notes on homological algebra - ucb mathematicswodzicki/253.f16/ha.pdf · 1.1.5 note that a cokernel...

Notes on Homological Algebra

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., additivecategories with kernels, cokernels, every epimorphism being a kernel, andevery monomorphism being a cokernel.

1.1.2

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

•

• •

flfl

ι

u α

[[

θ (1)

in which ι is an image of α , i.e., a kernel of a cokernel of α , and θ is acoimage 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

• • •u α uβ

(2)

is said to be exact if β factorizes

κ β′

with κ being a kernel of α and β′ 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 akernel of α form an extension

. • • •u u λ u xκ (3)

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 preciselywhen κ is an image of β and λ is a coimage of α

•

• • •

•

flflα

u α

[[

λ

uβ

flfl β^

[[κ

in which case coimage factorization factorizes α through a cokernel ofβ with α being a monomorphism while image factorization factorizes βthrough a kernel of α with β being an epimorphism. Let us record thisfact for future use.

Lemma 1.1 For any composable pair of arrows, the following three conditions areequivalent

(a) β factorizes through a kernel of α

• •

•

uβ

ffffflfl β′^

[[κ

(b) α factorizes being a cokernel of β

•

• •

flffffflα′

uα

[[

λ

(c) a cokernel of β and a kernel of α form an extension.

1.1.6 Condition α β = 0

A weaker condition, α β = 0 , is equivalent to existence of a factorizationof β

• •

• •uu

λ

uβ

u

vκ

u xµ

(4)

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 epimor-phism.

1.1.7 Homology

A cokernel χ of µ is called a homology of composable pair (2) and it formsan extension

• • •u uχ

u xµ

. (5)

1.1.8

The target of χ is usually referred to as the homology (group) of the com-posable pair. This terminology originally was applied to homomorphismsof abelian groups but the tradition is so well established that it continuesto 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 thefact that χ is an epimorphism, its target is zero, precisely when µ is anisomorphism, i.e., when (2) is exact.

1.1.10

Composable pairs (2) satisfying the condition α β = 0 form a full subcat-egory of the category of composable pairs A2,0 .

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

χ : A2,0 −→ ArrA.

1.1.11

The compositions

Z˜ s χ and H˜ t χ

5

with the source and, respectively, the target functors

s : ArrA −→ A and t : ArrA −→ 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 ofα .

1.1.12 A dual approach to homology

Based on the corresponding factorization of α instead of β one can developa dual approach to homology.

1.1.13 Chain complices

We shall use the term chain complex in two ways: either in a loose sense, todenote any sequence of composable morphisms of any length such thatthe 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 ∂ ofdegree −1 such that

∂ ∂ = 0.

The graded endomorphism ∂ is referred to as the boundary morphism oroperator.

1.1.14 Notation

We may use notation (C•, ∂•) , (C, ∂) , C• or, simply, C , to denote a chaincomplex (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 forepimorphisms, more precisely, if for any diagram

P

• •uu u

ε

6

with ε being an epimorphism, there exists an arrow

P

•

\\]

such that the diagramP

• •u

\\\\]

u uε

commutes.

1.2.2

The above definition makes sense in any category, in general categories,however, there may be “too many” epimorphisms making the LiftingProperty 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, epimor-phisms are mappings with dense image. For any nonsurjective epimor-phism f : Y −→ Z and any nonempty space X the maping that sends allpoints in X to a point in Z \ f (Y) cannot be lifted to Y . Thus, the onlytopological space that has Lifting Property for epimorphisms is the emptyspace.

1.2.4 Projective objects in a general category

For the above reasons, projective objects in general categories are definedby 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 formonomorphisms, more precisely, if for any diagram

I

• •

u

v wµ

with µ being a monomorphism, there exists an arrow

I

•

\\

such that the diagramI

• •

u

v wµ

\\

\\

commutes.

Exercise 3 Show that P is projective if and only if for any commutative diagramwith an exact row

P

• • •

fl

0

uφ

uα

uβ

there exists an arrowP

•

\\]φ

such that the diagramP

• • •u

φ

\\\\]φ

uα

uβ

commutes.

8

1.2.6

Dually, I is injective if and only if for any commutative diagram with anexact row

I

• • •

uψ

uα

uβ

[[

[[0

there exists an arrowI

•fffffiψ

such that the diagram

I

• • •fffffffffiψu

ψ

uα

uβ

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,

P • •u u π u xι ,

splits.

Proof. Consider the diagram

P

P •u

idP

u u π

.

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

9

Vice-versa, suppose that a diagram

P

A′′ Au

α

u u ε

is given. Consider a pullback by α of the extension

A′′ A A′u u ε u xκ

where κ is a kernel of ε ,

P A A′

A′′ A A′u

α

uα

u u ε u xκ

u u ε u xκ

If P Awι splits the pullback extension, then α ι lifts α . Indeed,

ε (idA−ι ε) = 0

implies thatidA−ι ε = κ π′

for a unique π′ and

α ε = ε α = ε α idA = ε α ι ε + ε α κ π′ = ε α ι ε

which in turn implies that α = e (α ι) since ε 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 itsobjects and Z -sequences of morphisms (φq)q∈Z ,

A′q Aquφq

, (6)

as morphisms. In order to avoid confusion with morphisms in the categoryof Z -graded chain complices Ch(A) , we shall denote such sequences gradedmaps.

