speciﬁc class of epimorphisms, e.g., the class of all strong epimorphisms, see Notes on Category Theory.

7 1.2.5 Injective objects in an abelian category An object I is said to be injective if it possesses the Extension Property for monomorphisms, more precisely, if for any diagram

I u

••v µ w with µ being a monomorphism, there exists an arrow

I \^ \ • such that the diagram I u \^\ \ \ ••v µ w commutes. Exercise 3 Show that P is projective if and only if for any commutative diagram with an exact row P 0   φ ﬂ u •••u u α β there exists an arrow P \ ˜ \]φ • such that the diagram P \ \φ˜ φ \\] u •••u u α β commutes.

8 1.2.6 Dually, I is injective if and only if for any commutative diagram with an exact row I u [^[ [0 ψ [ •••u u α β there exists an arrow I ψ˜ fffi ff • such that the diagram

I u ψ˜ fffffi ff ψ ff •••u u α β commutes.

1.2.7 Characterization of projectivity in terms of extensions Lemma 1.2 An object P is projective if and only if every extension of P,

π ι P u u ••u , x splits.

Proof. Consider the diagram

idP . u π P u u •

If P is projective, then there exists ιP such that π ◦ ιP = idP .

9 Vice-versa, suppose that a diagram

α u e A00 u u A is given. Consider a pullback by α of the extension

e κ A00 u u A u A0 x where κ is a kernel of e, e¯ κ¯ P u u A¯ u A0 x

α α¯ u u e κ A00 u u A u A0 x ι If P A¯ splits the pullback extension, then α¯ ◦ ι lifts α. Indeed, w

e¯ ◦ (idA¯ −ι ◦ e¯) = 0 implies that 0 idA¯ −ι ◦ e¯ = κ¯ ◦ π for a unique π0 and 0 α ◦ e¯ = e ◦ α¯ = e ◦ α¯ ◦ idA¯ = e ◦ α¯ ◦ ι ◦ e¯ + e ◦ α¯ ◦ κ¯ ◦ π = e ◦ α¯ ◦ ι ◦ e¯ which in turn implies that α = e ◦ (α¯ ◦ ι) since e¯ is an epimorphism.

1.3 Categories of chain complices 1.3.1 The (strict) category of graded objects

The strict category of Z-graded objects has Z-sequences (Aq)q∈Z as its objects and Z -sequences of morphisms (φq)q∈Z ,

0 φq Aq u Aq ,(6) as morphisms. In order to avoid confusion with morphisms in the category of Z-graded chain complices Ch(A), we shall denote such sequences graded maps.

10 1.3.2 Ch(A) Morphisms in the category of chain complices are graded maps that commute with the boundary operators,

0 ∂q ◦ φq = φq−1 ◦ ∂q (q ∈ Z).

1.3.3 The graded category of graded objects We shall also consider graded maps of degree d, i.e., sequences of morphisms (φq)q∈Z in A, 0 φq Aq+d u Aq (q ∈ Z),(7) and often will refer to them simply as maps of degree d. The resulting graded category will be referred to as the graded category of graded objects of A.

1.3.4 Subcategories of Ch(A) The category of chain complices Ch(A) of an abelian category A has several naturally deﬁned subcategories, for example, for any pair of subsets J ⊆ I ⊆ Z, let ChI,J(A) denote the full subcategory of chain complices with chain objects being zero in degrees outside I ,

Cl = 0 for l < I, and homology objects being zero in degrees outside J ,

Hl = 0 for l < J.

Thus, ChZ,∅(A) denotes the subcategory of acyclic chain complices. When I = J we shall use notation ChI (A).

1.3.5 Ch+(A)

The subcategory Ch+(A) of complices vanishing at −∞, i.e., satisfying

Cl = 0 for l 0, is the union of the subcatories Ch≥n(A) of complices satisfying

Cl = 0 for l < n.

11 1.3.6 Ch-(A)

The subcategory Ch-(A) of complices vanishing at ∞, i.e., satisfying

Cl = 0 for l 0, is the union of the subcatories Ch≤n(A) of complices satisfying

Cl = 0 for l > n.

1.4 Functors associated with chain complices 1.4.1 The category of chain complices is studied together with with a number of associated functors.

1.4.2 The shift functors The shift functors

[n] : Ch(A) −→ Ch(A), C 7−→ [n]C,(8) where n ([n]C)l ˜ Cl−n, ([n]∂)l ˜ (−1) ∂l−n,(9) and ([n] f )q ˜ fq−n,(10) provide an action of the additive group of integers Z on Ch(A) and those subcategories of Ch(A) that are invariant under shifts.

1.4.3 The q-chain functors

Cl : Ch(A) −→ A, C 7−→ Cq (q ∈ Z).(11)

1.4.4 q-cycles functors Given a functor

ζl : Ch(A) −→ Arr A, C 7−→ ζq, that assigns to a chain complex C a kernel of ∂q , its composition with the source functor yields a q-cycles functor

Zq : Ch(A) −→ A, C 7−→ s(ζq).(12)

12 1.4.5 q-boundaries functors Given a functor

βq : Ch(A) −→ Arr A, C 7−→ βq, that assigns to a chain complex C an image βq of ∂q+1 , its composition with the source functor yields a q-boundaries functor

Bq : Ch(A) −→ A, C 7−→ s(βq).(13)

1.4.6 q-homology functors

The condition ∂q ◦ ∂q+1 means that ∂q+1 uniquely factorizes through ζq ,

∂q+1 = ζq ◦ ∂˜ q+1.(14)

Factorization (14) should not be confused with the image factorization of ∂q+1 ∂q+1 = βq ◦ ∂¯ q+1.(15)

Assigning to a chain complex C the target of a cokernel of ∂˜ q+1 deﬁnes a q-homology functor Hq : Ch(A) −→ A .(16)

1.4.7 The q-homology extensions

Functors Hq , Zq and Bq form an extension

χq β¯q Hq u u Zq u Bq (17) x because the source of an image of ∂q+1 is also the source of an image of ∂¯ q+1 .

1.4.8 q-co-cycles functors Given a functor

0 0 ζq : Ch(A) −→ Arr A, C 7−→ ζq,

0 that assigns to a chain complex C a cokernel ζq of ∂q+1 , its composition with the target functor yields a q-co-cycles functor

0 0 Zq : Ch(A) −→ A, C 7−→ t(ζq).(18)

13 1.4.9 q-co-boundaries functors Given a functor

0 0 βl : Ch(A) −→ Arr A, C 7−→ βq,

0 that assigns to a chain complex C a coimage βq of ∂q , its composition with the target functor yields a q-co-boundaries functor

0 0 Bq : Ch(A) −→ A, C 7−→ t(βq).(19)

1.4.10

Since an image factorization of ∂q is also its coimage factorization, func- 0 tors Bq and Bq+1 are isomorphic by a unique isomorphism of functors compatible with the corresponding kernel and cokernel functors.

1.4.11 q-co-homology functors 0 The condition ∂q ◦ ∂q+1 means that ∂q uniquely factorizes through ζq ,

0 ∂q = ∂q ◦ ζq.

Assigning to a chain complex C the source of a kernel of ∂q deﬁnes a q-co-homology functor 0 Hq : Ch(A) −→ A .(20)

1.4.12 The q-co-homology extensions

Functors Hq , Zq and Bq form an extension

¯0 0 βq χq B0 u u Z0 u H0 (21) q q x q because the target of a coimage of ∂q is also the target of a coimage of ∂q .

14 1.4.13 The diagrams of functor extensions By deﬁnition the above seven sequences of functors from Ch(A)to A enter four extensions that form the following commutative diagram

0 Hq v 0 χq u 0 0 ζq βq Z u u Cq u Bq q x v ¯0 βq β¯q (22) u u 0 u u C u Z Bq 0 q q βq ζq x

χq u Hq

Using the diagram chasing techniques, one can construct a canonical iso- 0 morphism of Hq with Hq , see the chapter on Diagram chasing in Notes on Category Theory for details.

1.4.14 By taking into account this isomorphism, we may redraw diagram (22) in the form Hq k h 4774 h 4 h 4 h 4 h 4 hkh 4 0 Zq Zq (23) k [^^ ﬂ h [  474 h ﬂ 4 4 h Cq h ‚“ 4 ‚‚ h AA ‚ 4 hkh A ‚ 4 A ‚“ 0 DA 7 Bq AD Bq

15 2 Fundamental lemmata

2.1 First Fundamental Lemma 2.1.1 Consider a diagram whose columns are chain complices

. . . .

∂2 ∂2 u u Q1 P1

∂1 ∂1 u u Q0 P0

e e

u f u N u M

u u 0 0 with the left column acyclic and Pl projective for all l ∈ N.

2.1.2 Since P is projective and

e Q0 N (24) w

16 f ◦e is an epimorphism, P N factorizes through (24), w

. . . .

∂2 ∂2 u u Q1 P1

∂1 ∂1 u u φ0 Q0 u P0

e e

u f u N u M

u u 0 0

2.1.3 Since e ◦ (φ0 ◦ ∂1) = f ◦ e ◦ ∂1 = 0, and the left column is exact at Q0 , arrow φ0 ◦ ∂1 factorizes through ∂1 Q1 Q0 , w

17 . . . .

∂2 ∂2 u u φ1 Q1 u P1

∂1 ∂1 u u φ0 Q0 u P0

e e

u f u N u M

u u 0 0

∂2 Exercise 4 Show that φ1 ◦ ∂2 factorizes through Q2 Q1 . w

2.1.4 This inductive procedure yields a morphism from the right column to the left column. If we denote by

P• = (Pl, ∂l)l∈N and Q• = (Ql, ∂l)l∈N the corresponding chain complices, then we established the ﬁrst fundamental fact of Homological Algebra.

Lemma 2.1 (First Fundamental Lemma) Given a morphism f N u M in an abelian category A, and a diagram of chain complices in A

Q• P•

e e

u [0] f u [0]N u [0]M

18 with chain complices P• and Q• concentrated in nonnegative degrees, all Pl being projective, and the left arrow quasiisomorphism, there exists a morphism of chain complices making the diagram

φ Q• u P•

e e (25)

u [0] f u [0]N u [0]M commute.

2.1.5 In this case we say that φ covers f . The vertical arrows in diagram (25) are represented by the augmentations e of complices P• and Q• by M and N , respectively.

19 2.2 Second Fundamental Lemma 2.2.1

Suppose that f = 0. In this case e ◦ φ0 = 0 and φ0 factorizes uniquely, in ∂0 view of projectivity of P0 , through Q1 Q0 , w . . . .

∂2 ∂2 u u φ1 Q1 u P1 \^\ \χ0 ∂1 \ ∂1 u u Q0 u P0 φ0

e e u u u N 0 M

u u 0 0

Exercise 5 Show that ∂1 ◦ (φ1 − χ0 ◦ ∂1) = 0

χ1 and that there exists an arrow P Q2 such that 1 w

φ1 = χ0 ◦ ∂1 − ∂2 ◦ χ1.

2.2.2 This inductive procedure yields a sequence of morphisms

χl P Ql+ (l ∈ N) l w 1 such that ∂l ◦ (φl − χl−1 ◦ ∂l) = 0 and φl = χl−1 ◦ ∂l − ∂l+1 ◦ χl (26)

20 for all l > 0. Identity (26) expresses the fact that morphism φ is the boundary of χ, a map of degree +1, in the Hom•(P, Q) chain complex

φ = ∂Homχ = [χ, ∂].

2.2.3 Chain homotopy Morphisms of chain complices

φ BAsk φ0 are said to by homotopic if there exists a degree +1 map χ such that

φ0 − φ = ∂Homχ = [χ, ∂].(27)

We may denote this fact by employing notation

φ0 ∼ φ.

A degree +1 map χ that satisﬁes dentity (27) is said to be a homotopy from φ to φ0 . We may denote this fact by employing notation

0 φ ∼χ φ and represent it diagrammatically

~ χ BA_

φ0 or φ

BA^ χ

φ0 (note what side of the wavy line the label χ is located; the placement indicates that χ is a homotopy from φ to φ0 ).

21 2.2.4 Null homotopic morphisms A morphism homotopic to the zero morphism is said to be null homotopic. In this case, χ such that

Hom φ ∼χ 0 , φ = ∂ χ = [χ, ∂] , is called a contracting homotopy. We shall also say that χ contracts φ or that χ is a contraction of φ, and will represent this diagramatically

φ BAs χ or φ u B f χ A

2.2.5 Two morphisms are homotopic if and only if their difference is null homotopic.

2.2.6 φ We proved above that a morphism of chain complices Q• u P• that 0 covers the zero arrow N u M is null homotopic. Its corollary is the following fact.

Lemma 2.2 (Second Fundamental Lemma) Any two morphisms of chain complices that make diagram (25) commute are homotopic.

2.3 Homotopy categories of an abelian category 2.3.1 Properties of the homotopy relation The zero arrow of degree +1 from A to B is a homotopy from φ to φ.

2.3.2 If χ is a homotopy from φ to φ0 , then it is also a homotopy from −φ0 to −φ and −χ is a homotopy from φ0 to φ.

22 2.3.3 If 00 0 0 φ ∼χ0 φ and φ ∼χ φ, then χ0 + χ is a chain homotopy from φ to φ00 .

2.3.4 In particular, the chain homotopy relation is an equivalence relation on HomCh(A)(A, B). Exercise 6 Show that if 0 0 φ1 ∼χ1 φ1 and φ2 ∼χ2 φ2 then 0 0 φ1 + φ2 ∼χ1+χ2 φ1 + φ2. Exercise 7 Suppose that β α B0 u B and A0 u A are morphisms of chain complices and 0 φ ∼χ φ. Show that 0 0 φ ◦ α ∼χ◦α φ ◦ α and β ◦ φ ∼β◦χ φ.

2.3.5 In other words, the chain homotopy relation is a congruence on any full subcategory of Ch(A). The corresponding quotient categories are referred to as the homotopy categories of A and are denoted K(A), K+(A), K−(A) and so on (“Ch” being replaced by K).

2.4 Projective resolution functors 2.4.1 A category with sufﬁciently many projectives We say that A has sufﬁciently many projectives if, for any object M, there exists an extension e κ M u u P u M (28) x 1 with P projective. The kernel arrow in (28) is called a 1-st syzygy of M and M1 is referred to as a 1-st syzygy object of M.

23 2.4.2 In particular, there exists an extension

e κ M u u 1 P u 1 M 1 1 x 2 and so on. There exists a sequence of extensions

el κl M u u P u Ml+1 (l ≥ 0) l l x with M0 ˜ M, P0 = P, and all Pl projective. By splicing them we obtain an acyclic augmented complex

e ∂1 ∂2 ... u 0 u M u P0 u P1 u ... (29) where ∂l ˜ κl−1 ◦ el (l > 0). It is referred to as an (augmented) projective resolution of M.

2.4.3 Syzygy objects

The objects Ml are referred to as l -th (projective) syzygy objects of M.

Exercise 8 Given a projective resolution P• of M, show that  HomA(Ml, N) for l ≥ 0 HomCh(A)(P•, [l]N) = (30) 0 for l < 0

Exercise 9 Show that null-homotopic morphisms P• −→ [l]N correspond to morphisms f ∈ HomA(Ml, N) which extend to Pl−1 . In particular,  HomA(Ml, N)  for l ≥ 0 { f ∈ Hom (M , N) which extend to P } HomK(A)(P•, [l]N) = A l l−1  0 for l < 0 (31)

24 2.4.4 According to Lemmata 2.1 and 2.2, any assignment on the class of objects of A, M 7−→ P• of its projective resolution together with an augmentation

P•

e u [0]M admits a unique extension to an additive functor P from A to the homotopy category K≥0,{0}(A) equipped with a natural transformation of functors

e u [0] which, for each object M, induces an isomorphism of the 0-th homology of PM with M,

H0(PM)

' . u M

2.4.5 Moreover any two such augmented functors are isomorphic by a unique isomorphism of functors

' P0 u P

e0 e (32) u u [0][0]

25 2.5 Third Fundamental Lemma 2.5.1 Consider a commutative diagram

A00 u u A u A0 πA ιA x

α00 α0 u u B00 u u B u B0 πB ιB x

β00 β β0 u u u C00 u u C u C0 πC ιC x

γ γ0 u u D u D0 ιD x with chain complices in columns, extensions in rows, and a monomorphism in the bottom row, such that A00 is projective and the right column is exact at C0 .

2.5.2 00 00 In view of projectivity of A , arrow α factorizes through πB ,

00 α = πB ◦ α˜ for some α˜ . Since

00 00 00 πC ◦ β ◦ α˜ = β ◦ πB ◦ α˜ = β ◦ α = 0, exactness of the C-row implies that

β ◦ α˜ = ι ◦ δ

26 for a unique arrow δ

A00 u u A u A0 \ πA ιA x 4 \ α00 4 \ α0 4 \ u 4 u 00 \δ 0 B β◦α˜ 4 \ B 4 \ β00 4 \ β0 4 \] u 46 u C00 u u C u C0 πC ιC x

γ γ0 u u D u D0 ιD x

2.5.3 Noting that 0 ιD ◦ γ ◦ δ = γ ◦ β ◦ α˜ = 0 and ιD is a monomorphism, we deduce

γ0 ◦ δ = 0.

In view of projectivity of A00 and exactness of the right column at C0 , arrow δ factorizes through β0 , δ = β0 ◦ δ˜ for some arrow δ˜, πA ιA A00 u u A u A0 [‹ x 4 ‹‹ [ ‹ ˜ 00 4 δ 0 α [ ‹‹ α 4 [ ‹ u 4 ‹› u 00 [δ 0 B β◦α˜ 4 [ B 4 [ β00 4 [ β0 4 [] u 46 u C00 u u C u C0 πC ιC x

γ γ0 u u D u D0 ιD x

27 2.5.4 In view of projectivity of A00 , the top row extension is split. Let π0 a left inverse of ιA . It is a unique arrow such that

00 0 idA −ι ◦ πA = ιA ◦ π

00 where ι is a right inverse of πA . Set 0 0 α ˜ α˜ ◦ πA + ιB ◦ α ◦ π .(33)

Exercise 10 Show that the diagram

πA ιA A00 u u A u A0 x

α00 α α0 u u u B00 u u B u B0 πB ιB x

commutes and β ◦ α = 0.(34)

Lemma 2.3 (Third Fundamental Lemma) Given a diagram whose columns

28 are chain complices . . . .

00 0 ∂2 ∂2 u u P1 Q1

00 0 ∂1 ∂1 u u P0 Q0 (35)

e00 e0 u u π ι M00 u u M u M0 x

u u u 0 0 0 with the right column acyclic and Pl projective for all l ∈ N, there exists a complex (R•, ∂•) augmented by M which is an extension of augmented complex

P•

e00 u M00 by augmented complex

Q•

e0 u M0

29 i.e., diagram (35) can be extended to a commutative diagram of chain complices

......