10

1.3.2 Ch(A)

Morphisms in the category of chain complices are graded maps that com-mute with the boundary operators,

∂′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 ,

A′q+d Aquφq

(q ∈ Z), (7)

and often will refer to them simply as maps of degree d . The resulting gradedcategory 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 hasseveral naturally defined subcategories, for example, for any pair of subsetsJ ⊆ I ⊆ Z , let ChI,J(A) denote the full subcategory of chain compliceswith 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. WhenI = 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 ofassociated functors.

1.4.2 The shift functors

The shift functors

[n] : Ch(A) −→ Ch(A), C 7−→ [n]C, (8)

where([n]C)l˜ Cl−n, ([n]∂)l˜ (−1)n∂l−n, (9)

and([n] f )q ˜ fq−n, (10)

provide an action of the additive group of integers Z on Ch(A) and thosesubcategories 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) −→ ArrA, C 7−→ ζq,

that assigns to a chain complex C a kernel of ∂q , its composition with thesource 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) −→ ArrA, C 7−→ βq,

that assigns to a chain complex C an image βq of ∂q+1 , its compositionwith 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 aq -homology functor

Hq : Ch(A) −→ A . (16)

1.4.7 The q -homology extensions

Functors Hq , Zq and Bq form an extension

Hq Zq Bqu uχq

u xβq

(17)

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

ζ′q : Ch(A) −→ ArrA, C 7−→ ζ ′q,

that assigns to a chain complex C a cokernel ζ ′q of ∂q+1 , its compositionwith the target functor yields a q-co-cycles functor

Z′q : Ch(A) −→ A, C 7−→ t(ζ ′q). (18)

13

1.4.9 q -co-boundaries functors

Given a functor

β′l : Ch(A) −→ ArrA, C 7−→ β′q,

that assigns to a chain complex C a coimage β′q of ∂q , its composition withthe target functor yields a q-co-boundaries functor

B′q : Ch(A) −→ A, C 7−→ t(β′q). (19)

1.4.10

Since an image factorization of ∂q is also its coimage factorization, func-tors Bq and B′q+1

are isomorphic by a unique isomorphism of functorscompatible with the corresponding kernel and cokernel functors.

1.4.11 q -co-homology functors

The condition ∂q ∂q+1 means that ∂q uniquely factorizes through ζ ′q ,

∂q = ∂q ζ ′q.

Assigning to a chain complex C the source of a kernel of ∂q defines aq -co-homology functor

H′q : Ch(A) −→ A . (20)

1.4.12 The q -co-homology extensions

Functors Hq , Zq and Bq form an extension

B′q Z′q H′qu uβ′q

u xχ′q

(21)

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 enterfour extensions that form the following commutative diagram

H′q

Z′q Cq Bq

B′q Cq Zq

Hq

v

uχ′q

uuβ′q

u uζ ′q

v

uβq

u xβq

u uβ′q

uuχq

u xζq

(22)

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

1.4.14

By taking into account this isomorphism, we may redraw diagram (22) inthe form

Hq

Z′q Zq

Cq

B′q Bq

khhhhhhhk

khhhhhhhk

444444477

flfl

[[

AAAAAADD “‚

‚‚‚‚

‚“

744444447

(23)

15

2 Fundamental lemmata

2.1 First Fundamental Lemma

2.1.1

Consider a diagram whose columns are chain complices

......

Q1 P1

Q0 P0

N M

0 0

u∂2

u∂2

u∂1

u∂1

uε

uε

u u

uf

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

2.1.2

Since P is projective and

Q0 Nwε (24)

16

is an epimorphism, P Nwf ε

factorizes through (24),

......

Q1 P1

Q0 P0

N M

0 0

u∂2

u∂2

u∂1

u∂1

uε

uε

uφ0

u u

uf

2.1.3

Sinceε (φ0 ∂1) = f ε ∂1 = 0,

and the left column is exact at Q0 , arrow φ0 ∂1 factorizes throughQ1 Q0w

∂1 ,

17

......

Q1 P1

Q0 P0

N M

0 0

u∂2

u∂2

u∂1

u∂1

uφ1

uε

uε

uφ0

u u

uf

Exercise 4 Show that φ1 ∂2 factorizes through Q2 Q1w∂2 .

2.1.4

This inductive procedure yields a morphism from the right column to theleft 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 fundamen-tal fact of Homological Algebra.