00 0 ∂2 ∂2 ∂2 u u u π1 ι1 P u u R u Q1 1 1 x

00 0 ∂1 ∂1 ∂1 u u u π0 ι0 P u u R u Q0 0 0 x

e00 e e0 u u u π ι M00 u u M u M0 x

u u u 0 0 0 with an extension in each row.

Exercise 11 Prove the Third Fundamental Lemma.

30 3 The cone of a morphism

3.1 Direct sum and matrix calculus 3.1.1 A direct sum of a family of objects

A direct sum of a family (ai)i∈I is, by deﬁnition, a coproduct c, (ιj)j∈I ,(36) equipped with a second family of morphisms

ιj aj u c , uniquely deﬁned by the identities ( idai when i = j πi ◦ ιj = (37) 0 when i , j

3.1.2 Matrix morphisms between direct sums

Given a matrix (αij)i∈I,j∈J of morphisms

αij bi u aj such that the set {i ∈ I | αij , 0} is ﬁnite for every j ∈ I , the family of morphisms from aj to d,

∑ ιi ◦ αij, i∈I deﬁnes a unique morphism α from a coproduct c of a family (aj)j∈j to a coproduct d of a family (bi)i∈I . For obvious reasons, we shall denote this morphism by ∑ ιi ◦ αij ◦ πj i∈I and call it the morphism associated with the matrix (αij)i∈I,j∈J .

3.1.3 Composition and addition of matrix morphisms corresponds to multiplication and addition of their matrices. Accordingly, we shall be representing such morphisms by their matrices and performing all calculations involving the morphisms by employing those matrices.

31 3.1.4 From now on we shall adopt the direct sum notation M aj (38) j∈I to denote a direct sum of (aj)j∈J equipped with the two families of morphisms (ιj)j∈J and (πj)j∈J . Note that (38) should be treated as a generic notation.

3.2 Cone( f ) 3.2.1 Given a map f of degree 0 from a chain complex A to a chain complex B, the direct sum of graded objects B and [1]A,

Cone( f )q ˜ Bq ⊕ Aq−1, equipped with the degree −1 morphism

B ! ∂q fq−1 ∂Cone ˜ q A 0 −∂q−1 is called a cone of f .

Exercise 12 Show that ! 0 ∂B ◦ f − ∂A ◦ f Cone Cone q q q−1 q−1 ∂q ◦ ∂q+1 = (39) 0 0

In particular,

∂Cone ◦ ∂Cone = 0 if and only if [ f , ∂] = 0.

3.2.2 By deﬁnition, a cone of the zero morphism Cone(0) is a direct sum of B and [1]A in the category of chain complices. The cone of a morphism of chain complices is thus an inﬁnitesimal deformation of B ⊕ [1]A.

32 3.2.3 The Cone Extension Exercise 13 Show that the maps

π[1]A ιB [1]ABu ⊕ [1]A and B ⊕ [1]A u B are morphisms of chain complices

[1]A u Cone( f ) and Cone( f ) u B .

Since the composable pair

π[1]A ιB [1]A u Cone( f ) u B (40) is in every degree a split extension associated with a direct sum Bq ⊕ Aq−1 , it is an extension in the category of chain complices. We shall refer to (40) as the cone extension.

3.2.4 A particularly important feature of the cone extension is that the morphisms Hq( f ) induced by f in homology coincide with the connecting homomorphisms δq+1 of the homology long exact sequence associated with extension (40).

3.2.5 The cone functor If we ﬁx a binary direct sum functor on the underlying category A, we obtain a functor, denoted Cone, from the category of arrows Arr Ch(A) to Ch(A).

Exercise 14 Indeed, given a square of morphisms of complices

φss A0 o A (41)

f 0 f

B0 o B φtt show that 0 ! 0 φtt ◦ f − f ◦ φss [φ, ∂Cone] = 0 0

33 where ! φtt 0 φ ˜ (42) 0 φss In particular, (42) is a chain complex morphism if and only if square (41) commutes.

3.2.6 Note that

A [1]A = Cone 0 and B = Cone 0B , where

A 0 ˜ 0 u A and 0B ˜ B u 0 , and the commutative diagram

AA u 0

f u u u 0 u BB is an extension in the category of arrows Arr Ch(A),

A u u f u 0B (43) 0 x which yields the Cone extension, cf. (40), when one applies the cone functor to (43).

3.2.7 ↓ If we denote by 0 and 0↓ the functors Ch(A) −→ Arr Ch(A) that send a complex C to C 0 and, respectively, 0C , then extensions (43) give rise to an extension of functors on Arr Ch(A),

↓ ◦ u u idArr Ch(A) u 0↓ ◦ t 0 s x where s and t denote the source and the target functors from Arr Ch(A) to Ch(A).

34 3.2.8 A morphism of extensions in Ch(A)

π ι A00 o o A A o A o A0

f 00 f f 0

B00 o o B o o B0 πB ιB induces an extension of the corresponding cones

Cone( f 00) o o π Cone( f ) o ι o Cone( f 0) , the cone extension, (40), for example, being induced by the morphism of trivial extensions A A o 0

0 o B B

3.2.9 It follows that Cone induces a functor from the category of arrows of the category of extensions of chain complices to the category of extensions of chain complices

Cone Arr Ext Ch(A) Ext Ch(A) . w In particular, Cone Arr Ch(A) Ch(A) w is an exact functor. We shall see an explanation of this fact later.

3.2.10 Consider the extension [1]A o o Cone(ιB) o o Cone idB (44)

35 induced by the morphism of extensions

o 0 B B (45)

ιB [1]A o o Cone( f ) o o B π[1]A ιB

Exercise 15 Find an explicit isomorphism Cone(ιB) ' Cone idB ⊕ [1]A by ﬁnding ﬁrst a splitting of extension (44) in the category of chain complices.

3.2.11 Consider the extension

o o o o Cone id[1]A Cone π[1]A [1]B (46) induced by the morphism of extensions

π[1]A ι [1]A o o Cone( f ) o B o B (47)

π[1]A [1]A [1]A o 0

Exercise 16 Find an explicit isomorphism Cone π[1]A ' [1]B ⊕ Cone id[1]A by ﬁnding ﬁrst a splitting of extension (46) in the category of chain complices.

3.3 The cone diagram associated with an anticommutative square 3.3.1 An anticommutative square of chain complices

g1 A01 o A11 (48)

f0 + f1 A00 o A10 g0

36 is a 3-complex with

Apqr = Apq,r (p, q, = 0, 1, r ∈ Z),

whose boundary operators in p-direction are provided by gp , in q-direction — by fq , and in r-direction — by the boundary operators of the corresponding complices.

3.3.2 Its total complex in degree n is the direct sum

A00,n ⊕ A10,n−1 ⊕ A01,n−1 ⊕ A11,n−2

equipped with the boundary operators

 00  ∂n g0,n−1 f0,n−1    10   −∂n−1 f1,n−2      (49)  − 01 g   ∂n−1 1,n−2   11 ∂n−2

By exchanging A10 and A01 in the direct sum, one obtains an alternate form of the matrix representation of the boundary operator

 00  ∂n f0,n−1 g0,n−1    10   −∂n−1 g1,n−2     (50)  − 01 f   ∂n−1 1,n−2    11 ∂n−2

3.3.3

Let us denote the total complex of (48) as Cone(). Formulae (49)–(50) mean that it is canonically identiﬁed with

Cone(f) and Cone(g)

where f f f ˜ 0 = 0 −[1] f1 ~1 f1

37 is the induced morphism between the g-cones

Cone(g0) u Cone(g1) and g g g ˜ 0 = 0 −[1]g1 ~1g1 is the induced morphism between the f -cones

Cone( f0) u Cone( f1) .

Here and below ~l f denotes (−1)l[l] f . This, as we shall soon see, is a proper way to deﬁne the shift functors on the category of arrows Arr Ch(A). The main advantage at this point of using ~ versus [], is that Cone ~l f = [l] Cone( f ).

3.3.4 A consequence of this observation is the existence of the following diagram

g1 [1]A11 o o Cone(g1) o o A01 o A11 (51)

−[1] f1 f f0 + f1

g0 [1]A10 o o Cone(g0) o o A00 o A10

g [1] Cone( f1) o o Cone() o o Cone( f0) o Cone( f1)

−[1]g1 [2]A11 o o [1] Cone(g1) o o [1]A01 o [1]A11

38 or, using the ~ notation,

g1 [1]A11 o o Cone(g1) o o A01 o A11 (52)

~1 f1 f f0 + f1

g0 [1]A10 o o Cone(g0) o o A00 o A10

g [1] Cone( f1) o o Cone() o o Cone( f0) o Cone( f1)

~1g1 [2]A11 o o [1] Cone(g1) o o [1]A01 o [1]A11

3.3.5 All squares commute except the original one that generated the whole picture, and located in the right top corner. That single square anticomm- mutes.

3.3.6 The diagram contains 8 extensions. In the left 3 × 2 subdiagram all 4 rows are the cone extensions associated with the 4 morphisms on the right. In the bottom 2 × 3 subdiagram all 4 columns are the cone extensions that are similarly associated with the 4 morphisms at the top of the diagram.

3.3.7 One should think of the 3 × 3 subdiagram of extensions located in the bottom left corner as being the 2-dimensional version of the cone extension.

39 The latter relates the cone of an arrow (a 1-dimensional ‘cell’ of the category of complices) to its target and source (0-dimensional ‘faces’). The former relates the cone of an anticommutative square Cone() to the cones of its 1- and 0-dimensional faces, if we agree to consider the cone functor on 0-diemsional ‘cells’, i.e., objects of Ch(A), to be the identity functor

0 Cone ˜ idCh(A) .

3.3.8 n-dimensional cone functors Totalization of the n-dimensional cube involving 2n chain complices with anticommuting 2-dimensional faces can be regarded as the n-dimensional cone functor. For n = 1, we obtain the original cone functor on arrows of Ch(A), for n = 0, we obtain the identity functor on Ch(A).

3.3.9 The role of the n-dimensional cone extension is played by the n-dimensional diagram involving 3n complices forming 3n−1n one-dimensional cone extensions.

3.3.10 Cone(g ◦ f ) Given a composable pair of morphisms of chain complices

g f CBo o A the above 4 × 4-diagram associated with the anticommutative square

− id A o A A

g◦ f + f

o CBg

40 has the following form

− idA [1]A o o Cone(− idA) o o A o A (53)

−[1] f f g◦ f + f

g [1]B o o Cone(g) o o C o B

g [1] Cone( f ) o o Cone() o o Cone(g ◦ f ) o Cone( f )

id[1]A o o o o o [2]A Cone id[1]A [1]A [1]A where g ◦ f g ◦ f f ˜ = (54) −[1] f ~1 f is a morphism between the cones

Cone(g) u Cone(− idA) and g g g ˜ = id[1]A −~1 idA is a morphism between the cones

Cone(g ◦ f ) u Cone( f ) .

41 3.3.11 One can consider diagram (53) as expressing the relation between the cone of a composable pair of arrows (a ‘2-simplex’ of the category Ch(A)), to the cones of its 1-dimensional ‘faces’ f , g and g ◦ f . In this respect the most important are the second row from the bottom and the second column from the left. This is the essence of the so called Octahedron Axiom of triangulated categories.

Exercise 17 Find an explicit isomorphism Cone() ' Cone id[1]A ⊕ Cone(g) by ﬁnding ﬁrst a splitting of the cone extension of f, given by (54),

Cone(g)

Cone()

Cone id[1]A in the category of chain complices.

φ 3.4 Morphisms Cone( f 0) u Cone( f ) 3.4.1 Given a pair of arrows in the category of chain complices

A0 A

f 0 f

B0 B

φ a morphism Cone( f 0) u Cone( f ) is represented by a matrix ! φtt φts φ = (55) φst φss

42 or, diagramatically, by the square with 6 arrows

φss 0 o A _ A (56)

φst

f 0 f

φts ~ B0 o B φtt where φst is of degree −1, arrows φtt and φss are of degree 0, and φts is of degree +1.

Exercise 18 Show that   [φ , ∂] − f 0 ◦ φ φ ◦ f − f 0 ◦ φ − [φ , ∂] tt st tt ss ts φ, ∂Cone =   (57) [φst, ∂] φst ◦ f − [φss, ∂] where [ , ] denotes the supercommutator of graded maps.

3.4.2 The meaning of the condition φ, ∂Cone = 0 The integrability condition

φ, ∂Cone = 0 translates into 4 separate conditions.

3.4.3

The st-condition says that φst is a morphism of chain complices

φst [1]A0 u B .

3.4.4 0 The tt-condition says that φtt is a contracting homotopy for f ◦ φst while the ss-condition says that φss is a contracting homotopy for φst ◦ f . Dia-

43 grammatically,

0 0 φss A _ and A ` A

φst f 0 f φst B0 B B φtt

3.4.5 0 It follows that both φtt ◦ f and f ◦ φss contract the triple composite

0 f ◦ φst ◦ f .

0 Finally, the ts-condition says that φts is a homotopy from f ◦ φss to φtt ◦ f . Diagrammatically, φss A0 o A

f 0 f φts B0 o B φtt

3.5 The matrix homotopy category of arrows M(A) 3.5.1 Deﬁne M(A) to be the category whose objects are arrows of Ch(A) and a morphism φ from f to f 0 consists of a morphism from the target of the source arrow to the source of the target arrow (shifted by 1),

υ : t( f ) −→ [1]s( f 0),

0 a pair of homotopies ϕt and ϕs contracting f ◦ υ and υ ◦ f , respectively, and a further homotopy ψ between the homotopies

0 f ◦ϕs

~ ψ BA_

ϕt◦ f

44 3.5.2

We shall refer to ϕt and ϕs as the primary homotopies, and to ψ as the secondary homotopy.

3.5.3 Fixing A, B, A0 and B0 , yields a full small subcategory

M(A)B0 A0|BA which besides being preadditive has an abelian group structure on the set of objects.

3.5.4 The category of arrows UT(A) The category of arrows and commutative squares UT(A) is a subcategory of M(A) with morphisms being precisely the quadruples (0; ϕt, ϕs; ψ).

3.5.5 Note that the set of morphisms

φ f 0 u f with ﬁxed (υ; ϕt, ϕs) is a torsor over the group HomCh(A)(A, [−1]B) while 0 the set of morphisms in M(A) with ﬁxed υ is a torsor over HomUT(A)( f , f ).

3.5.6 The shift functors

If we deﬁne the shift functors ~l on M(A) by the correspondence

A [l]A (58)

f / (−1)l f B [l]B on objects, and the correspondence

(υ; ϕt, ϕs; ψ) 7−→ ([l]υ; [l]ϕt, [l]ϕs; [l]ψ) (59)

45 on morphisms, then the correspondence

f 7−→ Cone( f ) on objects and ! ϕt ψ (υ; ϕt, ϕs; ψ) 7−→ υ ϕs on morphisms, deﬁnes an epifunctor,

Cone : M(A) −→ Ch(A),(60) i.e., a functor that is surjective on the class of objects and on the class of arrows, which commutes with the shift functors,

Cone(~l f ) = [l] Cone( f ). This functor extends the cone functor from subcategory Arr Ch(A).

3.5.7 Caveat Note that ~l f = (−1)l[l] f . On the left hand side f is an object of M(A), on the right hand side f is a morhism of Ch(A). The sign difference is necessitated by the following considerations. Each of the shift functors must preserve the identity morphisms, as a consequence the shift functors cannot reverse the sign of morphisms of shifted objects. At the same time shifts of morphisms in M(A) remain morphisms only if the shifts of arrows consider as objects of M(A), change sign simultaneusly with the boundary operators ∂. This is compatible with the fact that objects of M(A) are constituent parts of the boundary operators of their cones.

3.5.8 The category of arrows and homotopy commutative squares UT(A)

Morphisms with υ = 0 are triples (φt, φs; ψ) that describe homotopy commutative squares φs A0 o A

f 0 f ψ B0 o B φt

46 They correspond to those matrices (55) that are upper triangular (note that morphisms in Arr Ch(A) correspond to diagonal matrices).

3.5.9 The special case A A (61)

f 0 f ψ BB corresponds to ψ being a homotopy from f 0 to f or, equivalently, −ψ being a homotopy from f to f 0 .

Exercise 19 Show that A A

f f 0 −ψ BB is the inverse of (61).

3.5.10 It follows that the cones of homotopic morphisms of chain complices are isomorphic in Ch(A), an isomorphism functorially depending on a particular homotopy from f 0 to f .

A 3.6 Morphisms between 0 and 0B 3.6.1 We shall investigate morphisms in M(A) and in UT(A) between arrows whose source or target is 0. From now on we adopt the convention that the component arrows of a morphism not indicated in the morphism diagram are tacity assumed to be zero.

47 3.6.2 Note that

A _ 0 υ

0 B is a morphism A 0B −→ 0 in M(A) precisely when υ is a morphism

B −→ [1]A in Ch(A). In particular,

A A HomM(A)(0B, 0 ) ' HomCh(A)(B, [1]A) and HomUT(A)(0B, 0 ) = 0.

3.6.3 Similarly, 0 A

ψ B 0 is a morphism A 0 −→ 0B in M(A) precisely when ψ is a morphism

[1]A −→ B in Ch(A). In particular,

A A HomM(A)(0 , 0B) = HomUT(A)(0B, 0 ) ' HomCh(A)([1]A, B).

3.6.4 In the category of arrows one has, of course,

A A HomArr Ch(A)(0B, 0 ) = 0 and and HomArr Ch(A)(0B, 0 ) = 0.

48 3.6.5 Note that

A A0 A A0 0 HomM(A)(0 , 0 ) = HomArr Ch(A)(0 , 0 ) ' HomCh(A)(A, A ) and

0 HomM(A)(0B, 0B0 ) = HomArr Ch(A)(0B, 0B0 ) ' HomCh(A)(B, B ).

3.7 Adjunctions 3.7.1 The functor identities

↓ Cone ◦0↓ = idCh(A) = [0] and Cone ◦0 = [1]

↓ mean that 0↓ is a right inverse of Cone while 0 is a right inverse of

[−1] ◦ Cone = Cone ◦ ~−1 .

↓ In particular, 0↓ and ~−1 ◦ 0 are both right inverses of the cone functor.

3.7.2 The canonical identiﬁcation

0 0 HomM(A)( f , f ) u Hom (A) Cone( f ), Cone( f ) w Ch

↓ means that 0↓ and ~−1 ◦ 0 are both right and left adjoint to Cone.

3.7.3 ↓ This, in turn, means that 0↓ and ~−1 ◦ 0 are isomorphic as functors

Ch(A) −→ M(A).

There exists a canonical isomorphism

↓ 0↓ ' ~ − 1 ◦ 0 .

49 It is provided by the families of mutually inverse natural transformations given by the diagrams

[−1]C 0 and 0 [−1]C (62) a id id } 0 C C 0

Exercise 20 Show that the two morphisms in (62) are mutually inverse.

3.7.4 Isomorphism 0↓ ◦ Cone ' idM(A) Exercise 21 Show that the diagrams

A c 0 and 0 A (63)

π[1]A f f ι[1]A { B o Cone( f ) Cone( f ) o B πB ιB deﬁne morphisms in M(A) and that these morphisms are mutually inverse.

It follows that the cone functor provides an equivalence of categories between M(A) and Ch(A), with the functor 0↓ (or isomorphic to it functor ~−1 ◦ 0↓ ) providing an “inverse” equivalence.

3.7.5 Noting that

HomUT(A)( f , 0C) = HomM(A)( f , 0C), but HomUT(A)(0C, f ) , HomM(A)(0C, f ), and

C C C C HomUT(A)(0 , f ) = HomM(A)(0 , f ) but HomUT(A)( f , 0 ) , HomM(A)( f , 0 ), we observe that if we consider Cone as a functor UT(A) −→ Ch(A), then ↓ 0↓ is its right but not left adjoint while [−1] ◦ 0 is its left but not right adjoint.

50 3.8 Graded-split extensions 3.8.1 An extension of chain complices

π ι C00 u u C u C0 (64) x is said to be graded-split if it is split in the category of Z-graded objects, i.e., in every degree is a split extension in A.

3.8.2 If ι00 : C00 −→ C is a map of degree 0 that supplies a right inverse to π in the category of graded objects AZ , then let π0 : C −→ C0 be the corresponding graded projection onto C0 . Recall that it is deﬁned by the identity

00 0 idC −ι ◦ π = ι ◦ π .

The quartet 00 00 ι / o ι 0 C o C / C π π0 represents C as a direct sum of C00 and C0 in the category of Z-graded objects.

3.8.3 Graded-split epimorphisms and monomorphisms Morphisms that occur as π in a graded-split extension (64) are called graded-split epimorphisms. Morphisms that occur as ι are called graded-split monomorphisms. The former are precisely those morphisms that admit a graded right inverse ι00 , while the latter are those morphisms that admit a graded left inverse π0 .

3.8.4 Exercise 22 Show that π ◦ [ι00, ∂] = 0.

In particular, [ι00, ∂] = ι ◦ φ

51 for a unique morphism of chain complices

φ 0 C00 [1]C .(65) w

We can represent (65) as [1] f = −~1 f for

00 [−1]φ f ˜ [−1]C C0 . w Exercise 23 Show that

[−1]C00 0 and 0 [−1]C00 (66) b π f f ι00 | C0 o C CCo 0 π0 ι deﬁne morphisms in M(A) and that these morphisms are mutually inverse.

3.8.5 It follows that extension (64) is isomorphic to the cone extension of f

π ι C00 u u C u C0 x

u 00 [1] [−1]C u u Cone( f ) u C0 x

3.8.6 A graded splitting σ [1]A Cone( f ) w of the cone extension is represented by the 2 × 1 matrix ! h

id[1]A for some map h : [1]A −→ B of degree 0.

52 Exercise 24 Show that [σ, ∂] is represented by the matrix ! [h, ∂] − f (67) 0 i.e., σ is a morphism of complices if and only if h is a homotopy that contracts f .

3.8.7 In particular, the cone extension of f is split if and only if f is null- homotopic, and the set of splittings is in a natural bijective correspondence with the set of homotopies contracting f .

53 4 Contractible complices

4.1 Basic properties 4.1.1

A complex C is said to be contractible if idC is null-homotopic, i.e., if

idC = [h, ∂] (68) for some degree +1 graded endomorphism h of C. The latter is referred to as a contracting homotopy or a contraction of C.

4.1.2 It follows from the discussion in Section 3.8.7 that a complex C is contractible if and only if the cone extension

π[1]C ιC [1]C u u Cone(id ) u C C x splits. Exercise 25 If h is a contracting homotopy, then the following identities hold (h ◦ ∂)2 = h ◦ ∂, (∂ ◦ h)2 = ∂ ◦ h and ∂ ◦ h2 ◦ ∂ = 0.(69)

4.1.3 Direct sums of contractible complices

A direct sum of a family (Ci)i∈I of contractible complices is contractible with ∑ ιi ◦ hi ◦ πi i∈I being a contracting homotopy where hi is a homotopy that contracts complex Ci .

4.1.4 Direct summands of contractible complices An object a in a preadditive category is said to be a direct summand of an object a0 if there exist morphisms

ι / 0 a o a π 0 such that π ◦ ι = ida . We shall refer to ι as the inclusion into a and to π as the projection onto a.

54 Exercise 26 If C is a direct summand of a contractible complex C0 with a contracting homotopy h0 , then h ˜ ι ◦ h0 ◦ π contracts C. It is integrable if h0 is integrable.

4.2 Characterizations of contractible complices.

4.2.1 Example: Cone(idC) Exercise 27 Show that 0 0 h ˜ (70) idC 0 contracts Cone(idC).

Note that the idC in (70) occurs as a degree +1 map from C to [1]C.

4.2.2 The ﬁrst characterization of contractible complices. By combining this with the result of Section 4.1.2, we obtain the following characterization of contractible complices in any additive actegory.

Lemma 4.1 A complex is contractible if and only if it is a direct summand of Cone(idA) for some complex A.

4.2.3 The second characterization of contractible complices. All the considerations of the chapter devoted to the cone construction so far were valid in any additive category. Under an additional hypothesis we shall establish that a contractible complex C is isomorphic to Cone(idZC) for the complex of cycles of C, the latter being equipped with zero boundary operators.

Proposition 4.2 Let C be a complex such that kernels and coimages of ∂C exist. If C is contractible, then it is isomorphic to Cone(idZC).

Proof. Under the hypothesis, C is an extension of the complex B0C of co-boundaries by the complex of cycles

β0 ζ B0C u u CZCu ,(71) x

55 both equipped with zero boundary operators. Given a contraction h of C, the graded map h ◦ ∂ uniquely factorizes through β0 ,

h ◦ ∂ = σ ◦ β0 and

0 0 0 0 0 0 β ◦ σ ◦ β = β ◦ h ◦ ∂ = β ◦ (h ◦ ∂ + ∂ ◦ h) = β ◦ idC = idB0C ◦β demonstrates, in view of β0 being epi, that

0 β ◦ σ = idB0C, i.e., σ is a graded splitting of extension (71).

4.2.4 By calculating

0 [σ, ∂] ◦ β0 = (σ ◦ ∂B C − ∂C ◦ σ) ◦ β0 = −∂ ◦ h ◦ ∂ = −∂ ◦ σ ◦ β0, we determine that extension (71) is isomorphic to the cone extension

B0C u u Cone( f ) u CZ x for f being the unique morphism from [−1]B0C to ZC such that

ζ ◦ [1] f = −∂ ◦ σ.

4.2.5 Note that

ζ ◦ [1] f ◦ (−β0 ◦ h ◦ ζ) = ∂ ◦ h ◦ ∂ ◦ h ◦ ζ = ∂ ◦ h ◦ ζ = (∂ ◦ h + h ◦ ∂) ◦ ζ = ζ implies, in view of ζ being mono, that

0 [1] f ◦ (−β ◦ h ◦ ζ) = id[1]ZC . Similarly,

(−β0 ◦ h ◦ ζ) ◦ [1] f ◦ β0 = β0 ◦ h ◦ ∂ ◦ σ ◦ β0 = β0 ◦ h ◦ ∂ ◦ h ◦ ∂ = β0 ◦ h ◦ ∂ = β0 ◦ (h ◦ ∂ + ∂ ◦ h) = β0

56 implies, in view of β0 being epi, that

0 (−β ◦ h ◦ ζ) ◦ [1] f = idB0C .

Thus, [1] f is an isomorphism between B0C and [1]ZC. This, in turn, induces an isomorphism between Cone( f ) and Cone(idZC)

f ZC o [−1]B0C

ZCZC which ﬁts into the following diagram of isomorphisms of extensions

β0 ζ B0C u u CZCu x

' u B0C u u Cone( f ) u ZC x

[1] f ' ' u u [1]ZC u u Cone(id ) u ZC ZC x

4.3 Characterization of null-homotopic morphisms 4.3.1 Exercise 28 If a morphism f factorizes through a complex C with a contractible homotopy h, f o BA[

f 0 f 00 Ó C

57 then f 0 ◦ h ◦ f 00 is a contracting homotopy for f .

4.3.2 Vice-versa, if h is a homotopy that contracts a morphism f , then the diagram of composable morphisms in M(A) 0 A 0 (72)

h BAo o A f idA represents a canonical factorization of f through Cone(idA).

4.3.3 Alternatively, the diagram

0 [−1]B 0 (73) a f id } B [−1]BAo h represents a canonical factorization of f through Cone(idB). Exercise 29 Explain why diagram (73) represents morphisms in M(A).

4.3.4

Note that factorization of f through Cone(idA) is realized via morphisms in UT(A). The two factorizations are equivalent in the sense that they are obtained from a single triple factorization in M(A)

0 [−1]B o h A 0 (74) a f id } B [−1]BAo o A h idA

58 by composing the ﬁrst two, or the last two morphisms in M(A)

4.3.5 We arrive at the following characterization of null-homotopic morphisms in Ch(A).

Lemma 4.3 A morphism of complices is null-homotopic if and only if it factorizes through a contractible complex.

4.3.6 A very important consequence is that the homotopy category of A is a quotient of Ch(A) by a subcategory of contractible complices.

4.4 Graded-split projective and graded-split injective complices 4.4.1 We say that a complex P is graded-split projective if it has the Lifting Property for the class of graded-split epimorphisms.

4.4.2 Graded-split injective complices are deﬁned dually as those complices that have the Extension Property for the class of graded-split monomorphisms.

4.4.3 Consider a diagram

Cone(idC)

[1]A o Cone( f ) π[1]A

59 The two arrows are represented by the following morphisms in M(A)

ϕ o A A and A _ C υ f

0 B 0 C and ϕ o A A _ C υ f

0 BCo f ◦ϕ represents a canonical factorization of g through Cone( f ).

4.4.4 Dually, consider a diagram

B Cone( f ) o i B

Cone(idC) The two arrows are represented by the following morphisms in M(A)

A 0 and C _ 0 υ f

o BB CBϕ and ϕ◦ f o C _ A 0 υ f o CBϕ B represents a canonical factorization of g through Cone( f ).

60 4.4.5 According to Section 3.8.5 every graded-split epimorphism is isomorphic to [1]A [1]A o π Cone( f ) , for some morphism of complices f , and every graded-split monomorphism is isomorphic to B Cone( f ) o i B , thus we arrive at the following characterization of graded-split projective and graded-split injective complices.

Lemma 4.4 The following three properties of a chain complex are equivalent

(a) C is graded-split projective;

(b) C is graded-split injective;

4.5 Contractions and higher homotopies 4.5.1 Integrable contracting homotopies Let h be a contracting homotopy of a contractible complex C.

Exercise 30 Show that also h0 ˜ h ◦ ∂ ◦ h (75) is a contracting homotopy.

Exercise 31 Show that h0 deﬁned in (75) satisﬁes the identity

h0 ◦ h0 = 0.(76)

A contracting homotopy satisfying (76) will be said to be integrable.

61 4.5.2 The set of integrable contracting homotopies In Exercises 30 and 31 we established that, for every contractible complex C, the set of integrable contracting homotopies is nonempty. The set of all contracting homotopies is a torsor over the group

HomCh(A)(C, [−1]C), i.e., the difference between any two contracting homotopies

h0 − h = φ is a morphism from C to [−1]C.

Exercise 32 Suppose that h is an integrable contracting homotopy. Show that h + φ is integrable if and only if the morphism φ : C −→ [−1]C satisﬁes the Maurer-Cartan equation [φ, h] + φ2 = 0.(77)

4.5.3 The integrability property of a contracting homotopy is equivalently expressed ia the identity 2 (∂ + h) = idC .(78)

4.5.4 When performing calculations involving supercommutators one is fre- quently using the supercommutator Leibniz Rule.

Exercise 33 (Supercommutator Leibniz Rule) Prove the following identity valid in any associative ring-type structure involving even and odd elements

˜ [ab, c] = a[b, c] + (−1)bc˜[a, c]b (79) where a˜ is the parity of a, understood to be an element of Z/2Z.

Exercise 34 Show that  0 if n is even [hn, ∂] = .(80) hn−1 if n is odd

62 4.5.5 Thus, for any contracting homotopy, h2 : C −→ [−2]C is a morphism of chain complices.

Exercise 35 Show that h2 is null-homotopic and ﬁnd a homotopy contracting h2 .

Exercise 36 If h3 is a graded endomorphism of C of degree 3 such that

2 [h3, ∂] + h = 0, then [h3, h] is a morphism of complices C −→ [−4]C.

4.5.6

The supercommutator [h3, h] is, in fact, again null-homotopic.

4.5.7 Consider a formal series

h ˜ h1 + h3 + h5 + ··· (81) where hi is a a degree i graded endomorphism of C, i.e., a map C −→ C of degree i. The equation

2 (∂ + h) = idC (82) expresses the inﬁnite sequence of equations

idC = [h1, ∂] 2 0 = [h3, ∂] + h1

0 = [h5, ∂] + [h3, h1] ··· (83)

0 = [h2n+1, ∂] + ∑ hihj i+j=2n−1 i, j odd ···

63 or, equivalently, of chain homotopies

idC ∼h1 0 2 0 ∼h3 h1

0 ∼h5 [h3, h1] ··· (84) ∼ 0 h2n+1 ∑ hihj i+j=2n−1 i, j odd ···

4.5.8 A contraction h of C is integrable precisely when

h = h + 0 + ··· is a solution of equation (82).

4.5.9 We shall demonstrate that, for any contraction h of C, there exists a sequence of higher homotopies h3 , h5 , ... , such that formal series (81), with h1 = h, is a solution of equation (82). We shall seek the solution in the form h = F(h) where F(h) is a formal power series in h

i F(h) = h + ∑ aih (ai ∈ Z). i>2, odd The sequence of equations (83), for i > 1, becomes the sequence of equations in unknown coefﬁcients ai ,

0 = a2n+1 + ∑ aiaj (n > 0) (85) i+j=2n−1 i , j odd which are equivalent to a single functional equation F − h 0 = + F2 h

64 or, equivalently, hF2 + F − h = 0.(86) The sole solution of (82) of the form F(h) = h + ··· is given by the Taylor power series expansion at 0 of the function p −x + 1 + (2x)2 F(x) ˜ 2x (−2)n−1(2n − 3)!! = x + ∑ x2n−1 n>1 n! = x − x3 + 2x5 − 5x7 + 14x9 + ··· .(87)

4.5.10

A degree 3 graded map h3 that satisﬁes the second equation in (83) differs 3 from −h by a morphism of chain complices h¯ 3 : C −→ [−3]C, 3 h3 = −h + h¯ 3. Exercise 37 Show that 5 h5 ˜ 2h + hh¯ 3h satisﬁes the third equation in (83).

4.5.11 (2n − 1)-contractions Let us call h2n−1 = h1 + h3 + ··· + h2n−1 a (2n − 1)-contraction of a complex C if the ﬁrst n equations (83) are satisﬁed.

4.5.12 A homotopy contraction of C is the same as a 1-contraction. We showed above that any 1-contraction extends to an ∞-contraction.

4.5.13 Question Does any (2n − 1)-contraction extend to a (2n + 1)-contraction? Above we showed that the answer is positive for 1- and 3-contractions.

65 5 Contractions of Cone( f ) and homotopy equivalences

φ 5.1 Null-homotopic morphisms Cone( f 0) u Cone( f ) 5.1.1

χ A degree +1 graded map Cone( f 0) u Cone( f ) is represented by a matrix ! χtt χts χ = .(88) χst −χss with entries being degree +1 graded maps

χtt χts B0 u B , B0 u [1]A , and χst −χss [1]A u B , [1]A0 u [1]A .

5.1.2 φ Given a null-homotopic morphism Cone( f 0) u Cone( f ) contracted by a homotopy χ, the identity

φ = [χ, ∂] translates into 4 identities:

0 0 φtt = f ◦ χst + [χtt, ∂], φts = f ◦ χss − χtt ◦ f + [χts, ∂] (89) and φst = [χst, ∂], φss = χst ◦ f + [χss, ∂] .(90)

5.1.3 In those 4 identities we take into account that, in terms of the input complices A, B, A0 and B0 , the ‘off-diagonal’ entries of (88) have degrees 0 and 2, and thus are even, while the diagonal entries have degree 1, and thus are odd. This affects the signs in the correspoding suppercommutators.

66 5.2 Null-homotopic morphisms in M(A) 5.2.1 The above identities translate into the following notion of homotopy in the matrix homotopy category M(A). A morphism (υ; ϕt, ϕs; θ) is null- homotopic precisely when the following occurs: υ is null-homotopic in Ch(A), i.e., υ ∼η 0 ,(91) 0 for some map η of degree 0 from B to A , the primary homotopies ϕt and 0 ϕs are homotopic to f ◦ η and η ◦ f , respectively, 0 ϕt ∼χt f ◦ η and ϕs ∼χs η ◦ f ,(92) 0 and the secondary homotopy ψ is homotopic to f ◦ χs − χt ◦ f , 0 ψ ∼θ f ◦ χs − χt ◦ f .(93)

A quartet (η; χt, χs; θ) will be referred to as a contracting homotopy for a morphism (υ; ϕt, ϕs; ψ) in M(A).

5.2.2 Homotopy split commutative squares We shall say that a strictly commutative square

ϕs A0 o A (94)

f 0 f

B0 o B ϕt is homotopy split if there exists a morphism of chain complices η such that both triangles in ϕs 0 o A _ A (95) η f 0 f

B0 o B ϕt are homotopy commutative. A homotopy splitting of the commutative square consists of the morphism η and the corresponding homotopies 0 ϕt ∼χt f ◦ η and ϕs ∼χs η ◦ f .

67 5.2.3 The resulting 2 homotopies between 0 0 ϕt ◦ f = f ◦ ϕs and f ◦ η ◦ f may or may not be homotopic. If they are, and 0 χt ◦ f ∼θ f ◦ χs , we say that a homotopy splitting is strong and the quartet (η; χt, χs; θ) is then referred to as a strong homotopy splitting data. If the square admits a strong homotopy splitting, we say that it is strongly homotopy split.

5.2.4 Commutative squares null-homotopic in M(A) A commutative square is null-homotopic in M(A) precisely when it is strongly homotopy split.

5.2.5

Providing a contracting homotopy for (0; ϕt, ϕs; 0) in M(A) is equivalent to supplying the splitting morphism η , a pair of primary homotopies χt 0 and χs between ϕt and f ◦ η , and between ϕs and η ◦ f and, ﬁnally, a 0 secondary homotopy θ between f ◦ χs and χt ◦ f .

5.3 Null-homotopic morphisms in UT(A) 5.3.1 In the category of arrows and homotopy commutative squares UT(A), a morphism (ϕt, ϕs; θ) is null-homotopic if ϕt and ϕs are null-homotopic,

ϕt ∼χt 0 and ϕs ∼χs 0,(96) 0 and the homotopy ψ is homotopic to f ◦ χs − χt ◦ f , exactly as in the case of M(A), cf. (93).