Lemma 2.1 (First Fundamental Lemma) Given a morphism

N Muf

in an abelian category A , and a diagram of chain complices in A

Q• P•

[0]N [0]Mu

ε

u

ε

u[0] f

18

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

Q• P•

[0]N [0]Mu

ε

u

ε

uφ

u[0] f

(25)

commute.

2.1.5

In this case we say that φ covers f . The vertical arrows in diagram (25) arerepresented by the augmentations ε 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 ε φ0 = 0 and φ0 factorizes uniquely, inview of projectivity of P0 , through Q1 Q0w

∂0 ,

......

Q1 P1

Q0 P0

N M

0 0

u∂2

u∂2

u∂1

u∂1

uφ1

uε

uε

uφ0

\\\\

χ0

u u

u0

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

and that there exists an arrow P1 Q2wχ1 such that

φ1 = χ0 ∂1 − ∂2 χ1.

2.2.2

This inductive procedure yields a sequence of morphisms

Pl Ql+1w

χl(l ∈ N)

such that∂l (φl − χl−1

∂l) = 0

andφl = χl−1

∂l − ∂l+1 χl (26)

20

for all l > 0 . Identity (26) expresses the fact that morphism φ is theboundary 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

B Aφ

ss

φ′kk

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

φ′ − φ = ∂Homχ = [χ, ∂]. (27)

We may denote this fact by employing notation

φ′ ∼ φ.

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

φ′ ∼χ φ

and represent it diagrammatically

χ

B A

φ

~~

φ′

__

or

B A

φ

χ

φ′

^^

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

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

φ ∼χ 0 , φ = ∂Homχ = [χ, ∂] ,

is called a contracting homotopy. We shall also say that χ contracts φ or thatχ is a contraction of φ , and will represent this diagramatically

B Aφ

ss

χ

orB Au

φ

f χ

2.2.5

Two morphisms are homotopic if and only if their difference is null homo-topic.

2.2.6

We proved above that a morphism of chain complices Q• P•uφ

that

covers the zero arrow N Mu 0 is null homotopic. Its corollary isthe following fact.

Lemma 2.2 (Second Fundamental Lemma) Any two morphisms of chain com-plices 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 φ′ , then it is also a homotopy from −φ′ to−φ and −χ is a homotopy from φ′ to φ .

22

2.3.3

Ifφ′′ ∼χ′ φ′ and φ′ ∼χ φ,

then χ′ + χ is a chain homotopy from φ to φ′′ .

2.3.4

In particular, the chain homotopy relation is an equivalence relation onHomCh(A)(A, B) .

Exercise 6 Show that if

φ′1∼χ1 φ1 and φ′

2∼χ2 φ2

thenφ′

1+ φ′

2∼χ1+χ2 φ1 + φ2.

Exercise 7 Suppose that

B′ Buβ

and A′ Au α

are morphisms of chain complices and

φ′ ∼χ φ.

Show thatφ′ α ∼χα φ α and β φ′ ∼βχ φ.

2.3.5

In other words, the chain homotopy relation is a congruence on any fullsubcategory of Ch(A) . The corresponding quotient categories are referredto 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 , thereexists an extension

M P M1u u ε u x

κ (28)

with P projective. The kernel arrow in (28) is called a 1-st syzygy of M andM1 is referred to as a 1-st syzygy object of M .

23

2.4.2

In particular, there exists an extension

M1 P1 M2u u ε1 u x

κ1

and so on. There exists a sequence of extensions

Ml Pl Ml+1u uεl u x

κl(l ≥ 0)

with M0˜M , P0 = P , and all Pl projective. By splicing them we obtainan acyclic augmented complex

. . . 0 M P0 P1. . .u u u ε u ∂1 u ∂2 (29)

where∂l ˜ κl−1

εl (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

HomCh(A)(P•, [l]N) =

HomA(Ml , N) for l ≥ 0

0 for l < 0

(30)

Exercise 9 Show that null-homotopic morphisms P• −→ [l]N correspond tomorphisms f ∈ HomA(Ml , N) which extend to Pl−1

. In particular,

HomK(A)(P•, [l]N) =

HomA(Ml , N)

f ∈ HomA(Ml , N) which extend to Pl−1 for l ≥ 0

0 for l < 0

(31)

24

2.4.4

According to Lemmata 2.1 and 2.2, any assignment on the class of objectsof A ,

M 7−→ P•

of its projective resolution together with an augmentation

P•

[0]Mu

ε

admits a unique extension to an additive functor P from A to the homotopycategory K≥0,0(A) equipped with a natural transformation of functors

P

[0]u

ε

which, for each object M , induces an isomorphism of the 0-th homology ofPM with M ,

H0(PM)

Mu' .

2.4.5

Moreover any two such augmented functors are isomorphic by a uniqueisomorphism of functors

P′ P

[0] [0]u

ε′

uε

u '

(32)

25

2.5 Third Fundamental Lemma

2.5.1

Consider a commutative diagram

A′′ A A′

B′′ B B′

C′′ C C′

D D′

uα′′

u uπA

uα′

u xιA

uβ′′

uβ

u uπB

uβ′

u xιB

uγ

u uπC

uγ′

u xιC

u xιD

with chain complices in columns, extensions in rows, and a monomorphismin the bottom row, such that A′′ is projective and the right column is exactat C′ .

2.5.2

In view of projectivity of A′′ , arrow α′′ factorizes through πB ,

α′′ = πB α

for some α . Since

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

A′′ A A′

B′′ B′

C′′ C C′

D D′

uα′′

4444444446

βα

\\\\\\\\\]

δ

u uπA

uα′

u xιA

uβ′′

uβ′

uγ

u uπC

uγ′

u xιC

u xιD

2.5.3

Noting thatιD γ′ δ = γ β α = 0

and ιD is a monomorphism, we deduce

γ′ δ = 0.

In view of projectivity of A′′ and exactness of the right column at C′ , arrowδ factorizes through β′ ,

δ = β′ δ

for some arrow δ ,A′′ A A′

B′′ B′

C′′ C C′

D D′

uα′′

4444444446

βα

[[[[[[[[[]

δ

‹‹‹‹‹‹‹‹›

δ

u uπA

uα′

u xιA

uβ′′

uβ′

uγ

u uπC

uγ′

u xιC

u xιD

27

2.5.4

In view of projectivity of A′′ , the top row extension is split. Let π′ a leftinverse of ιA . It is a unique arrow such that

idA−ι′′ πA = ιA π′

where ι′′ is a right inverse of πA .Set

α ˜ α πA + ιB α′ π′. (33)

Exercise 10 Show that the diagram

A′′ A A′

B′′ B B′u

α′′

uα

u uπA

uα′

u xιA

u uπB

u xιB

commutes andβ α = 0. (34)

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

28

are chain complices...

...

P1 Q1

P0 Q0

M′′ M M′

0 0 0

u∂′′

2

u∂′

2

u∂′′

1

u∂′

1

uε′′

uε′

u u

u u π

u

u xι

(35)

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

P•

M′′u

ε′′

by augmented complexQ•

M′u

ε′

29

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

......

...

P1 R1 Q1

P0 R0 Q0

M′′ M M′

0 0 0

u∂′′

2

u∂2

u∂′

2

u∂′′

1

u∂1

u u π1

u∂′

1

u xι1

uε′′

uε

u u π0

uε′

u xι0

u u

u u π

u

u xι

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

aj cuιj

,

uniquely defined by the identities

πi ιj =

idai when i = j

0 when i , j(37)

3.1.2 Matrix morphisms between direct sums

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

bi ajuαij

such that the seti ∈ I | αij , 0

is finite for every j ∈ I , the family of morphisms from aj to d ,

∑i∈I

ιi αij,

defines a unique morphism α from a coproduct c of a family (aj)j∈j to acoproduct d of a family (bi)i∈I . For obvious reasons, we shall denote thismorphism by

∑i∈I

ιi αij πj

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 multiplica-tion and addition of their matrices. Accordingly, we shall be representingsuch morphisms by their matrices and performing all calculations involvingthe morphisms by employing those matrices.

31

3.1.4

From now on we shall adopt the direct sum notation⊕j∈I

aj (38)

to denote a direct sum of (aj)j∈J equipped with the two families of mor-phisms (ιj)j∈J and (πj)j∈J . Note that (38) should be treated as a genericnotation.

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

∂Coneq ˜

(∂B

q fq−1

0 −∂Aq−1

)

is called a cone of f .

Exercise 12 Show that

∂Coneq ∂Cone

q+1=

(0 ∂B

q fq − ∂Aq−1 fq−1

0 0

)(39)

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 Band [1]A in the category of chain complices. The cone of a morphism ofchain complices is thus an infinitesimal deformation of B⊕ [1]A .

32

3.2.3 The Cone Extension

Exercise 13 Show that the maps

[1]A B⊕ [1]Au π[1]A

and B⊕ [1]A Bu ιB

are morphisms of chain complices

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

Since the composable pair

[1]A Cone( f ) Bu π[1]A

u ιB(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 mor-phisms Hq( f ) induced by f in homology coincide with the connectinghomomorphisms δq+1 of the homology long exact sequence associatedwith extension (40).

3.2.5 The cone functor

If we fix a binary direct sum functor on the underlying category A , weobtain a functor, denoted Cone, from the category of arrows Arr Ch(A) toCh(A) .

Exercise 14 Indeed, given a square of morphisms of complices

A′

f ′

Aφss

oo

f

B′ Bφtt

oo

(41)

show that

[φ, ∂Cone] =

(0 φtt f − f ′ φss

0 0

)

33

where

φ˜

(φtt 0

0 φss

)(42)

In particular, (42) is a chain complex morphism if and only if square (41) commutes.

3.2.6

Note that

[1]A = Cone(0

A) and B = Cone(0B),

where

0A˜ 0 Au and 0B ˜ B 0u ,

and the commutative diagram

A A 0

0 B Bu u

f

u

u

u

is an extension in the category of arrows Arr Ch(A) ,

0A f 0Bu u u x (43)

which yields the Cone extension, cf. (40), when one applies the cone functorto (43).