5.3.2

In UT(A) a commutative square is null-homotopic precisely when both ϕt and ϕs are null homotopic and the resulting 2 homotopies that contract 0 ϕt ◦ f = f ◦ ϕs are themselves homotopic. In particular, a morphism null-homotopic in M(A) is generally not null-homotopic in UT(A).

68 5.4 Arrows contractible in M(A) 5.4.1

Let us call an object f of M(A) contractible if id f is null-homotopic, i.e., when the square A A (97)

f f BB is strongly homotopy split. Identities (89)–(90) become in this case

idB − f ◦ η = [χt, ∂] , 0 = [ f , χ] + [θ, ∂] (98) and 0 = [η, ∂] , idA −η ◦ f = [χs, ∂] .(99) where [ f , χ] denotes f ◦ χs − χt ◦ f .

5.4.2 A homotopy equivalence data in Ch(A) A pair of morphisms of chain complices

f / A o B (100) g together with a pair of homotopies

idB ∼hB f ◦ g and idA ∼hA g ◦ f will be referred as a homotopy equivalence (data) between complices A and B. By deﬁnition, A and B appear in it on equal footing. If complices A and B admit such data, we say that they are homotopy equivalent.

5.5 Homotopy equivalence of chain complices 5.5.1

We say that a morphism f is a homotopy equivalence if there exist g, hB and hA , that form a homotopy equivalence data. In this case g is said to be a

69 homotopy inverse of f . These are precisely the morphisms in Ch(A) that correspond to isomorphisms in the homotopy category K(A). In contrast with the deﬁnition of a homotopy equivalence data where f and g appear on equal footing, f is primary data while the homotopy inverse g and the two homotopies are secondary. One needs to be aware of these two related but different uses of the term homotopy equivalence.

5.5.2 This is very similar to the two uses of the term equivalence of categories.

5.5.3 If f is contractible in M(A), then f is a chain homotopy equivalence. More precisely, if (η; χt, χs; θ) contracts f in M(A), then η is a homotopy inverse of g while χt and χs are the corresponding homotopies hB and hA . A homotopy equivalence data with ﬁxed f is the same as a homotopy splitting of (97) while a contraction of f in M(A) is a strong homotopy splitting.

Exercise 38 Show that the monomorphism

ιB A ⊕ B u A x is a homotopy equivalence if A is a contractible complex.

5.5.4 We shall now investigate the structure of a homotopy equivalence in greater detail. We start by making a few observations. Exercise 39 Show that

[ f ◦ hA − hB ◦ f , ∂] = 0 and [g ◦ hB − hA ◦ g, ∂] = 0.(101)

5.5.5 Thus, [ f , h] ˜ f ◦ hA − hB ◦ f is a morphism of complices. A given homotopy equivalence can be extended to a contraction of f in M(A) precisely when [ f , h] is null- homotopic in Ch(A).

70 Exercise 40 Prove the identities

2 2 hB, ∂ = [ f ◦ g, hB] and hA, ∂ = [g ◦ f , hA].(102)

5.5.6

Given a homotopy equivalence ( f , g; hB, hA) let

h¯ B ˜ [ f , h] ◦ g = ( f ◦ hA − hB ◦ f ) ◦ g.

Since h¯ B : B −→ [−1]A is a morphism of chain complices, 0 ¯ hB ˜ hB + hB is a homotopy that contracts idB − f ◦ g.

0 5.5.7 Calculating f ◦ hA − hB ◦ f In order to simplify calculations we shall omit the composition symbol ◦,

0 f hA − hB f = ( f hA − hB f )(idA −g f ) = ( f hA − hB f )[hA, ∂]

= [( f hA − hB f )hA , ∂].

5.5.8 An alternative method is to utilize the other commutator morphism [g, h] to modify hB . Let ¯ h¯ B ˜ f ◦ [g, h] = f ◦ (g ◦ hB − hA ◦ g). ¯ Since h¯ B : B −→ [−1]A is a morphism of chain complices,

00 ¯ hB ˜ hB − hB is again a homotopy that contracts idB − f ◦ g.

71 00 5.5.9 Calculating f ◦ hA − hB ◦ f : the ﬁrst method Aided by the identities established in Exercises 39, 33 and 40, we calculate 00 f hA − hB f = f hA(idA −g f ) − (idB − f g)hB f

= ( f hA − hB f )(idA −g f ) + hB f (idA −g f ) − (idB − f g)hB f

= ( f hA − hB f )[hA, ∂] + f ghB f − hB f g f 2 = [( f hA − hB f )hA , ∂] + [hB f , ∂] 2 = [( f hA − hB f )hA + hB f , ∂] 2 2 = [hB f − hB f hA + f hA , ∂] .

00 5.5.10 Calculating f ◦ hA − hB ◦ f : the second method

00 f hA − hB f = f hA(idA −g f ) − (idB − f g)hB f

= f hA(idA −g f ) + (idB − f g)( f hA − hB f ) − (idB − f g) f hA

= f g f hA − f hAg f + [hB, ∂]( f hA − hB f ) 2 = [ f hA, ∂] − [hB( f hA − hB f ) , ∂] 2 = [ f hA − hB( f hA − hB f ) , ∂] 2 2 = [hB f − hB f hA + f hA , ∂] .

5.5.11 We established the following important result.

Proposition 5.1 Given a homotopy equivalence ( f , g; hB, hA), let 0 hB ˜ hB + [ f , h] ◦ g = hB + ( f ◦ hA − hB ◦ f ) ◦ g (103) and 0 θ ˜ −( f ◦ hA − hB ◦ f ) ◦ hA (104) and, also, let

00 hB ˜ hB − f ◦ [g, h] = hB − f ◦ (g ◦ hB − hA ◦ g) (105) and 00 2 2 θ ˜ −hB ◦ f + hB ◦ f ◦ hA − f ◦ hA .(106)

72 Then both 0 0 00 00 (g; hB, hA; θ ) and (g; hB, hA; θ ) are contractions of f in M(A).

Corollary 5.2 A morphism of chain complices f is a homotopy equivalence if and only if Cone( f ) is contractible.

5.5.12 We found that by modifying just one of the two homotopies in a homotopy equivalence data, one is able to produce a strong homotopy splitting of square (97) or, equivalently, a homotopy that contracts Cone( f ) in Ch(A).

5.5.13 In Sections 5.5.9 and 5.5.10 we established, using two slighly different methods, the identity

2 2 [ f , h] + f [g, h] f = [hB f − hB f hA + f hA , ∂].(107)

By exchanging f and g we obtain a symmetric identity:

2 2 [g, h] + g[ f , h]g = [hAg − hAghB + ghB,, ∂].(108)

5.5.14 It follows that if 0 = [g, h] + [g2, ∂], for some degree 2 graded map g2 , then

2 2 0 = [ f , h] + [ f g2 f − hB f + hB f hA − f hA , ∂] and vice-versa, if 0 = [g, h] + [ f2, ∂], then 2 2 0 = [ f , h] + [g f2g − hAg + hAghB − ghB , ∂]. This yields the following result.

73 Proposition 5.3 A homotopy equivalence data ( f , g; hB, hA) provides a strong homotopy splitting of square (97) if and only if it provides a strong homotopy splitting of a similar square for g

B B (109)

g g

5.5.15

If we ﬁx f , g and hA , then the set of homotopy equivalence data ( f , g; hB, hA) is a torsor over the group HomCh(A)(B, [−1]B), whereas the set of homotopy equivalence data that provide a strong homotopy splitting of square (97) is a torsor over the subgroup ˜ ˜ hB ∈ HomCh(A)(B, [−1]B) such that hB ◦ f is null-homotopic . Exercise 41 If f is a homotopy equivalence, then a morphism ν in Ch(A) is null-homotopic if and only if ν ◦ f is null-homotopic. More precisely, ﬁnd a formula for a contracting homotopy of ν in terms of a given contracting homotopy of ν ◦ f and a given homotopy equivalence data ( f , g; hB, hA).

In particular, the set of homotopy equivalence data that provide a strong homotopy splitting of square (97) is a torsor over the subgroup of null-homotopic morphisms: ˜ ˜ hB ∈ HomCh(A)(B, [−1]B) such that hB is null-homotopic . Corollary 5.4 Every homotopy splitting of square (97) is strong if and only if every morphism B −→ [−1]B is null-homotopic.

5.6 ∞-contractions of Cone( f ) and homotopy ∞-equivalences 5.6.1 According to Section 4.5.9, any contraction (88) of Cone( f ) extends to an ∞-contraction χ = χ1 + χ3 + ···

74 where ! χtt,2n−1 χts,2n−1 χ2n−1 = .(110) χst,2n−1 −χss,2n−1 If we denote

χtt,2n−1 by hB,2n−1, χts,2n−1 by f2n,

χst,2n−1 by g2n−2, χss,2n−1 by hA,2n−1, then B ! ∂ + hB f ∂Cone ( f ) + χ = (111) A g −(∂ + hA) where

hB = hB,1 + hB,3 + ··· , f = f0 + f2 + ··· (112)

g = g0 + g2 + ··· , hA = hA,1 + hA,3 + ··· .(113) and

hB,1 = hB, f0 = f

g0 = g, hA,1 = hA.

5.6.2 The equation Cone ( f ) 2 ∂ + χ = idCone( f ) (114) is equivalent to 4 equations

B 2 idB −f ◦ g = ∂ + hB) 0 = [f, ∂ + h] (115)

A 2 0 = [g, ∂ + h] idA −g ◦ f = ∂ + hA) .(116)

5.6.3 Homotopy n-equivalences Each of the 4 equations in (115)–(116) has inﬁnitely many component equations according to their degree. The ‘diagonal’ equations have only components of even degree, the off-diagonal equations have only components of odd degree.

75 5.6.4 If we retain in expansions (112)–(113) only terms of degree less or equal n, we shall say that they constitute a homotopy n-equivalence between complices A and B if all the component equations not involving terms of degree greater than n are satisﬁed.

5.6.5 Homotopy 0- and 1-equivaences Thus, a 0-equivalence is simply a pair (100) of morphisms of chain complices, a 1-equivalence is a homotopy equivalence data ( f , g; hB, hA).

5.6.6 Homotopy 2-equivalences A 2-equivalence consists of a sextet

( f , f2; g, g2; hB, hA) such that (g; hB, hA; f2) contracts f in M(A) and ( f ; hA, hB; g2) contracts g in M(A).

5.6.7 General pairs of morphisms, of course, are not homotopy equivalences. In particular, 0-equivalences do not extend, in general, to 1-equivalences.

5.6.8 We know that a homotopy equivalence does not extend, in general, to contracting homotopies of the corresponding cone complices. In particular, 1-equivalences do not, in general, extend to 2-equivalences. They do, however, after possibly replacing one of the homotopies hB or hA , see Proposition 5.1.

5.6.9 In view of Section 4.5.9, any contraction of Cone( f ) extends to an ∞- contraction, hence any homotopy equivalence data ( f , g; hB, hA), after possibly replacing either hB or hA by another homotopy, extends to an ∞-equivalence.

76 5.6.10 Integrable contractions of Cone( f )

An integrable contraction of Cone( f ) consists of a quartet (g; hB, hA; f2) such that B !2 ∂ + hB f + f2 = id .(117) A B⊕[1]A g −(∂ + hA)

Equation (117) is equivalent to ( f , g; hB, hA) being a homotopy equivalence data satisfying 0 = [ f , h] + [ f2, ∂] (118) and 4 additional equations

2 0 = f2g + hB 0 = [ f2, h] (119) 2 0 = [g, h] 0 = hA + g f2 .(120)

5.6.11 In view of Exercise 30, Cone( f ) admits an integrable contraction when f is a homotopy equivalence. In particular, any any homotopy equivalence admits a homotopy inverse g, a pair of primary homotopies hB and hA , and a secondary homotopy f2 such that equations (118) and (119)–(120) are satisﬁed.

77 6 Projectives in the categories of complices

6.1 Projective objects in Ch(A) 6.1.1 Given a complex C, let π o o n 0 Cn Pn 0 be a family of arbitrary epimorphisms with Pn projective. It induces the corresponding family of morphisms of complices

u 0 C Cone id[n]Pn (121) given by . . . .

∂n+2

Cn+1 o 0

∂n+1 π o o n 0 Cn Pn

∂n

∂ ◦π o n n 0 Cn−1 Pn

∂n−1

Cn−2 o 0

∂n−2 . . . . whose coproduct provides an epimorphism

CPo o

78 from the contractible complex P ˜ 0 122 ä Cone idPn ( ) n∈Z with projective terms 0 0 Pq = Pq ⊕ Pq+1.

6.1.2

If Pn , 0 only for Cn , 0 and the support of C is contained in the interval

[l, m],

( ) with l and m possibly being inﬁnite, then the support of Cone idZ Cone(idC) is contained in the interval [l−1, m] .

Exercise 42 Show that for any projective object Q of an additive category A, the complex Cone(id[n]Q)

··· o 0 o Q Q o 0 o ··· n−1 n is a projective object in Ch(A).

6.1.3 It follows that the complex P constructed above is a projective object of Ch(A) and the category of complices Ch(A) has sufﬁciently many projectives if A has sufﬁciently many projectives.

6.1.4 It also follows that any projective object in the category of chain complices of an abelian category with sufﬁciently many projectives is isomorphic to a direct summand of a contractible complex with projective terms: all terms of such a complex being direct summands of projective objects are projective.

79 6.1.5

A contractible complex P with projective terms is isomorphic to Cone(idZP). 0 0 Since each Pq is isomorphic to Pq ⊕ Pq−1 , where

0 Pq ˜ ZqP

0 is the object of q-cycles of P, each Pq is projective, and P is isomorphic to the coproduct Cone(id 0 ) ä [n]Pn n∈Z in the category of chain complices.

6.1.6 A coproduct of any family of projective objects being projective, we obtain the following description of projective objects in several categories of chain complices.

Proposition 6.1 (a) If A is an abelian category with sufﬁciently many projectives, then a chain complex is a projective object in Ch(A) if and only if it is a contractible complex with projective terms.

(b) The category Ch(A) has sufﬁciently many projectives if and ony if A has sufﬁciently many projectives.

6.2 Projective objects in Ch≥l(A) 6.2.1 Exercise 43 Let Q be a projective object of a preadditive category A. Show that [l]Q is a projective object of Ch≥l(A).

Exercise 44 If A is an abelian category with sufﬁciently many projectives, then for any complex C in Ch≥l(A), there exists a epimorphism

C o o [l]Q ⊕ P where P is a contractible complex with projective terms with Pq = 0 for q < l .

80 6.2.2 We obtain the corresponding version of Proposition 6.1

Proposition 6.2 (a) If A is an abelian category with sufﬁciently many projectives, then a chain complex is a projective object in Ch≥l(A) if and only if it is a contractible complex with projective terms.

(b) The category Ch≥l(A) has sufﬁciently many projectives if and ony if A has sufﬁciently many projectives.

Exercise 45 Find the analog of the above characterizations of projective objects in the subcategory of Ch≥l(A) formed by complices with Hl = 0.

Exercise 46 State the dual versions of Propositions 6.1 and 6.2 for injective objects in the corresponding categories of chain complices.

Exercise 47 For each of the chain complex categories C mentioned in Propositions 6.1 and 6.2 consider the n-homology functor

Hn : C −→ A

q and determine its left and right derived functors Lq Hn and R Hn .

Exercise 48 Show that any bounded below acyclic complex with projective terms is contractible.

6.2.3 Acyclic complices with projective terms may not be contractible if they are not bounded below. For example, if A is the category of unitary modules over the ring k[e] = k[t]/(t2) of dual numbers, the following complex P of rank 1 free, and therefore projective, k[e]-modules

×e ×e ×e ··· o k[e] o k[e] o ··· (123) is acyclic but not contractible. Indeed, contractibility of a complex is preserved by any additive functor. On the other hand, tensoring by k

81 over k[e], which is certainly an additive functor, transforms (123) into the complex with zero boundary operators

··· o 0 k o 0 k o 0 ··· .

6.3 Complices with projective terms 6.3.1 Consider a commutative diagram in a preadditive A with rows being extensions and compositions in columns being 0,

} 00 0 C o o Cq o o C q πq ιq q

P ∂q

00 0 ∂q Pq−1 ∂q ∂q

fq−1 f˜q−1

} ! 00 0 C o o Cq−1 o o C (124) q−1 πq−1 ιq−1 q−1

P ∂q−1

00 0 ∂q−1 Pq−2 ∂q−1 ∂q−1

fq−2 f˜q−2

} ! 00 0 C o o Cq−2 o o C q−2 πq−2 ιq−2 q−2

82 6.3.2

If Pq is projective, then there exists an arrow f¯q making the upper triangle in Pq

fq f¯q

} ! 00 0 C o o Cq o o C q πq ιq q

P ∂q

00 0 ∂q Pq−1 ∂q ∂q

fq−1 f˜q−1

} ! 00 0 C o o Cq−1 o o C q−1 πq−1 ιq−1 q−1 commute.

6.3.3 Let ˜ P ¯ δq ˜ fq−1 ◦ ∂q − ∂q ◦ fq.(125) Since P 00 ¯ πq−1 ◦ δq = fq−1 ◦ ∂q − ∂q ◦ πq ◦ fq = 0 , arrow (125) uniquely factorizes through ιq−1 ,

0 δq = ιq−1 ◦ δq and

0 0 0 ιq−2 ◦ ∂q−1 ◦ δq = ∂q−1 ◦ ιq−1 ◦ δq = ∂q−1 ◦ δq ˜ P P = ∂q−1 ◦ fq−1 ◦ ∂q = fq−2 ◦ ∂q−1 ◦ ∂q = 0 which means, in view of ιq−2 being a monomorphism, that

0 0 ∂q−1 ◦ δq = 0.

83 6.3.4 0 0 0 If the C column is exact at Cq−1 , then by using projectivity of Pq again, we obtain an arrow δ˜q such that 0 0 ˜ δq = ∂q ◦ δq and we set f˜q = f¯q + ιq ◦ δ˜q.

Exercise 49 Verify that ˜ ˜ P ∂q ◦ fq = fq−1 ◦ ∂q .

6.3.5 We established the following result.

Lemma 6.3 Consider a commutative diagram (124) with rows being extensions, 0 compositions in columns being zero, Pq being projective and the C -column exact 0 in Cq−1 . Then there exists an arrow

f˜q : Pq −→ Cq making the diagram

fq f¯˜q

} ! 00 0 C o o Cq o o C q πq ιq q

P ∂q

00 0 ∂q Pq−1 ∂q ∂q

fq−1 f˜q−1

} ! 00 0 C o o Cq−1 o o C q−1 πq−1 ιq−1 q−1 commute.

84 6.3.6 Epi quasiisomorphisms Let us call an epimorphism π occuring in an extension of chain complices

C00 o o π C o o C0 an epi quasiisomorphism if C0 is acyclic.