3.2.7

If we denote by 0↓ and 0↓ the functors Ch(A) −→ Arr Ch(A) that send a

complex C to0

C and, respectively, 0C ,

then extensions (43) give rise to an extension of functors on Arr Ch(A) ,

0↓ s idArr Ch(A) 0↓ tu u u 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)

A′′

f ′′

A

f

πAoooo A′

f ′

ooιA

oo

B′′ BπB

oooo B′ooιB

oo

induces an extension of the corresponding cones

Cone( f ′′) Cone( f )πoooo Cone( f ′)oo

ιoo ,

the cone extension, (40), for example, being induced by the morphism oftrivial extensions

A

A

f

0

oo

0 Boo B

3.2.9

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

Arr Ext Ch(A) Ext Ch(A)wCone .

In particular,

Arr Ch(A) Ch(A)wCone

is an exact functor. We shall see an explanation of this fact later.

3.2.10

Consider the extension

[1]A Cone(ιB)oooo Cone(idB)

oooo (44)

35

induced by the morphism of extensions

0

B

ιB

oo B

[1]A Cone( f )π[1]A

oooo BooιB

oo

(45)

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

Cone(id[1]A

)Cone

(π[1]A

)oooo [1]Boooo (46)

induced by the morphism of extensions

[1]A Cone( f )

π[1]A

π[1]Aoooo Boo

ιBoo

[1]A [1]A 0oo

(47)


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

A01

f0

+

A11

f1

g1

oo

A00 A10g0

oo

(48)

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 correspond-ing 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−1f1,n−2

−∂01

n−1g1,n−2

∂11

n−2

(49)

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

∂00

n f0,n−1 g0,n−1

−∂10

n−1g1,n−2

−∂01

n−1f1,n−2

∂11

n−2

(50)

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 ˜(

f0

−[1] f1

)=

(f0

~1 f1

)37

is the induced morphism between the g -cones

Cone(g0) Cone(g1)u

and

g ˜(

g0

−[1]g1

)=

(g0

~1g1

)is the induced morphism between the f -cones

Cone( f0) Cone( f1)u .

Here and below ~l f denotes (−1)l [l] f . This, as we shall soon see, is aproper 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

[1]A11

−[1] f1

Cone(g1)

f

oooo A01

f0

oooo

+

A11

f1

g1

oo

[1]A10

Cone(g0)

oooo A00

oooo A10

g0

oo

[1]Cone( f1)

Cone()

oooo Cone( f0)

oooo Cone( f1)

goo

[2]A11 [1]Cone(g1)oooo [1]A01oooo [1]A11

−[1]g1

oo

(51)

38

or, using the ~ notation,

[1]A11

~1 f1

Cone(g1)

f

oooo A01

f0

oooo

+

A11

f1

g1

oo

[1]A10

Cone(g0)

oooo A00

oooo A10

g0

oo

[1]Cone( f1)

Cone()

oooo Cone( f0)

oooo Cone( f1)

goo

[2]A11 [1]Cone(g1)oooo [1]A01oooo [1]A11

~1g1

oo

(52)

3.3.5

All squares commute except the original one that generated the wholepicture, 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 rowsare the cone extensions associated with the 4 morphisms on the right. Inthe bottom 2× 3 subdiagram all 4 columns are the cone extensions thatare 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 thebottom 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 categoryof complices) to its target and source (0-dimensional ‘faces’). The formerrelates the cone of an anticommutative square Cone() to the cones ofits 1- and 0-dimensional faces, if we agree to consider the cone functor on0-diemsional ‘cells’, i.e., objects of Ch(A) , to be the identity functor

Cone0˜ 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 -dimensionalcone functor. For n = 1 , we obtain the original cone functor on arrows ofCh(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 -dimensionaldiagram involving 3

n complices forming 3n−1n one-dimensional cone ex-

tensions.

3.3.10 Cone(g f )

Given a composable pair of morphisms of chain complices

C Bg

oo Af

oo

the above 4× 4-diagram associated with the anticommutative square

A

g f

+

A

f

− idAoo

C Bgoo

40

has the following form

[1]A

−[1] f

Cone(− idA)

f

oooo A

g f

oooo

+

A

f

− idAoo

[1]B

Cone(g)

oooo C

oooo B

goo

[1]Cone( f )

Cone()

oooo Cone(g f )

oooo Cone( f )

goo

[2]A Cone(id[1]A

)oooo [1]Aoooo [1]A

id[1]Aoo

(53)

where

f ˜(

g f−[1] f

)=

(g f

~1 f

)(54)

is a morphism between the cones

Cone(g) Cone(− idA)u

and

g ˜(

gid[1]A

)=

(g−~1 idA

)is a morphism between the cones

Cone(g f ) Cone( f )u .

41

3.3.11

One can consider diagram (53) as expressing the relation between the coneof a composable pair of arrows (a ‘2-simplex’ of the category Ch(A) ), to thecones of its 1-dimensional ‘faces’ f , g and g f . In this respect the mostimportant are the second row from the bottom and the second column fromthe left. This is the essence of the so called Octahedron Axiom of triangulatedcategories.


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.

3.4 Morphisms Cone( f ′) Cone( f )uφ

3.4.1

Given a pair of arrows in the category of chain complices

A′

f ′

A

f

B′ B

a morphism Cone( f ′) Cone( f )uφ

is represented by a matrix

φ =

(φtt φts

φst φss

)(55)

42

or, diagramatically, by the square with 6 arrows

A′

f ′

Aφss

oo

f

φts~~

B′ Bφtt

oo

φst

__(56)

where φst is of degree −1 , arrows φtt and φss are of degree 0, and φts isof degree +1 .


[φ, ∂Cone] =

[φtt, ∂]− f ′ φst φtt f − f ′ φss − [φts, ∂]

[φst, ∂] φst f − [φss, ∂]

(57)

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

[1]A′ Buφst

.

3.4.4

The tt -condition says that φtt is a contracting homotopy for f ′ φst whilethe ss -condition says that φss is a contracting homotopy for φst f . Dia-

43

grammatically,

A′

f ′

B′ Bφtt

φst

__and A′ A

φss

f

B

φst

``

3.4.5

It follows that both φtt f and f ′ φss contract the triple composite

f ′ φst f .

Finally, the ts -condition says that φts is a homotopy from f ′ φss to φtt f .Diagrammatically,

A′

f ′

Aφss

oo

f

φts

B′ Bφtt

oo

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 amorphism φ from f to f ′ consists of a morphism from the target of thesource arrow to the source of the target arrow (shifted by 1),

υ : t( f ) −→ [1]s( f ′),

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

ψ

B A

f ′ϕs

~~

ϕt f

__

44

3.5.2

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

3.5.3

Fixing A , B , A′ and B′ , yields a full small subcategory

M(A)B′A′|BA

which besides being preadditive has an abelian group structure on the setof objects.

3.5.4 The category of arrows UT(A)

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

3.5.5

Note that the set of morphisms

f ′ fuφ

with fixed (υ; ϕt, ϕs) is a torsor over the group HomCh(A)(A, [−1]B) whilethe 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

f

[l]A

(−1)l f

//

B [l]B

(58)

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

(υ; ϕt, ϕs; ψ) 7−→(

ϕt ψ

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 ofarrows, 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 fis a morhism of Ch(A) . The sign difference is necessitated by the follow-ing considerations. Each of the shift functors must preserve the identitymorphisms, as a consequence the shift functors cannot reverse the signof morphisms of shifted objects. At the same time shifts of morphisms inM(A) remain morphisms only if the shifts of arrows consider as objects ofM(A) , change sign simultaneusly with the boundary operators ∂ . This iscompatible with the fact that objects of M(A) are constituent parts of theboundary 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 homotopycommutative squares

A′

f ′

Aφs

oo

f

ψ

B′ Bφt

oo

46

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

3.5.9

The special caseA

f ′

A

f

ψ

B B

(61)

corresponds to ψ being a homotopy from f ′ to f or, equivalently, −ψbeing a homotopy from f to f ′ .

Exercise 19 Show thatA

f

A

f ′

−ψ

B B

is the inverse of (61).

3.5.10

It follows that the cones of homotopic morphisms of chain complices are iso-morphic in Ch(A) , an isomorphism functorially depending on a particularhomotopy from f ′ to f .

3.6 Morphisms between 0A and 0B

3.6.1

We shall investigate morphisms in M(A) and in UT(A) between arrowswhose source or target is 0. From now on we adopt the convention that thecomponent arrows of a morphism not indicated in the morphism diagramare tacity assumed to be zero.

47

3.6.2

Note thatA

0

0 B

υ

__

is a morphism0B −→ 0

A

in M(A) precisely when υ is a morphism

B −→ [1]A

in Ch(A) . In particular,

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

A) = 0.

3.6.3

Similarly,0

A

ψ

B 0

is a morphism0

A −→ 0B

in M(A) precisely when ψ is a morphism

[1]A −→ B

in Ch(A) . In particular,

HomM(A)(0A, 0B) = HomUT(A)(0B, 0

A) ' HomCh(A)([1]A, B).

3.6.4

In the category of arrows one has, of course,

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

A) = 0.

48

3.6.5

Note that

HomM(A)(0A, 0

A′) = HomArr Ch(A)(0A, 0

A′) ' HomCh(A)(A, A′)

and

HomM(A)(0B, 0B′) = HomArr Ch(A)(0B, 0B′) ' 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

HomM(A)( f , f ′) HomCh(A)

(Cone( f ), Cone( f ′)

)u w

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 transformationsgiven by the diagrams

[−1]C

0

0 C

id

aaand 0

[−1]C

id

C 0

(62)

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

f

0

B Cone( f )πB

oo

π[1]A

ccand 0

A

f

ι[1]A

Cone( f ) BιB

oo

(63)

define morphisms in M(A) and that these morphisms are mutually inverse.