6.3.7 An immediate corollary of Lemma 6.3 is the the following Lifting Property of complices with projective terms.

Proposition 6.4 (a) A bounded below complex with projective terms has the Lifting Property for epi quasiisomorphisms of arbitrary complices.

(b) An arbitrary complex with projective terms has the Lifting Property for epi quasiisomorphisms of bounded below complices.

6.3.8 In both situations one constructs a lift of a morphism P −→ C00 to C, 00 beginning in a degree m such that either Pq , or both Cq and Cq , are zero for all q < m, by induction on q and aided by Lemma 6.3. Proposition 6.4 can be called with justiﬁcation the Fourth Fundamental Lemma of Homological Algebra.

Proposition 6.5 A quasiisomorphism between bounded below complices with projective terms is a homotopy equivalence.

Proof. Suppose that f P0 u u P is an epi quasiisomorphism. The Lifting Property established in Proposition 6.4 shows that there exists a right inverse f 0 of f . For a general quasiisomorphism, consider the commutative diagram

Q ⊕ P a

(e f ) ιP

| | a P0 o P f

85 where e is an epimorphism and Q is a contractible complex with projective terms, e.g., the canonical projection

0 e P u u Cone id[−1]P0 .

Since ιP is, according to Exercise 38, a homotopy equivalence, every quasiisomorphism f between bounded below complices with projective terms has a homotopy right inverse f 0 ,

0 f ◦ f ∼ idP0 .

Any homotopy right inverse of a quasiisomorphism is a quasiisomorphism in view of the dentity

0 0 0 idHq P = Hq( f ◦ f ) = Hq f ◦ Hq f 00 combined with the fact that Hq f is an isomorphism. Let f be a homotopy right inverse of f 0 . Then

f ∼ f ◦ ( f 0 ◦ f 00) ∼ ( f ◦ f 0) ◦ f 00 ∼ f 00.

In other words, a homotopy right inverse of f is also a homotopy left inverse. An alternative proof. A morphism f is a quasiisomorphism precisely when its cone Cone( f ) is acyclic. The cone of a morphism between bounded below complices with projective terms is a bounded below projective complex with projective terms.

Exercise 50 Show that a bounded below complex with projective terms is contractible if it is acyclic. (Hint: construct a contraction inductively on q starting at the smallest q such that Pq , 0.)

Thus, Cone( f ) is contractible and therefore f is a homotopy equivalence.

6.4 Cartan-Eilenberg resolutions 6.4.1 Suppose that C is a complex in an abelian category A with sufﬁciently many projectives.

86 6.4.2 By applying the Third Fundamental Lemma to the q-homology extension and arbitrary projective resolutions of Hp = HpC and Bp = BpC, we obtain an extension of augmented projective resolutions

......

H Z B ∂p2 ∂p2 ∂p2 u u u πp1 ιp1 PH u u PZ u PB p1 p1 x p1

H Z B ∂p1 ∂p1 ∂p1 u u u πp0 ιp0 PH u u PZ u PB p0 p0 x p0

H Z B ep ep ep u u u πp ιp Hp u u Zp u Bp x

u u u 0 0 0

87 6.4.3 By applying again the Third Fundamental Lemma, we obtain an extension of augmented projective resolutions

......

B C Z ∂p−1,2 ∂p2 ∂p2

u 0 u u βp1 ζ p1 PB u u PC u PZ p−1,1 p1 x p1

B C Z ∂p−1,1 ∂p1 ∂p1

u 0 u u βp0 ζ p0 PB u u PC u PZ p−1,0 p0 x p0

B0 C Z ep ep ep

u 0 u u 0 βp ζ p B u u Cp u Zp p x

u u u 0 0 0

B0 B where ep is ep−1 composed with the canonical isomorphism of Bp with 0 Bp .

88 6.4.4 The First Fundamental Lemma yields morphisms of augmented resolutions

. . . .

Z B ∂p2 ∂p2 u u β¯ p1 PZ u PB p1 x p1

Z B ∂p1 ∂p1 u u β¯ p0 PZ u PB p0 x p0

Z B ep ep u u β¯ p Zp u Bp x

u u 0 0

89 6.4.5 By combining constructions of Sections 6.4.2, 6.4.3 and 6.4.4, we obtain the sequence of morphisms of augmented projective resolutions

......

C C C ∂p−1,2 ∂p2 ∂p+1,2 u u u ∂ ∂ ∂ ∂ p−1,1 C p1 C p+1,1 C p+2,1 ··· u Pp−1,1 u Pp1 u Pp+1,1 u ···

C C C ∂p−1,1 ∂p1 ∂p+1,1 u u u ∂ ∂ ∂ ∂ p−1,0 C p0 C p+1,0 C p+2,0 (126) ··· u Pp−1,0 u Pp0 u Pp+1,0 u ···

C C C ep−1 ep ep+1 u u u ∂p−1 ∂p ∂p+1 ∂p+2 ··· u Cp−1 u Cp u Cp+1 u ···

u u u 0 0 0 where ¯ 0 ∂pq ˜ ζ p−1,q ◦ βp−1,q ◦ βpq .

Exercise 51 Show that ∂p−1,q ◦ ∂pq = 0 .

6.4.6 If we set ← q ↓ C ∂pq = (−1) ∂pq and ∂pq = ∂pq, then ← ↓ Ppq, ∂pq, ∂pq p,q∈Z forms a double complex with Ppq = 0 for q < 0 and diagram (126) can be interpreted either as a double complex P augmented by [−1](C) or as a

90 morphism of double complices

e (127) u [0](C) where [m](C) denotes the double complex obtained by placing C in the m-th row (in order not to confuse it with the shifted complex [m]C we enclosed C in parentheses).

6.4.7 Cartan-Eilenberg resolutions We constructed a Cartan Eilenberg resolution of a complex C, i.e., an augmented double complex (127) such that the induced augmented complices

......

u u u u Hp1P Zp1P Bp1P Pp1

u u u u Hp0P Zp0P Bp0P and Pp0

e e e e u u u u HpC ZpC BpC Cp

u u u u 0 0 0 0 are augmented projective resolutions of the p-th chain, boundary, cycle, and homology objects of C. Here HpqP, ZpqP and BpqP, denote the p-th homology, cycle and boundary objects of the q-th row of the double complex.

91 6.4.8 For any Cartan-Eilenberg resolution of C, the induced morphism of complices e C u Tot P ,(128) where Tot P denotes the total complex of P, is an epi quaisiisomorphism (note that Tot P = TotNWP here).

6.4.9 Note that Tot P is a complex with projective terms if there are ﬁnitely many nonzero Ppq on each diagonal p + q = n. Our construction produces a Cartan-Eilenberg resolution of C such that

Ppq = 0 when Cp = 0.

We arrive at the following result.

Proposition 6.6 Let A be an abelian category with sufﬁciently many projectives. (a) A complex has the Lifting Property for epi quasiisomorphisms if and only if it is a complex with projective terms.

(b) For every complex C in Ch≥l A, there exists an epi quasiisomorphism

qis CPo o

from a complex P with projective terms in Ch≥l A.

Exercise 52 State the dual result for complices with injective terms.

6.4.10 Given an extension of chain complices

C00 o o π C o ι o C0 (129) and an epi quasiisomorphism P

e C00

92 the pullback extension

P o o π˜ Q o ι˜ o C0 (130)

e e˜ 00 o o o o 0 C π C ι C is graded-split in view of projectivity of terms of P, hence isomorphic to the cone extension of a certain morphism f : C0 −→ [−1]P. Since the kernel of e˜ is isomorphic to the kernel of e and the latter is acyclic, e˜ is an epi quasiisomorphism, and we arrive at the following corollary of Part (b) of the last Proposition.

Corollary 6.7 For any extension (129) with bounded below C00 , there exists an epi quasiisomorphism from a graded-split extension, cf. (130), where P denotes a bounded below complex with projective terms or, equivalently, from the cone extension of a certain morphism f from C0 to a bouded below complex with projective terms.

6.4.11 Thanks to the Lifting Property of Proposition 6.4, given a diagram

P0 P

qis qis

C0 o C f there exists a ‘lift’ of f to P and P0

f˜ P0 o P (131)

qis qis

C0 o C f

Lemma 6.8 A morphism from a complex P with projective terms P to a bounded below acyclic complex K is null-homotopic.

93 Proof. Let Q

K be an epi quasiisomorphism whose existence is secured by Proposition 6.6, with Q being a bounded below complex with projective terms. In view of the hypothesis, Q is acyclic and therefore contractible, cf. Proposition 6.5. A morphism f KPu factorizes through e, hence it is null-homotopic in view of contractibility of Q.

6.4.12 If f˜0 is another lift of to P and and P0 , cf. diagram (131) the difference f˜0 − f˜ factorizes through the acyclic kernel of the left epi quasiisomorphism in (131) and thus, according to Lemma 6.8, is null-homotopic. We arrive at the following proposition.

Proposition 6.9 Any assignment

C 7−→ e (C ∈ Ob A) u C of an epi quasiisomorphism from a bounded below complex with projective terms gives rise to a unique additive functor from Ch+(A) to the homotopy category K+(PA) of bounded complices with projective terms

P Ch+(A) K+(PA) , C 7−→ P,(132) w and an epi quasiisomorphism of functors

J+ ◦ P

e (133) u Q

94 where J+ denotes the embedding of K+(PA) onto a full subcategory of K+(A) and Q is the quotient functor

Q Ch+(A) K+(A) .(134) w

6.4.13 Note that the composition of P with the functor [0] that embeds A onto a full subcategory of Ch+(A) yields a projective resolution functor from A to K+(PA). Moreover, any such resolution functor admits an extension to Ch+(A) as we saw while constructing a Cartan-Eilenberg resolution of an arbitrary complex.

6.4.14 We shall continue to refer to any functor (132) described in Proposition 6.9 as a projective resolution functor.

6.4.15 Any additive functor sends a contractible complex to a contractible complex, in particular it sends homotopic morphisms to homotopic morphisms. It follows that any projective resolution functor passes to a functor

P K+(A) K+(PA) (135) w that will be denoted by the same symbol P.

6.4.16 Epi quasiisomorphism of functors (133) from the category of chain complices Ch+(A) now becomes a quasiisomorphism of endofunctors of the corresponding homotopy category K+(A)

J+ ◦ P

e .(136) u

idK+(A)

95 6.4.17 Recall that every morphism C −→ C0 in the homotopy category is represented by a composite

η ι C0 u u C00 u C x 00 where η is an epimorphism in Ch+(A), C is a direct sum of C and a contractible complex, and ι is the corresponding monomorphism embedding of C into C00 , cf. the ﬁrst proof of Proposition 6.5. We shall remember that the quasiisomorphism in (136) is represented by an epimorphism in Ch+(A).

6.4.18 If f is a quasiisomorphism, so it is its lift f˜, cf. diagram (131). The latter is a homotopy equivalence, according to Proposition 6.5, and homotopy classes of homotopy equivalences are precisely isomorphisms in K+(A).

6.4.19 In particular, the composite functor

P◦J+ K+(PA) K+(PA) w is isomorphic to the identity functor on K+(PA).

6.4.20

One can choose P so that P ◦ J+ is equal to idK+(PA) . If so, a projective resolution functor is a retraction of the homotopy category of bounded below chain complices onto the full subcategory of complices with projective terms.

6.4.21 Another consequence of the observation made in Section 6.4.18, is that any two resolution functors are isomorphic by a unique isomorphism of functors compatible with natural transformations (133). This is also a reﬂection of the fact that a projective resolution functor (135) is right adjoint to the embedding functor J+ .

96 Proposition 6.10 A projective resolution functor (135) is right adjoint to the embedding functor J+ with the unit of the adjunction being an isomorphism,

' idK (PA) P ◦ J+ , + η w and the counit being an epi quasiisomorphism,

qis id u u ◦ K+(A) e J+ P .

Proof. Given a pair of bounded below complices with projective terms P and P0 , and an epi quasiisomorphism of chain complices

e , u C the correspondence assigning to a homotopy class of a morphism f C u P0 the homotopy class of its lift f˜ P u P0

0 is bijective and natural in C, P and P .

6.4.22 We say that a functor F : C −→ C0 inverts arrows belonging to a certain subclass S ⊆ Arr C if Fσ is an isomorphism for every σ ∈ S.

Theorem 6.11 (a) Any projective resolution functor (135) inverts all quasiisomorphisms.

(b) For any additive functor F from K+(A) to an additive category B, epi quasiisomorphism (136) induces a natural transformation of functors

F ◦ J+ ◦ P

Fe (137) u F

97 which is an isomorphism of functors if and only if F inverts quasiisomorphisms.

Proof. Part (a) is a consequence of the argument preceding the statement of the theorem. If F inverts quasiisomorphisms, then Fe is an isomorphism itself. Vice- versa, for every additive functor F, the composite functor F ◦ J+ ◦ P inverts quasiisomorphisms, hence if Fe is an isomorphism, F inverts quasiisomorphisms.

Exercise 53 State the dual versions of Proposition 6.10 and of Theorem 6.11 for injective resolution functors.

6.5 The derived categories 6.5.1 Localization of a category Let Σ ⊆ Arr C be any subclass of the class of arrows of a category C. We call a functor Λ C C0 w a localization of C with respect to or at Σ, if any functor

F CD w that inverts arrows from Σ factorizes uniquely through Λ

F / C ? D

Λ F˜ C0

6.5.2 Localization of a category The functor Λ is referred to as the localization functor while its target, C0 , is often itsel called a localization of C at Σ.

98 6.5.3 If such a localization functor exists, it is unique up to a unique isomorphism of functors. In particular, its target C0 is unique up to a unique isomorphism that is compatible with the corresponding localization functors. Generic notation for C0 is C[Σ-1]. There is no generally agreed notation for the localization functor.

6.5.4 Any small category has a localization at any subset Σ ⊆ Arr C. This is demonstrated by the construction of C[Σ-1] as the quotient of the category F freely generated by the disjoint union of the set of all arrows of C and the set Σ Arr C t Σ . Elements of the second summand correspond to formal inverses of arrows belonging to Σ. Let us denote those elements by σ¯ .

6.5.5 One then considers the congruence ∼ generated by all relations occuring between arrows of C and additional relations

σ ◦ σ¯ = idc and σ¯ ◦ σ = idc0 where c is the source and c0 is the target of σ. The latter express the fact that σ¯ is an inverse of σ ∈ Σ.

6.5.6 The free category F has the same objects as C. The set theoretic inclusion of Arr C into Arr C t Σ induces a functor

C F/∼ w when composed with the quotient functor F −→ F/∼

6.5.7 For other categories a localization often can be implemented by the formal- ism of right, ασ−1 (α ∈ Arr C, σ ∈ Σ)

99 or left fractions, σ−1α . These methods impose conditions on the class Σ of arrows to be inverted, known as the right and, respectively, left Ore conditions, ﬁrst introduced by Oysteyn Ore in his study of formal inverses in rings of differential operators in early 1930-ties.

Lemma 6.12 A localization of the category K+(A) at the class of all quasiisomorphisms exists.

Proof. Let us ﬁx a projective resolution functor P. Let L be the category 0 with the same objects as Ch+(A) and the morphisms between C and C declared to be the morphisms between PC and PC0 ,

0 ˜ 0 HomL(C, C ) HomK+(A)(PC, PC ).

The identity correspondence on objects and P on arrows deﬁnes a functor

Λ K+(A) L w that inverts all quasiisomorphisms. If

F K+(A) B w is a functor inverting all quasiisomorphisms, we set F˜ to be F on the class of objects and −1 F˜(α) ˜ F(eC0 ) ◦ F(α) ◦ F(eC) (138) where eC and eC0 are the corresponding epi quasiisomorphisms from PC 0 0 to C and, respectively, from PC to C .

Exercise 54 Show that any functor F˜ from L to B such that

F˜ ◦ Λ = F satisﬁes identity (138).

100 6.5.8 D+(A)

The target of a localization of the homotopy category K+(A) at the class of all quasiisomorphisms is one of several categories called the derived category of A. The above is the fastest proof of its existence but the model of D+(A) our construction provides explicitly depends on a particular projective resolution functor P. We shall see soon advantages of this approach. If it is, however, desirable to have a model of D+(A) independent of any such functor, one can represent D+(A) as the category of right fractions with denominators being quasiisomorphisms. Veriﬁcation that the class of quasiisomorphisms in K+(A) satisﬁes right Ore conditions is an easy exercise in view of the results of this Chapter. Then the existence of the category of right fractions follows from general and rather lengthy considerations. The criterion of equality of morphisms in categories of fractions is unfortunately highly implicit.

6.5.9 Let us represent key categories and functors by the following commutative diagram

K+(PA) (139) 7 O P

[0] Q A / Ch+(A) / K+(A) Λ◦J+ P˜

Λ ' D+(A)

We are not using here any speciﬁc model for D+(A) but if it is the model constructed above with the aid of P, then P˜ has a very simple description: on objects it is P, on arrows it is the identity,

C 7−→ PC, α 7−→ α.

6.5.10

The functor P˜ is right adjoint to Λ ◦ J+ with both the unit and the counit of the adjunction

˜ ˜ idK+(PA) −→ P ◦ Λ ◦ J+ and Λ ◦ J+ ◦ P −→ idD+(A)

101 being isomorphisms. In particular, P˜ and Λ ◦ J+ provide an equivalence of the derived category D+(A) with its full subcategory of K+(PA).

6.5.11 The (total) left derived functor LF For an additive functor F : A −→ B, the unique functor that makes the diagram commute

K+(PA) (140) 7 O P F◦J+ ' K+(A) Λ◦J+ P˜ K+(B) Λ Λ ' & D+(A) / D+(B) LF is denoted LF and called the left derived functor of F. It is induced by the functor Λ◦F◦J+◦P K+(A) / D+(B) that inverts quasiisomorphisms since P does. On objects LF acts by sending a complex C to FPC.

6.5.12

In contrast with the fact that D+(A) can be constructed with no reference to any projective resolution functors, the derived functors LF are deﬁned explicitly in terms of such resolution functors. This is one more reason why the model of D+(A) we provided may be more convenient.

6.5.13 The (classical) left derived functors LqF

If B is an abelian category, then the classical left derived functors LqF are the composites Lq F AB w u

Q◦[0] Hq (141) u K+(A) K+(B) F◦J+◦P w

102 The homology functors invert quasiisomorphisms, hence they induce functors from the derived category D+(B) to B. We shall use the same symbol Hq to denote them. Diagram (141) can be rewritten as

Lq F AB w u

Λ◦Q◦[0] Hq (142) u D+(A) D+(B) LF w

6.5.14 The derived category D−(A) and the (total) right derived functor RF

The derived category D−(A) and the right derived functor are deﬁned in a dual way, by replacing projective by injective, epi quasiisomorphisms by mono quasiisomorphisms, bounded below by bounded above, the category K+(PA) by the homotopy category of bounded above complices with injective terms K−(IA), and the full embedding functor J+ by the full embedding functor

J− K−(IA) K−(A) . w

6.5.15

The derived category D−(A) can be realized as the category of left fractions with quasiisomorphisms as denominators.

6.5.16 The dual of the diagram deﬁning LF

K−(IA) (143) 7 O I F◦J− ' K−(A) I˜ Λ◦J− K−(B) Λ Λ ' & D−(A) / D−(B) RF

103 deﬁnes the (total) right derived functor RF. Here I denotes an injective resolution functor I K−(A) K−(IA) . w

Here I is left adjoint to J− while I˜ and Λ ◦ J− again provide an equivalence of categories.

6.5.17 The (classical) right derived functors RqF If B is an abelian category, then the classical right derived functors RqF are the composites Rq F AB w u

Q◦[0] H−q (144) u K−(A) K−(B) F◦J−◦I w or, equivalently, the composites Rq F AB w u

Λ◦Q◦[0] H−q (145) u D−(A) D−(B) RF w

6.6 The Ext groups 6.6.1 HomD+(A) [k]M, [l]N

For any objects M and N of A, the morphisms in D+(A) [k]M −→ [l]N form a group that up to a canonical isomorphism is equal to l−k ˜ ExtA (M, N) HomK(A)(PM, [l − k]N) (146)

' HomK(A)(PM, [l − k]PN) (147) where PM and PN denote any projective resolutions of M and N . The group on the right hand side of (146) was calculated in Exercise 9. In particular, HomD+(A) [k]M, [l]N = 0 when k > l .

104 6.6.2 The Ext-groups as derived functors Even though the Ext-groups are calculated in terms of a projective resolution of their first argument, they are right derived functors. This is so because HomA( , ) is a contravariant functor of the first argument. When viewed as a covariant functor, A is replaced by Aop , bounded below complices become bounded above complices and projective resolutions become injective resolutions. In follows that the Ext groups as functors of the first, contravariant argument, are the right derived functors

q q ExtA(M, N) = R HomA( , N)(M) (148) of the functor

op op A −→ Ab, M 7−→ HomA(M, N).

6.6.3 The Ext groups as functors of the second, covariant argument, are the right derived functors

q q ExtA(M, N) = R HomA(M, )(N) (149) of the functor

A −→ Ab, N 7−→ HomA(M, N). reﬂecting the fact that HomD+(A) [k]M, [l]N ' HomD−(A) [k]M, [l]N ' HomD−(A) [−l]M, [−k]N

' HomK−(A)([−(l − k)]M, IN) where IN is an injective resolution of N .

6.6.4

This property of the functor HomA to the effect that

q q R HomA( , N)(M) = R HomA(M, )(N), is a manifestation of HomA() being a right balanced functor of 2 arguments.

105 6.7 Classical derived functors for functors of n arguments

6.7.1 Left derived functors LqF The values LqF(M1,..., Mn) of classical left derived functors of additive functors of n arguments are deﬁned as homology groups Hq of the totalization of the n-complex obtained by replacing each covariant argument by its projective resolution and each contravariant argument by its injective resolution.

6.7.2 Right derived functors RqF The values q R F(M1,..., Mn) of classical right derived functors are deﬁned as homology groups H−q of the totalization of the n-complex obtained by replacing each covariant argument by its injective resolution and each contravariant argument by its projective resolution.

6.7.3 Balanced functors of n arguments A functor is said to be left balanced if replacing any argument by the corresponding resolution – projective, in the case of covariant, and injective, in the case of contravariant arguments, makes it an exact functor of the remaining arguments.

6.7.4 Dually, a functor is said to be right balanced if replacing any argument by the corresponding resolution – injective, in the case of covariant, and projective, in the case of contravariant arguments, makes it an exact functor of the remaining arguments.

6.7.5 Calculating left derived functors of left balanced functors of n arguments can be performed by replacing any single argument by the corresponding resolution. And similarly for calculating right derived functors of right balanced functors.

106 6.7.6

The HomA functor is right but not left balanced.

R 6.7.7 Torq groups The functor of tensor product

⊗R mod-R, R-mod Ab w and its bimodule variants are left but not right balanced. The left derived functors of ⊗R are denoted

R Torq (M, N) ˜ Lq ⊗R (M, N) .(150)

Notation comes from the fact that these groups were ﬁrst introduced for R = Z, i.e., for the tensor product of abelian groups, where

Z Tor1 (A, Z/nZ) happens to be the n-torsion group of an abelian grup A. Indeed,

×n ZZu is a projective resolution of the cyclic group Z/nZ, hence

A ⊗ Z/nZ = Tor0(A, Z/nZ) and Tor1(A, Z/nZ) are isomorphic, respectively, to the cokernel and kernel of the multiplication by n map ×n AAu .

6.7.8

Higher Torq groups of abelian groups identically vanish for all q > 1 because any subgroup of a free abelian group is free which means, in particular, that any projective abelian group is free and that any non free abelian group has a projective resolution of length 1.

107 7 Triangulated categories

7.1 Exact triangles in the homotopy category of chain complices 7.1.1 The cone triple associated with a morphism The cone extension associated to a morphism of chain complices f ‘hides’ f in the inﬁnitesimal structure of Cone( f ). If we augment (40) by f

[1]A B f [1]A o o π Cone( f ) o ι o B o A (151) we obtain a composable triple of morphisms that can be extended indeﬁ- nitely in both directions by replication with A, B and f replaced by

[l]A, [l]B and ~ f = (−1)l[l] f , (l ∈ Z).

7.1.2 If we pass to the homotopy category K(A), then we can say more: every consecutive three morphisms in this inﬁnite sequence of morphisms

fn−2 fn−1 fn Xn−3=[1]Xn o Xn−2 o Xn−1 o Xn is isomorphic to the triple of the form (151)

π[1]Xn ιXn−1 fn [1]Xn o o Cone( fn) o o Xn−1 o Xn (152) as follows from Exercises 15 and 16. Thus, any such triple determines the whole inﬁnite sequence up to an isomorphism and the corresponding shift in n.

7.1.3 Exact triangles A composable triple of chain complices

g f [1]ACo h o B o A (153) isomorphic in K(A) to the cone triple (151) will be referred to as an exact triangle. The terminology reﬂects the fact that such a triple can be

108 represented by the triangle diagram

f o B DA (154)

−1 g h C with −1 marking reﬂecting the fact that h is a morphism from C to A of degree −1, i.e., is, actually, a morphism from C to [1]A.

7.1.4 Shifts of exact triangles If we denote the composable triple (153) by E, then let [1]E be deﬁned as

~1 f g [1]B o [1]A o h C o B (155) and [−1]E as g f ~−1h CBo o A o [−1]C (156)

7.1.5 Morphisms between exact triangles Morphisms between exact triangles E and E0 are naturally deﬁned as triples of morphisms between the corresponding components that make the following diagram commute

g f [1]A o h C o B o A (157)

[1]φA φC φB φA [1]A0 o C0 o B0 o A0 h0 g0 f 0

7.1.6 A morphism between exact triangles extends to the corresponding morphism between inﬁnite sequences obtained by replication on both sides of shifted morphisms. Any such ‘3-periodic’ sequence of morphisms is uniquely determined, of course, by its restriction to any 3 consecutive arrows, i.e., to any of its embedded exact triangles.

109 7.1.7

Providing only φA and φB allows completion to a morphism between E and E0 , in view of (157) being up to an isomorphism essentially a morphism between the cone extensions of f and f 0 . In other words, any commutative square f B o A

φB φA B0 o A0 f 0 extends to a morphism of the corresponding exact triangles.

7.1.8 By appying this observation to [1]E and [1]E0 , we deduce that also any commutative square g C o B

φC φB C0 o B0 g0 extends to a morphism between E and E0 .

Exercise 55 Show that any commutative square

[1]A o h C

[1]φA φC [1]A0 o C0 h0 extends to a morphism between E and E0 .

7.1.9 A degree of rigidity of exact triangles is visible in the following fact that plays the role of ‘5-lemma’ in K(A).

110 Exercise 56 Show that any morphism φ between E and E0 is an isomorphism if and only if any two of the morphisms φA , φB or φC , are isomorphisms.

[1]A o h C

[1]φA φC [1]A0 o C0 h0 extends to a morphism between E and E0 .

7.1.10 An immediate corollary is that every exact triangle is up to an isomorphism determined by just one of the three component arrows.

7.1.11 For any composable pair of morphisms

g f CBo o A , the three cone triangles associated with f , g and g ◦ f , form a diagram that can be completed to a diagram, see (53), with rows and columns being exact triangles. The left edge of that diagram is a shift of the cone triangle of f , the top and and the bottom edges are ‘trivial in the sense that they are the cone triangles associated with − idA and id[1]A . This leaves four cone triangles of interest, three of which are the cone triangles of f , g, and g ◦ f . The fourth cone triangle is canonically isomorphic in K(A) to a triangle involving the cones of f , g and g ◦ f ,

[1] Cone( f ) o Cone(g) o Cone(g ◦ f ) o Cone( f ) .

This is the second row from the bottom in the 4 × 4 diagram (53) with Cone() replaced by a canonically homotopy equivalent complex Cone(g).

111 7.1.12 It follows that a commutative diagram involving any 3 exact triangles

A (158)

g◦ f f

Ô o C g B f

w [1]A0 A0

B0 o C0

Õ [1]A admits a completion to a commutative diagram involving 4 exact triangles

A (159)

g◦ f f

Ô o C g B f

w [1]A0 A0 g

x B0 o C0

Õ [1]A

112 7.1.13 If we modify the above diagrams by replacing [1]A and [1]A0 by A and A0 , respectively, and considering the morphisms with those targets as morphisms of degree −1, we can folding the opposite commutative triangles so that A and A0 form opposite lying vertices of an octahedron each half containing 2 exact and 2 commutative triangles. This is why the described property of the class of exact triangles was christened the Octahedron Axiom.

7.1.14 The ‘planar’ layout we employ seems to be signiﬁcantly more clear, however.

7.2 Triangulated categories 7.2.1 Let us call a composable triple of arrows in an additive category A equipped with an endofunctor T

g f TACo h o B o A ,(160) a triangle. Triangles admit a natural notion of a morphism,

g f TA o h C o B o A

TφA φC φB φA TA0 o C0 o B0 o A0 h0 g0 f 0

7.2.2 An additive category A equipped with an automorphism T and a class of triangles, referred to as distinguished (or exact) is called a triangulated category if the class of distinguished triangles satisﬁes the following properties

(Tr 1) the triangle 0 o A A o 0 is distinguished, any arrow f of A occurs in some distinguished triangle, and any triangle (160) isomorphic to a distinguished triangle is distinguished;

113 (Tr 2) a triangle (160) is distinguished if and only if

−T f g TBTAo o h C o B

is distinguished; (Tr 3) a commutative diagram

g f TA o h C o B o A

TφA φB φA TA0 o C0 o B0 o A0 h0 g0 f 0

with distinguished rows admits completion to a morphism of triangles; (Tr 4) any commutative diagram (158) involving 3 distinguished triangles admits completion to a commutative diagram (159) involving 4 distinguished triangles.

7.2.3 The automorphism T is part of the structure of a triangulated category and is referred to as the translation (or shift) functor. Axiom (Tr 2) provides an action of T on the class of distinguished triangles.

7.2.4 Axiom (Tr 4) is traditionally referred to as the Octahedron Axiom. The reasons were explained in Section 7.1.13.

7.2.5 Since the axioms are modelled on the class of exact triangles in the homotopy category of chain complices K(A) of an abelian category, the latter provides our ﬁrst example of a triangulated category.

7.2.6

Various shift-invariant subcategories of K(A), like K+(A), K−(A) and Kb(A), become triangulated subcategories of K(A).

114 7.2.7

The second example is provided by the derived categories D+(A), D−(A) and Db(A) with distinguished triangles deﬁned in the same way, namely as isomorphic in the derived category to the cone triangles of morphisms of chain complices in Ch(A).

7.2.8 In contrast with the homotopy category, every extension of bounded below or bounded above chain complices is, according to Corollary 6.7, isomorphic in the apprpriate derived category to the cone extension of some morphism of chain complices. Thus, every extension of bounded below chain complices gives rise to an exact triangle in D+(A), and every extension of bounded above chain complices gives rise to an exact triangle in D−(A).

7.2.9 There are other examples of triangulated categories besides various homotopy categories of chain complices and their localizations like the derived categories. The triangulated categories K+(A) and K−(A) ﬁt into a general scheme how triangulated structures arise in projective-injective homotopy theory.

7.3 Two homotopy theories associated with an abelian category 7.3.1 The projective homotopy category of an abelian category pr The projective homotopy category category KA of an abelian category A is defined as the quotient of A by the subcategory of projective objects PA. pr The morphisms M −→ N in KA are, by definition, equivalence classes of morphisms in A modulo the abelian subgroup of those morphisms that factorize through a projective object. The latter are ‘null-homotopic’, by definition, in projective homotopy theory in A.

7.3.2 Let us assume that A has sufﬁciently many projectives and let

N00 o o N o N0 (161)

115 be any extension in A.

Exercise 57 Show that any morphism f : M −→ N00 in A extends to a morphism of extensions in A

M o o P o o M1 (162)

f f˜ f1 NNo o o o N0 where the top row is an extension with projective P.

Exercise 58 Show that if

M o o P o o M1 .

˜0 0 f f f1 NNo o o o N0

0 is another morphism of those extensions, then f1 − f1 factorizes through P.

Exercise 59 Show that, if f factorizes through some projective object P0 , then there exists an extension of f to a morphism of extensions (162) such that f1 = 0.

7.3.3 00 It follows that assignment to f ∈ HomA(M, N ) of the projective homotopy equivalence class of f1 deﬁnes a homomorphism of abelian groups

00 0 HomKpr (M, N ) HomKpr (M1, N ) A w A and any assignment to each object M of a 1-st syzygy extension (28) induces an endofunctor

pr Ω pr K K , M 7−→ M1, [ f ] 7−→ [ f1],(163) A w A where [ f1] is the equivalence class of the morphism betweenn the 1-st syzygy objects of M and N induced by a morphism f ∈ Homc A(M, N).

116 7.3.4 For any two such assignments of 1-st syzygy extensions, there is a unique isomorphism of the corresponding functors compatible with the respective 1-st syzygy extensions.

7.3.5 The ‘loop’ functor Ω From now on, we shall ﬁx one such assignment and will denote the corresponding 1-st syzygy extension by

MPMo o o o ΩM .(164)

This is an abelian analogue of what in homotopy theory of spaces is known as the path ﬁbration associated to a topological space with a distinguished point.

7.3.6 The loop endofunctor, in general, is not an equivalence, its iterations pr provide one-sided translation functors on KA .

7.3.7 The Ω-sequence associated with an extension in A pr Any extension in A induces a half-inﬁnite sequence of morphisms in KA

N00 o π N o ι N0 o ∂ ΩN00 o Ωπ ΩN o Ωι ··· (165) where ∂ is the equivalence class of f1 associated with f = idM .

Exercise 60 Show that

NNo ι 0 o ∂ ΩN00 and N0 o ∂ ΩN00 o Ωπ ΩN pr 0 are both isomorphic in KA to some extensions of N and, respectively, N .

117 pr 7.3.8 Distinguished triangles in KA pr If we deﬁne ‘distinguished triangles’ in KA as composable triples of morphisms ACo o B o ΩA pr admitting an isomorphism in KA A o C o B o ΩA

φA ' φC ' φB ' ΩφA ' N00 o N o N0 o ΩN00 π ι ∂ with the initial triple of the Ω-sequence of some extension on A, then the class of distinguished triangles possesses, as we already saw, some properties required by the deﬁnition of a triangulated category.

7.3.9 In Topology the loop functor Ω has a left adjoint, the suspension functor Σ involved in the mapping cone coﬁbration. In our situation the ‘suspension functor’ is realized, unfortunately, as an endofunctor of the ‘dual’ homotopy category, for which ‘null-homotopic’ has the meaning: ‘factorizes through an injective object.

7.3.10 The injective homotopy category of an abelian category If we express the projective homotopy category of the opposite category abelian category Aop as well as the related notions, in terms of the original in category, we obtain the injective homotopy category KA of A and the so called, mapping cone coﬁbration, ΣMIMo o o o M .(166) i.e., a ﬁxed 1-st injective syzygy co-extension of M in A.

Exercise 61 State the injective analogs of statements in Exercises 57–59.

in Exercise 62 Give a deﬁnition of the Σ-sequence in KA

··· o Σπ N o Σι ΣN0 o δ N00 o o π N o ι N0 (167) associated with an extension (161) in A.

118 Exercise 63 State the injective analog of Exercise 60.

in Exercise 64 Give an explicit deﬁnition of distinguished triangles in KA .

7.3.11

in The class of distinguished triangles in KA possesses again some properties required by the deﬁnition of a triangulated category. In certain abelian categories, e.g., in the categories of modules over Frobenius algebras, the classes of projective and injective objects coincide. In this case, both homotopy categories coincide and if we ﬁx a realization of the ‘loop’ and ‘suspension’ functors so that they are inverse to each other, then they provide a translation automorphisms on the homotopy category, and we pr in indeed obtain a traingulated category structure on KA = KA .

7.3.12 We experience precisely this situation in the case of various categories of chain complices, if we extend previous considerations from abelian to exact categories. The class of graded-split extensions is an exact structure with the property that a complex is graded-split projective or graded-split injective precisely when it is contractible, cf. Lemma 4.4. Recall, as well, that the homotopy category of complices is precisely the quotient of Ch(A) by the subcategory of contractible complices, cf. Section 4.3.6. Thus K(A) and its bounded variants K+(A) and K−(A) provide examples of triangulated categories discussed here.

7.4 Satellites 7.4.1 The iterated ‘loop’ and ‘suspension’ functors provide an alternative approach, avoiding use of chain complices altogether, in homological study of failure of left or right exactness of additive functors. In this approach, one associates with any additive functor on A a series of its left satellites q SqF and of right satellites S F.

7.4.2 Left satellites The 1-st left satellite of an additive functor F with values in any abelian category or, more generally, in an additive category with kernels, is deﬁned

119 as S1F(M) ˜ Ker( FPMFo ΩM ) (168) and the higher left satellites are deﬁned by iteration:

SqF ˜ S1(Sq−1F), S0F ˜ F.(169)

Exercise 65 Show that q−1 SqF = S1F ◦ Ω .

7.4.3 Right satellites The 1-st right satellite of an additive functor F with values in any abelian category or, more generally, in an additive category with cokernels, is deﬁned as S1F(M) ˜ Coker( FΣMFIMo ) (170) and the higher right satellites are deﬁned by iteration:

SqF ˜ S1(Sq−1F), S0F ˜ F.(171)

Exercise 66 Show that SqF = S1F ◦ Σq−1.