It follows that the cone functor provides an equivalence of categoriesbetween 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

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

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

C),

we observe that if we consider Cone as a functor UT(A) −→ Ch(A) , then0↓ is its right but not left adjoint while [−1] 0

↓ is its left but not rightadjoint.

50

3.8 Graded-split extensions

3.8.1

An extension of chain complices

C′′ C C′u u π u xι (64)

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 ι′′ : C′′ −→ C is a map of degree 0 that supplies a right inverse to π in thecategory of graded objects AZ , then let π′ : C −→ C′ be the correspondinggraded projection onto C′ . Recall that it is defined by the identity

idC−ι′′ π = ι π′.

The quartet

C′′ι′′

// Cπ

oo

π′// C′

ιoo

represents C as a direct sum of C′′ and C′ in the category of Z -gradedobjects.

3.8.3 Graded-split epimorphisms and monomorphisms

Morphisms that occur as π in a graded-split extension (64) are calledgraded-split epimorphisms. Morphisms that occur as ι are called graded-splitmonomorphisms. The former are precisely those morphisms that admit agraded right inverse ι′′ , while the latter are those morphisms that admit agraded left inverse π′ .

3.8.4

Exercise 22 Show thatπ [ι′′, ∂] = 0.

In particular,[ι′′, ∂] = ι φ

51

for a unique morphism of chain complices

C′′ [1]C′wφ

. (65)

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

f ˜ [−1]C′′ C′w[−1]φ

.


[−1]C′′

f

0

C′ Cπ′

oo

π

bband 0

[−1]C′′

f

ι′′

||

C C′ι

oo

(66)

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

C′′ C C′

[1]([−1]C′′

)Cone( f ) C′

u

'

u u π u xι

u u u x

3.8.6

A graded splitting

[1]A Cone( f )wσ

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

0

)(67)

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 correspondencewith 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 referredto 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 con-tractible if and only if the cone extension

[1]C Cone(idC) Cu u π[1]Cu x

ιC

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 contractiblewith

∑i∈I

ιi hi πi

being a contracting homotopy where hi is a homotopy that contractscomplex Ci .

4.1.4 Direct summands of contractible complices

An object a in a preadditive category is said to be a direct summand of anobject a′ if there exist morphisms

aι

// a′π

oo

such that π ι = ida . We shall refer to ι as the inclusion into a′ and to π asthe projection onto a .

54

Exercise 26 If C is a direct summand of a contractible complex C′ with a con-tracting homotopy h′ , then

h˜ ι h′ π

contracts C. It is integrable if h′ is integrable.

4.2 Characterizations of contractible complices.

4.2.1 Example: Cone(idC)


h˜(

0 0

idC 0

)(70)

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 followingcharacterization of contractible complices in any additive actegory.

Lemma 4.1 A complex is contractible if and only if it is a direct summand ofCone(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 sofar were valid in any additive category. Under an additional hypothesis weshall establish that a contractible complex C is isomorphic to Cone(idZC)for the complex of cycles of C , the latter being equipped with zero boundaryoperators.

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 B′C ofco-boundaries by the complex of cycles

B′C C ZCu uβ′

u xζ

, (71)

55

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

h ∂ = σ β′

and

β′ σ β′ = β′ h ∂ = β′ (h ∂ + ∂ h) = β′ idC = idB′C β′

demonstrates, in view of β′ being epi, that

β′ σ = idB′C,

i.e., σ is a graded splitting of extension (71).

4.2.4

By calculating

[σ, ∂] β′ = (σ ∂B′C − ∂C σ) β′ = −∂ h ∂ = −∂ σ β′,

we determine that extension (71) is isomorphic to the cone extension

B′C Cone( f ) CZu u u x

for f being the unique morphism from [−1]B′C to ZC such that

ζ [1] f = −∂ σ.

4.2.5

Note that

ζ [1] f (−β′ h ζ) = ∂ h ∂ h ζ = ∂ h ζ = (∂ h+ h ∂) ζ = ζ

implies, in view of ζ being mono, that

[1] f (−β′ h ζ) = id[1]ZC .

Similarly,

(−β′ h ζ) [1] f β′ = β′ h ∂ σ β′ = β′ h ∂ h ∂

= β′ h ∂ = β′ (h ∂ + ∂ h)

= β′

56

implies, in view of β′ being epi, that

(−β′ h ζ) [1] f = idB′C .