7.5 Certain properties of triangulated categories 7.5.1 We shall examine general triangulated categories in more detail.

Exercise 67 Show that composition of any arrows in every distinguished triangle is 0. (Hint: Consider a morphism from a distinguished triangle containing the identity morphism.)

Exercise 68 Show that a distinguished triangle with one of the morphisms being 0 is isomorphic with a shift of the direct sum triangle

π ι [1]ACo 0 o C C ⊕ A o A A .(172)

Exercise 69 Show that in a triangulated category every epi and monomorphism is split. More precisely, every epimorphism is isomorhic to a projection π of certain direct sum, and every monomorphism is isomorphic to an injection ι of certain direct sum.

120 7.5.2 It follows that every extension in a triangulated category is split, i.e., A is semisimple. In particular, a triangulated category is abelian only if it is semisimple and this does not happen in essentially all interesting cases. Every additive functor from a semisimple additive category is, of course, exact.

7.5.3 As we see the notions of epimorphism, monomorphism, extension, are all of little use in a triagulated category. This perhaps surprising conclusion is further strengthened by the following observations.

Exercise 70 Show that any morphism in an abelian category with sufﬁciently many projectives is projective-homotopic to an epimorphism. Dually, show that any morphism in an abelian category with sufﬁciently many injectives is injective- homotopic to a monomorphism.

7.5.4 ‘Exactness’ becomes an intrinsic notion in a triangulated category, not ex- pressible in terms of extensions, kernels and cokernels. In particular, the notion of an exact functor acquires a new meaning: a functor between triangulated categories is declared to be exact, if it respects distinguished triangles. This does not contradict anything we did before, since a triangulated category is not an additive category with certain properties, it is, like an exact category, an additive category enriched by a structure of a new kind.

7.5.5 Recall that the correspondence

A 7−→ [0]A (A ∈ Ob A) embeds an abelian category A onto a full subcategory of various categories of chain complices, their homotopy category quotients and the corresponding derived categories. This full embedding is exact if the target is either the abelian category of complices or the derived category, if by ‘exact’ we understand that any extension on A becomes an distinguished triangle in D+(A) or D−(A).

121 Considered to be embedding of A into either of the homotopy categories, this embedding is not exact, in general, since non-split extensions of A do not give rise to distinguished triangles in the homotopy category.

7.5.6 Derived equivalence of abelian categories When two abelian categories A and B have equivelent derived categories, we talk about derived equivalence between A and B. In the early 1980- ies several important examples of such an equivalence were discovered and investigated between categories that are not equivalent as abelian categories. Various abelian categories of D-modules and constrible sheaves are involved in such an equivalence. This led to studying abelian subcategories cA of a given triangulated category T for which ⊗ is up to equivalence its derived category. Theory of t-structures was developed as means to investigate this problem.

122 8 Categories of fractions

8.1 The calculus of fractions 8.1.1 The right Ore conditions Let Σ ⊆ Arr C be a class of arrows in a category C that satisﬁes the following conditions

(O1) Σ is closed under composition and contains the identity arrows;

(O2) any diagram in C · · (173)

Σ · Ð admits completion to a commutative square

(174) Σ

· ·

Σ · Ð

(O3) if a parallel pair of arrows is coequalized by a member of Σ, it is also equalized by a member of Σ.

8.1.2 Denote by Σ-1Σ the class of arrows

Σ-1Σ ˜ {α ∈ Arr C | σα ∈ Σ for some σ ∈ Σ}.(175)

8.1.3 Notation We adopt the following convention: member arrows of Σ in diagrams will be marked with symbol Σ. Similarly, member arrows of Σ-1Σ will be marked with Σ-1Σ.

123 8.1.4 The following observation is a strengthening of Property (O2).

Lemma 8.1 Any diagram in C

· ·

Σ-1Σ · Ð admits completion to a commutative square

· ·

Σ-1Σ · Ð

Exercise 71 Prove Lemma 8.1.

8.1.5 Summits Diagrams ·

Ð a b will be called summits from b to a.

8.1.6 The relation ‘above’ We say that a summit φ = (α, σ)

α σ

Ð a b

124 is above a summit φ0 = (α0, σ0),

α0 σ0

Ð a b or, equivalently, that φ0 is below φ, if the diagram

α · σ

α0 σ0 × Ð a b admits completion to a commutative diagram

α · σ

α0 σ0 × Ð a b

Note that the inserted arrow is necessarily a member of Σ-1Σ.

Exercise 72 Show that (σ, σ) is above (idb, idb) where b denotes the target of σ ∈ Σ.

Lemma 8.2 If there is a common summit below φ and φ0 , then there is a common summit above.

8.1.7 Preordered classes A class S equipped with a reﬂexive and transitive relation ≺ is said to be preordered (by ≺).

125 8.1.8 Filtered preordered classes A preordered class with any two members φ and φ0 admitting another member ψ such that

φ0 ≺ ψ and φ ≺ ψ (176) is said to be ﬁltered (by ≺).

8.1.9 Sometimes, for greater precision, one may say that a preordered class (S, ≺) is ﬁltered by a subclass S0 ⊆ S if for any φ and φ0 in S, there exists ψ ∈ S0 that satisﬁes (176).

8.1.10 The relation ‘above’ preorders the class of summits. Lemma 8.2 can be visualised using the standard practice to represent preordered structures by placing the member that is ‘greater’ on a higher level than the member that is ‘smaller’ and connecting them by an edge. Using this convention we can rephrase Lemma 8.2 as follows:

any triple of summits

φ0 φ

admits completion to a quadruple

φ0 φ

In other words, the class of summits above a given summit is ﬁltered by the relation ‘above’.

126 Exercise 73 Use Lemma 8.1 to demonstrate Lemma 8.2.

Corollary 8.3 The relation

φ0 ∼ φ if φ and φ0 have a common summit above them (177) is an equivalence relation on the class of summits.

8.1.11 The relation ‘spans’ Given a commutative diagram · β¯ ρ¯ Σ · ·

α ρ β σ Σ Σ Ð Ð a b c we say that the summit ω = (α ◦ β¯, σ ◦ ρ¯) spans the summits φ = (α, ρ) and ψ = (β, σ). In this case we shall also say that ω is a span of φ and ψ.

8.1.12 In view of Property (O2), the class of spans of φ and ψ is always nonempty.

Lemma 8.4 For any two spans ω1 and ω2 of φ and ψ, there exists a span above them.

Exercise 74 Prove Lemma 8.4.

Lemma 8.5 If φ0 is above φ, then above every span of φ and ψ there exists a span of φ0 and ψ. Similarly, if ψ0 is above ψ, then above every span of φ and ψ there exists a span of φ and ψ0 .

Exercise 75 Prove Lemma 8.5.

Lemma 8.6 If φ0 ∼ φ and ψ0 ∼ ψ, then every span of φ and ψ is equivalent to every span of φ0 and ψ0 .

Exercise 76 Prove Lemma 8.6.

127 8.2 The category of right fractions 8.2.1 Given a class Σ ⊆ Arr C of arrows satisfying the right Ore conditions (O1)–(O3), the category of right fractions C[Σ-1] is deﬁned as follows. It has the same objects as C. Morphisms from b to a are equivalence classes of summits φ = (α, σ) from b to a with composition

[φ] ◦ [ψ] given by the equivalence class of summits that span φ and ψ and the identity morphisms given by (ida, ida) .(178)

Exercise 77 Show that composition of morphisms in C[Σ-1] is associative and the equivalence classes (178) are identity morphisms.

8.2.2 The category C[Σ-1] introduced above exists provided the equivalence classes of summits from any object of C to any other object form a set.

8.2.3 The localization functor The correspondence on arrows

α 7−→ [(α, idsα)] , where sα denotes the source of α, deﬁnes the so called localization functor

Λ : C −→ C[Σ-1].(179)

Exercise 78 Show that, for any arrow in Σ,

σ a b , w one has [(ida, σ)] ◦ [(σ, ida)] = [(ida, ida)] and [(σ, idb)] ◦ [(idb, σ)] = [(σ, σ)] = [(idb, idb)].

128 8.2.4 In particular, members of Σ become isomorphisms in C[Σ-1] and the localization functor Λ inverts all arrows from Σ. Given a functor F : C −→ D (180) that inverts all arrows from Σ let F˜[(α, σ)] ˜ F(α) ◦ F(σ)−1.(181)

Exercise 79 Show that the right hand side of (181) does not depend on a representative of the equivalence class of summits.

Exercise 80 Show that two functors

F0 -1 / C[Σ ] / D F00 are equal if and only if F0 ◦ Λ = F00 ◦ Λ.

8.2.5 By combining the above observations, we arrive at the following result.

Proposition 8.7 Any functor (180) that inverts all arrows from Σ uniquely factorizes through the localization functor (179),

F / C = D

Λ F˜ ! C[Σ-1]

8.2.6 Left Ore conditions The right Ore conditions for the opposite category Cop when expressed in terms of C become the so called left Ore conditions.

Exercise 81 State the left Ore conditions.

Exercise 82 Sketch the construction of the category of left fractions [Σ-1]C by going step by step and dualizing all concepts and arguments from Section 8.1.2 up to Proposition 8.7.

129 8.2.7 Initial objects

For an initial object i in C, let ιa denote the unique morphism from i to an object a.

Exercise 83 Show that any summit (α, σ) from i to a is below the summit -1 (ιa, idi). In particular, initial objects of C remain initial in C[Σ ].

8.2.8 Terminal objects

For a terminal object t in C, let τa denote the unique morphism from an object a to t.

Exercise 84 Show that any summit (α, σ) from a to t is above the summit -1 (τa, ida). In particular, terminal objects of C remain terminal in C[Σ ].

8.2.9 Binary products Exercise 85 Show that a pair of morphisms φ and φ0 in C[Σ-1] with a common source and arbitrary targets is represented by a pair of summits (α, σ) and (β, σ) with common σ.

Exercise 86 Use the previous exercise to demonstrate that if a diagram

πa πb

Ð a b represents a product of a and b in C, then the diagram

[(πa,idc)] [(πb,idc)]

Ð a b represents a product of a and b in C[Σ-1].

130 8.2.10 Exactness of the localization functor In a similar vein one can demonstrate that the localization functor preserves binary coproducts, equalizers and coequalizers. Each case is a worthy exercise. By combining these partial results we establish the following important proposition.

Proposition 8.8 The localization functor is exact, i.e., it preserves all ﬁnite direct and inverse limits.

8.3 The category of right fractions of a preadditive category 8.3.1 Consider the category of right fractions A[Σ-1] of a preadditive category A. We shall demonstrate that A[Σ-1] is canonically equipped with a preadditive structure making Λ an additive functor.

8.3.2 Addition of morphisms To add two morphisms φ and ψ from b to a in A[Σ-1], one ﬁnds a representation of both by summits (α, σ)(β, σ) with common ‘denominator’ σ, then one forms the summit

(α + β, σ).

Existence of such a representation with common denominators is guar- anteed by Property (O2). It remains to demonstrate that the result up to equivalence of summits depends only on φ and on ψ.

8.3.3 Property (O2), when both arrows are members Σ, guarantees that any ‘valley’ (173) with both slopes in Σ is covered by a ‘peak’ (174) with the

131 composite arrow being a member of Σ. Suppose two such peaks are given

· · (182)

ρ0 σ0 0 ρ σ00

· ·

Σ Σ ρ σ · Ð forming a commutative diagram with the composite arrows being members of Σ.

Exercise 87 Show that a diagram (182) can be completed to a commutative diagram ·

ρ˜ σ˜

· Ð ·

ρ0 σ0 0 ρ σ00

· ·

ρ σ · Ð with all composites being members of Σ.

8.3.4

In other words, the class Σb , consisting of members of Σ with target b, is ﬁltered by the factorization relation

σ ≺ σ0 if σ0 factorizes through σ.

132 8.3.5 Thus, above any pair of summits (α, ρ) and (β, σ) from an object b to an object a, there is a pair of summits (α0, τ) and (β0, τ) with a common to both τ ∈ Σ. Furthermore, for any two such pairs with common ‘denominator’ (α, σ) and (β, σ) or (α0, σ0) and (β0, σ0), representing morphisms φ and ψ, respectively, there is a pair (α00, σ00) and (β00, σ00) with common denominator which is above either of these pairs.

8.3.6 It follows that the summit (α00 + β00, σ00) is above both (α + β, σ) and (α0 + β0, σ0).

8.3.7 The above is succinctly expressed by saying ﬁrst that the class of pairs of summits from b to a, naturally preorderd by the relation ‘above’, is ﬁltered by the subclass of pairs ‘with common denominator’. Secondly, addition of summits ‘with common denominator’ preserves the relation ‘above’.

8.3.8 This completes demonstration the proof that addition of equivalence classes of summits from b to a is well defined. Its commutativity is built into the definition. Associativity follows immediately from the fact that the class of triples of summits is filtered by the subclass of triples ‘with common denominator’. The latter is an immediate consequence of a similar statemnt for pairs. Similar statemt is valid for the class of n-tuples of summits from b to a.

Exercise 88 Show that the class of (0ab, idb) is a neutral element of so deﬁned addition of classes and that (−α, σ) represents the additive inverse of [(α, σ)].

133 8.3.9 All of this together means that the class of equivalence classes of summits from b to a is an ‘abelian group’ class without necessarily being a set. In the case, however, when all such classes are sets, we obtain the category of fractions A[Σ-1] equipped with the canonical structure of a preadditive category.

8.3.10 The localization functor is clearly additive.

8.3.11 If A has a zero object, so does A[Σ-1], if A has binary products, so does A[Σ-1]. In fact, if A is abelian, so is A[Σ-1].

8.4 The derived categories as categories of fractions 8.4.1 Let A be an abelian category and Σ = Epiqis be the class of epi quasiisomorphisms in the category of chain complices ChI,J(A), cf. Section 1.3.4.

Exercise 89 Show that Epiqis satisﬁes right Ore conditions (O1)–(O2).

Exercise 90 Show that the class of mono quasiisomorphisms in ChI, J(A) satis- ﬁes the ﬁrst two left Ore conditions.

Exercise 91 Show that the class of quasiisomorphisms in K+(A) satisﬁes the third right Ore condition while the class of quasiisomorphisms in K−(A) satisﬁes the third left Ore condition.

−1 8.4.2 The existence of the category of fractions K+(A)[Qis ] Let q CPo o (183) be a quasiisomorphism of bounded below complices.

134 Exercise 92 Show that a summit φ = ( f , g),

C00 ,

f g

~ C0 C with g being an epi quasiisomorphism, is below

( f˜, q)

0 for some morphism f˜: P −→ C in Ch+(A).

8.4.3 Since every morphism of chain complices factorizes as a homotopy equivalence followed by an epimorphism, cf. Section 6.4.17, we deduce that every equivalence class of summits for Σ = Qis ⊂ Arr K+(A) is represented by a summit ( f , q) with a ﬁxed quasiisomorphism q, as in (183). This not only shows existence −1 of the category of right fractions K+(A)[Qis ] but also demonstrates that

0 0 ( ) −→ −1 ( ) 7−→ [( )] HomK+(A) P, C HomK+(A)[Qis ] C, C , f f , q ,(184) is surjective.

Exercise 93 Show that map (184) is injective.

8.4.4

This completes a construction of the derived category D+(A) as the cate- −1 gory of right fractions K+(A)[Qis ]. Dually, one obtains a construction of the derived category D−(A) as the category of left fractions

−1 [Qis ]K−(A).

Exercise 94 Deﬁne the left and the right derived functors LF and RF as functors from the corresponding categories of fractions.

135 9 Categories of ﬁltered objects

9.1 Subobjects 9.1.1 The class of monomorphisms with target b ∈ Ob C is preordered by the relation µ ≺ µ0 “µ factorizes through µ0 ”, i.e., there exists ν ∈ Arr C such that

{ · µ

{ b d ν

0 µ d ·

Such an arrow is a monomorphism and is unique when it exists.

Exercise 95 Show that

µ ≺ µ0 and µ0 ≺ µ (185) implies that the above ν is an isomorphism.

9.1.2 In other words, (185) holds if and only if monomorphisms µand µ0 are isomorphic by an isomorphism

' · o · ν

µ0 µ b b in the category of arrows of C. We shall write in this case µ ∼ µ0 . Equiva- lence classes of monomorphisms with target b are thought of as subobjects of b.

136 9.1.3 The class Sub b of subobjects of b may be a set even if C is not a small category. For example, subobjects of a set S in the category of sets are in bijective correspondence with subsets of S. The latter form a set according to one of the axioms of Set Theory.

9.1.4 Categories in which the class of subobjects of every object forms a set are called locally small or well powered.

9.1.5 Given subobjects M and M0 of b, if

if µ ≺ µ0 for some µ ∈ M and µ0 ∈ M0 , then µ ≺ µ0 for all µ ∈ M and µ0 ∈ M0 . We shall write in this case M ⊆ M0 and say that M is contained in M0 , or will write

M0 ⊇ M and say that M0 contains M. The relation of being contained is a partial order on Sub c.

9.1.6 The largest member of Sub b is given by the isomorphism class of the identity morphism B ˜ [idb]. We shall often identify the largest subobject of b with b itself.

9.1.7 Intersection and union of a family of subobjects

The inﬁmum and supremum of a family (Mi)i∈I of subobjects of c are referred to as, respectively, the intersection and the union of the family.

137 9.1.8 Notation We employ the notation \ [ Mi ˜ inf{Mi | i ∈ I} and Mi ˜ sup{Mi | i ∈ I}. i∈I i∈I

Lemma 9.1 If Sub b is a set and any family of subobjects of b has intersection, then any family of subobjects of b has union.

Proof. Given a family (Mi)i∈I , let U denote the set of all subobjects of b that contain every Mi . Since C ∈ U, this set is nonempty and its inﬁmum exists in view of the hypothesis. This inﬁmum is automatically the supremum of {Mi | i ∈ I}.

9.1.9 When C has an initial object, then morphisms from initial objects to b form the smallest subobject of b. We shall refer to it as the zero or trivial subobject.

9.1.10 The preimage of a subobject under morphism If µ represents a subobject M of b and φ is a morphism from a to b, then a pullback λ of µ by φ is a monomorphism,

o · ·

µ λ

bo a φ and its class, denoted α−1(M) and called the preimage of M under φ, depends only on M, not on a particular representative µ. The preimage of a subobject of b is a subobject of a.

9.1.11 The kernel of a morphism If C has a zero object, then morphisms that serve as a kernel of φ form a single equivalence class of monomorphisms with target a. The corresponding subobject of a will be denoted Ker φ and will be referred to as the kernel of φ.

138 Exercise 96 Show that the kernel of φ is the preimage of the zero subobject φ−1(0) of b.

9.1.12 The cokernel of a morphism Quotient-objects of b are deﬁned dually, as equivalence classes of epimorphisms with source b. Morphisms that serve as a cokernel of a morphism φ form a single isomorphism class of epimorphisms whose source is b, the target of φ. The corresponding quotient object of b will be denoted Coker φ and referred to as the cokernel of φ.

9.1.13 Any morphism in the category of arrows Arr C between monomorphisms µ and µ0 φs o · · (186)

µ λ

bo a φt is uniquely determined by its target component φt . In particular, HomArr C(λ, µ) naturally identiﬁes with the subset

HomC(a, b; λ, µ) ⊆ HomC(a, b) (187) consisting of those morphisms φ for which the diagram

· · (188)

µ λ

bo a φ admits completion to a commutative square

o · ·

µ λ

bo a φ

139 Exercise 97 Show that

0 0 HomC(a, b; λ , µ ) = HomC(a, b; λ, µ) if λ0 is isomorphic to λ and µ0 is isomorphic to µ.

9.1.14 Morphisms between subobjects

Independence of HomC(a, b; λ, µ) of particular representatives allows one to introduce the following notion of a morphism from a subobject L of a to a subobject M of b: it is

a morphism φ : a −→ b for which diagram (188) admits completion to (186) for some representatives λ and µ of the respective subobjects.

We shall say in this case that φ preserves the corresponding morphisms and express this by writing φ(L) ⊆ M.

9.1.15 Equipped with this deﬁnition of a morphism, subobojects of arbitrary objects of C form a category that will be denoted Sub C.

9.1.16 Naturally, there is a functor

I C Sub C , c 7−→ [idc],(189) w and a functor T C u Sub C (190) assigning object c to its subobject M. Note that T ◦ I = idC , i.e., C is a retract of Sub C.

Exercise 98 Show that T is left adjoint to I .

140 9.1.17 There is also an obvious functor from the full subcategory Mono C of Arr C consisting of monomorphisms in C to Sub C, [] Sub C u Mono C ,(191) that assigns to a monomorphism µ its equivalence class [µ].

Exercise 99 Show that any assignment to each M ∈ Sub C of a monomorphism µ ∈ M gives rise to a unique functor

S Sub C Mono C .(192) w Exercise 100 Show that S is both left and right adjoint to [].

9.1.18

By design [] ◦ S = idSub C , i.e., Sub C is a retract of Mono C, and the isomorphism idMono C ' S ◦ [] is implemented as follows: given a monomorphism µ in C, if µ0 = S([µ]) is the assigned representative of the class [µ], then there is a unique isomorphism ' o · ·

µ0 µ

·· Let us record this fact in the form of the following proposition.

Proposition 9.2 The ‘projection’ of the category Mono C of monomorphisms in C onto the category Sub C of subobjects of objects of C admits a functorial ‘section’ S that (arbitrarily) assigns to each class M its representative. The resulting pair of functors sets up an equivalence of categories between Mono C and Sub C.

9.1.19 Composing S with the source-of-an-arrow functor

s Mono C C w

141 produces a ‘source’ functor

Sub C C w that assigns to a subobject M the source of the monomorphism-representative assigned to M.

Exercise 101 Show that s ◦ S is right adjoint to I .

9.2 Filtrations 9.2.1 n-step ﬁltrations The class of subobjects of a given object a is just the ﬁrst of a sequence of similar equivalence classes associated with a. Let us consider a composable sequence of n monomorphisms

ι ι − ι ι : · / 1 / ··· / n 1 / · / n / a

Given another such n-sequence terminating at c, there exists at most one completion of the diagram

ι ι − ι · / 1 / ··· / n 1 / · / n / a

/ / / / / / · 0 ··· 0 · 0 a ι1 ιn−1 ιn to a diagram ι ι − ι · / 1 / ··· / n 1 / · / n / a

ν1 νn

/ / / / / / · 0 ··· 0 · 0 a ι1 ιn−1 ιn If this is so, we write ι ≺ ι0. As before, if ι ≺ ι0 and ι0 ≺ ι, then ν gives rise to an isomorphism of n-sequences. The equivalence classes of n-sequences terminating at a are called n-step ﬁltrations of a. Subobjects of a are precisely 1-step ﬁltrations of a.

142 9.2.2 Filtrations of a given object can be equivalently described in terms of equivalence classes of chains of monomorphisms

µ1 ≺ · · · ≺ µn−1 ≺ µn with common target a, i.e., chains of subobjects

M1 ⊆ · · · ⊆ Mn−1 ⊆ Mn of a.

9.2.3 Infinite filtrations Infinite filtrations, whose terms are indexed by natural numbers or by integers, can be defined as equivalence classes of sequences of monomorphisms

· · · ≺ µp−1 ≺ µp ≺ µp+1 ≺ · · · with common target a, or as inﬁnite chains of subobjects of a.

9.2.4 Considerations analogous to the ones for n = 1, show that such ﬁltrations form in each case a category that can be denoted

Fltrn C, FltrN C or FltrZ C. It is customary to refer to objects of any of these categories as ﬁltered objects of C. So, Fltr1 C = Sub C and, for n = 0, it is natural to consider Fltr0 C to be C itself.

9.2.5 All of these categories of ‘ﬁltered objects’ are related to each other by a number of obvious functors that either discard certain terms from ﬁltration, or repeat some terms. The simplest examples we already considered for n = 0 and 1 in Section 9.1.16.

9.2.6 Notation

Capital letter Fp with a subscript indicating the term of ﬁltration is fre- quently eployed as notation for the p-th term of a ﬁltration

... ⊆ Fp−1 ⊆ Fp ⊆ Fp+1 ⊆ ....(193)

143 9.2.7 Decreasing filtrations So far we have considered only increasing filtrations. Decreasing filtrations are sequences of subobjects

... ⊇ Fp−1 ⊇ Fp ⊇ Fp+1 ⊇ ....

The difference between increasing and decreasing ﬁltrations is reﬂected in notation: in the latter the indices are superscripts. Reindexing

−p Fp ˜ F transforms a decreasing ﬁltration into an increasing one (and vice-versa).

9.2.8 Complete Z-ﬁltrations If [ Fp = [ida], p∈Z we say that the ﬁltration is complete of a.

9.2.9 Separable Z-ﬁltrations If \ Fp p∈Z is the smallest subobject of a, we say that the ﬁltration is separable of a.

9.3 Submorphisms 9.3.1 Partial morphisms Let us call a diagram φ o b · (194)

a a partial morphism from a to b.

144 9.3.2 The class of partial morphisms from a to b is preordered by the relation

(φ, λ) ≺ (φ0, λ0) “(φ, λ) is produced from (φ0, λ0)”, i.e., there exists ν ∈ Arr C such that the diagram

0 · φ0 ν Ð b ou · φ λ

a commutes. Such an arrow is a monomorphism and is unique when it exists.

9.3.3 Like before for subobjects, if (φ, λ) ≺ (φ0, λ0) and vice-versa, then ν is an isomorphism. Equivalence classes of partial morphisms from a to b will be called submorphisms.

9.3.4 Composition of submorphisms Given a composable pair of partial morphisms

ψ o c · (195)

φ o b ·

145 the partial morphism (ψ ◦ φ˜, λ ◦ µ˜) is said to be their composite if the commutative square in

ψ φ˜ o o c · · (196)

µ µ˜

φ o b ·

a is Cartesian.

Exercise 102 If (φ, λ) is equivalent to (φ0, λ0) and (ψ, µ) is equivalent to (ψ0, µ0), then a composite of (φ, λ) and (ψ, µ) is equivalent to a composite of (φ0, λ0) and (ψ0, µ0).

9.3.5 If (φ, λ) represents a submorphism Φ from a to b and (ψ, µ) represents a submorphism Ψ from b to c, then the equivalence class of their composites which, according to Exercise 102 is independent of the particular representatives, will be denoted Ψ ◦ Φ.

9.3.6 Note that subobjects of a become identiﬁed with submorphisms represented by symmetric pairs (λ, λ).

Exercise 103 Show that under the correspondence

[µ] ←→ [(µ, µ)] intersection corresponds to composition:

[λ] ∩ [µ] ←→ [(λ, λ)] ◦ [(µ, µ)].

146 9.3.7 Morphisms between partial morphisms A triple of arrows making the diagram

φ o @ b ·

ft λ fm Ð d o · a ψ

µ fs c commute is naturally regarded as a morphism from (φ, λ) to (ψ, µ). Note that the middle component fm is uniquely determined by the source component fs . Supplying the target component ft provides an additional constraint on existence fm , thus the set of morphisms from (φ, λ) to (ψ, µ) is identiﬁed with the the subset of

HomC(a, c) × HomC(b, d) (197) consisting of those pairs of fs and ft such that fs ‘preserves’ the monomorphisms in question, i.e., fs [λ] ⊆ [µ], and whose restriction to their respective sources forms a commutative square with ft . Denote this subset of (197) by

Hom(φ, λ), (ψ, µ).

Exercise 104 Show that

Hom(φ, λ), (ψ, µ) = Hom(φ0, λ0), (ψ0, µ0) if (φ, λ) ∼ (φ0, λ0) and (ψ, µ) ∼ (ψ0, µ0).

9.3.8 The category of submorphisms Thus, submorphisms in C, i.e., the equivalence classes of partial morphisms, form a category. We shall denote it Subarr C and refer to its objects as submorphisms or, informally, ‘subarrows’ of C.

147 9.3.9 The domain functor Assigning to a submorphism Φ = [(φ, λ)] the subobject [λ] deﬁnes the domain functor Dom Subarr C Sub C , Φ 7−→ Dom Φ ˜ [λ].(198) w The submorphisms whose domains ‘coincide’ with their source, i.e.,

Dom Φ = [ida], form a subcategory in Subarr C that is naturally identiﬁed with the category of arrows Arr C.

9.3.10 The preimage of a subobject under a submorphism Given a subobject M = [µ] of b and a submorphism Φ from a to b, we deﬁne the preimage of M under Φ as

Φ−1 M ˜ Dom(M ◦ Φ) (199) where we identify M with the corresponding submorphism [(µ, µ)].

Exercise 105 Show that if M ⊆ M0 , then

Φ−1(M) ⊆ Φ−1(M0).

9.3.11 The preimage of a filtration under a submorphism Given a filtration (193) of b and a submorphism Φ from a to b, the preimages of the terms of that filtration form, according to Exercise 105,

−1 −1 −1 ... ⊆ Φ (Fp−1) ⊆ Φ (Fp) ⊆ Φ (Fp+1) ⊆ .... a ﬁltration of a.

9.3.12 The kernel functor Suppose that C is a category with zero and, for any object b in C and any submorphism Φ with target b, the preimage of the zero subobject 0 ⊆ b exists. Let us denote it Ker Φ,

Ker Φ ˜ Φ−1(0) , and call it the kernel of Φ.

148 Exercise 106 Show that assignment on objects

Φ 7−→ Ker Φ gives rise to a functor

Subarr C Sub C . w

149 10 Spectral sequences

10.1 The category of spectral sequences: the ungraded case 10.1.1 The concept of a spectral sequence is sufﬁciently complex so that it is prudent to introduce it in successive steps, each step adding one more layer of complexity.

10.1.2 In its simplest form, it is a sequence of objects of an additive category A equipped with a square-zero endomorphism

r r r r (E , d ), d ◦ d = 0, (r ≥ r0), connected to each other by a sequence of identiﬁcations of the homology of term Er with Er+1 ,

φr+1 Er+1 o H(Er, dr) .

10.1.3 The differentials of a spectral sequence Morphisms dr are referred to as the differentials.

10.1.4 Morphisms between spectral sequences Morphisms between spectral sequences are deﬁned naturally as sequences of morphisms Er −→ 0Er , one for each term, that commute with the differentials and are compatible with the φ-identiﬁcations.

10.1.5 For each r we assume that a kernel and an image of dr

Er o o Zr and Er o o Br exist. Either we ﬁx both or each r (or we employ a kernel and image functors, assuming every arrow in A has a kernel and a cokernel).

150 10.1.6 The image of dr factorizes through the kernel of dr in view of the hypothesis (dr)2 = 0, i.e., each term (Er, dr) is equipped with a composable pair of monomorphisms Er o o Zr o o Br and an extension ϕr+1 Er+1 o o Zr o o Br

10.1.7 If we employ the following notation:

Zr0 ˜ Er, Zr1 ˜ Zr, Br0 ˜ 0 and Br1 ˜ Br. and the subobject notation and terminology, then each term Er is equipped with a 4-step ﬁltration

Zr0 ⊇ Zr1 ⊇ Br1 ⊇ Br0.(200)

10.1.8 Let us consider ϕr+1 r+1 o r E Z

Er as a partial morphism from Er to Er+1 . We shall denote the corresponding submorphism by Φr+1 .

10.1.9 The preimage under Φr+1 of the corresponding 4-step ﬁltration of Er+1 induces a 6-step ﬁltration on Er ,

Zr0 ⊇ Zr1 ⊇ Zr2 ⊇ Br2 ⊇ Br1 ⊇ Br0 (201) where

−1 −1 Zrl = Φr+1 Zr+1,l−1 and Brl = Φr+1 Br+1,l−1 (1 ≤ l ≤ 2).

151 10.1.10 Thus, every term Er is equipped with a 6-step ﬁltration and taking its preimage under Φr+1 produces this time an 8-step ﬁltration Zr0 ⊇ Zr1 ⊇ Zr2 ⊇ Zr3 ⊇ Br3 ⊇ Br2 ⊇ Br1 ⊇ Br0 (202) where −1 −1 Zrl = Φr+1 Zr+1,l−1 and Brl = Φr+1 Br+1,l−1 (1 ≤ l ≤ 3).

10.1.11 If we apply this argument m times, we obtain a (2m + 4)-step ﬁltration on each term Er Zr0 ⊇ Zr1 ⊇ Zr2 ⊇ · · · ⊇ Zrm ⊇ Brm ⊇ · · · ⊇ Br2 ⊇ Br1 ⊇ Br0 (203) where −1 −1 Zrl = Φr+1 Zr+1,l−1 and Brl = Φr+1 Br+1,l−1 (1 ≤ l ≤ m).

10.1.12 By noting that −1 −1 Zrl = Φr+l ◦ · · · ◦ Φr+1 Zr+1,0 = Φr+l ◦ · · · ◦ Φr+1 Er+l (204) and, similarly, −1 −1 Brl = Φr+l ◦ · · · ◦ Φr+1 Br+1,0 = Φr+l ◦ · · · ◦ Φr+1 (0) (205) we obtain at once two inﬁnite ﬁltrations on each term Er Er = Zr0 ⊇ Zr1 ⊇ Zr2 ⊇ · · · ⊇ Br2 ⊇ Br1 ⊇ Br0 = 0 (206) whose Zrl and Brl terms are the preimages under l times iterated submorphism Φ of the largest and the smallest subbjects of the term l levels up from Er .

10.1.13 Note that each Zrl is by deﬁnition the domain of the l times iterated submorphism Φ Φr+l ◦ · · · ◦ Φr+1 with Brl appearing as its kernel.

152 10.1.14 If pullbacks of epimorphisms in A are epimorphisms, then, in fact, we have a double sequence ofextensions

r+l r+1 r+l Φ ◦···◦Φ rl rl E o o Z o o B (r ≥ r0, l ∈ N).

10.1.15 Note that the B-filtration is increasing and the Z-filtration is decreasing, and every term of the B-filtration is contained in every term of the Z- filtration. In particular, [ Br∞ ˜ Brl (207) l≥0 is contained in \ Zr∞ ˜ Zrl.(208) l≥0

10.1.16 One can think of ‘elements’ of Zr∞ as those ‘elements’ of Er that are in the domain of every iterated Φ-submorphim, and of the ‘elements’ of Br∞ as those ‘elements’ of Er that are eventually annihilated by some iteration of Φ.

10.1.17 By making one more step in the considerations of the previous chapter, we could introduce the category of subquotients. This would allow us to see that Φr+1 induces a morphism from the subquotient Zr∞/Br∞ of Er to the subquotient Zr+1,∞/Br+1,∞ of Er+1 .

10.1.18 The E∞ -term of a spectral sequence Under favorable circumstances (met in many abelian categories), the above morphisms between the ∞-subquotients are isomorphisms. In any case, E∞ is deﬁned as a direct limit of Zr∞/Br∞ provided the latter exists.

153 10.1.19 An alternative is to consider the direct limits

Z∞,∞ and B∞,∞ of sequences of morphisms

Zr0∞ / ··· / Zr∞ / Zr+1,∞ / ··· and Br0∞ / ··· / Br∞ / Br+1,∞ / ··· induced by Φr+1 .

10.1.20 All of the above structures are functorial in the sense that they are respected by arbitrary morphisms of spectral sequences.

10.2 The category of spectral sequences: the graded case 10.2.1 In this scenario, each term Er is Z-graded, the r-th differential is a graded- morphism of degree −r, and the φ-identifactions have degree 0. In other words, a spectral sequence is a triple

r r+1 dp φp r r / r r r / r+1 E = Ep, Ep Ep−r , Hp(E , d ) ' Ep (r ≥ r0, p ∈ Z) (209) where r r dp ◦ dp+r = 0 for all r ≥ r0 and p ∈ Z.

10.2.2 All the considerations of the ungraded case give can be retraced paying r attention to the presence of grading. For example, each term Ep is equipped with a double ﬁltration

r r0 r1 r2 r2 r1 r0 Ep = Zp ⊇ Zp ⊇ Zp ⊇ · · · ⊇ Bp ⊇ Bp ⊇ Bp = 0 (210)

154 10.2.3 The reason for a perhaps surprising requirement that the degrees −r of consective differentials follow descending pattern is due to the fact that the main source of graded spectral sequences, namely the spectral sequences associated with ﬁltered differential objects of A,

(A, F, ∂), i.e., objects A of A equipped with an increasing Z-ﬁltration (193) and a square-zero endomorphim of A that preserves F,

∂(Fp) ⊆ Fp (p ∈ Z), exhibit this pattern.

10.3 The category of spectral sequences: the bigraded case 10.3.1 In this scenario, each term Er is Z-bigraded, the r-th differential is a bigraded-morphism of bidegree (−r, r − 1), and the φ-identifactions have bidegree (0, 0). In other words, a spectral sequence is a triple

r r+1 dpq φpq r r / r r r / r+1 E = Epq, Epq Ep−r,q+r−1 , Hpq(E , d ) ' Epq (r ≥ r0, p ∈ Z) (211) where r r dpq ◦ dp+r,q−r+1 = 0 for all r ≥ r0 and p ∈ Z.

10.3.2 All the considerations of the ungraded case give can be retraced paying r attention to the presence of bigrading. For example, each term Epq is equipped with a double ﬁltration

r r0 r1 r2 r2 r1 r0 Epq = Zpq ⊇ Zpq ⊇ Zpq ⊇ · · · ⊇ Bpq ⊇ Bpq ⊇ Bpq = 0 (212)

10.3.3 The reason for the requirement that the bidegrees (−r, r − 1) of consective r differentials dpq follow that pattern is due to the fact that the main source

155 of bigraded spectral sequences, namely the spectral sequences associated with ﬁltered chain complices

(C, F, ∂), i.e., Z-ﬁltered objects of the category Ch(A) of chain complices, exhibit this pattern.

156