Thus, [1] f is an isomorphism between B′C and [1]ZC . This, in turn, inducesan isomorphism between Cone( f ) and Cone(idZC)

ZC [−1]B′C

f

foo

ZC ZC

which fits into the following diagram of isomorphisms of extensions

B′C C ZC

B′C Cone( f ) ZC

[1]ZC Cone(idZC) ZC

u'

u uβ′

u xζ

u[1] f '

u'

u u u x

u u u x

4.3 Characterization of null-homotopic morphisms

4.3.1

Exercise 28 If a morphism f factorizes through a complex C with a contractiblehomotopy h,

B Af

oo

f ′′

C

f ′

[[

57

thenf ′ h f ′′

is a contracting homotopy for f .

4.3.2

Vice-versa, if h is a homotopy that contracts a morphism f , then thediagram of composable morphisms in M(A)

0

A

h

0

B Af

oo AidA

oo

(72)

represents a canonical factorization of f through Cone(idA) .

4.3.3

Alternatively, the diagram

0

[−1]B

id

0

B [−1]B Ah

oo

f

aa(73)

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 morphismsin UT(A) . The two factorizations are equivalent in the sense that they areobtained from a single triple factorization in M(A)

0

[−1]B

id

Ahoo

0

B [−1]B Ah

oo

f

aa

AidA

oo

(74)

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 morphismsin Ch(A) .

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

4.3.6

A very important consequence is that the homotopy category of A is aquotient 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 Propertyfor the class of graded-split epimorphisms.

4.4.2

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

4.4.3

Consider a diagram

Cone(idC)

g

[1]A Cone( f )π[1]A

oo

59

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

A

A

f

0 B

and A

Cϕ

oo

0 C

υ

__

andA

A

f

Cϕ

oo

0 B Cf ϕ

oo

υ

__

represents a canonical factorization of g through Cone( f ) .

4.4.4

Dually, consider a diagram

Cone( f ) BiBoo

υ

Cone(idC)

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

A

f

0

B B

and C 0

C Bϕ

oo

υ

__

and

C A

f

ϕ foo

0

C Bϕ

oo

υ

__

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 isomorphicto

[1]A Cone( f )π[1]Aoo ,

for some morphism of complices f , and every graded-split monomorphismis isomorphic to

Cone( f ) BiBoo ,

thus we arrive at the following characterization of graded-split projectiveand 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;

(c) C is contractible.

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 alsoh′˜ h ∂ h (75)

is a contracting homotopy.

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

h′ h′ = 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 complexC , the set of integrable contracting homotopies is nonempty. The set of allcontracting homotopies is a torsor over the group

HomCh(A)(C, [−1]C),

i.e., the difference between any two contracting homotopies

h′ − h = φ

is a morphism from C to [−1]C .

Exercise 32 Suppose that h is an integrable contracting homotopy. Show thath + φ is integrable if and only if the morphism φ : C −→ [−1]C satisfies theMaurer-Cartan equation

[φ, h] + φ2 = 0. (77)

4.5.3

The integrability property of a contracting homotopy is equivalently ex-pressed ia the identity

(∂ + h)2 = 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 identityvalid 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 .


[hn, ∂] =

0 if n is even

hn−1 if n is odd. (80)

62

4.5.5

Thus, for any contracting homotopy, h2 : C −→ [−2]C is a morphism ofchain 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

[h3, ∂] + h2 = 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 −→ Cof degree i . The equation

(∂ + h)2 = idC (82)

expresses the infinite sequence of equations

idC = [h1, ∂]

0 = [h3, ∂] + h2

1

0 = [h5, ∂] + [h3, h1]

· · · (83)

0 = [h2n+1, ∂] + ∑i+j=2n−1

i, j odd

hihj

· · ·

63

or, equivalently, of chain homotopies

idC ∼h10

0 ∼h3h2

1

0 ∼h5[h3, h1]

· · · (84)

0 ∼h2n+1 ∑i+j=2n−1

i, j odd

hihj

· · ·

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 se-quence of higher homotopies h3 , h5 , . . . , such that formal series (81), withh1 = 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

F(h) = h + ∑i>2, odd

aihi (ai ∈ Z).

The sequence of equations (83), for i > 1 , becomes the sequence of equa-tions in unknown coefficients ai ,

0 = a2n+1 + ∑i+j=2n−1

i , j odd

aiaj (n > 0) (85)

which are equivalent to a single functional equation

0 =F− h

h+ F2

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

F(x)˜−x +

√1 + (2x)2

2x

= x + ∑n>1

(−2)n−1(2n− 3)!!n!

x2n−1

= x− x3 + 2x5 − 5x7 + 14x9 + · · · . (87)

4.5.10

A degree 3 graded map h3 that satisfies the second equation in (83) differsfrom −h3 by a morphism of chain complices h3 : C −→ [−3]C ,

h3 = −h3 + h3.

Exercise 37 Show thath5˜ 2h5 + hh3h

satisfies the third equation in (83).

4.5.11 (2n− 1) -contractions

Let us callh2n−1 = h1 + h3 + · · ·+ h2n−1

a (2n − 1) -contraction of a complex C if the first n equations (83) aresatisfied.

4.5.12

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

4.5.13 Question

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

65

5 Contractions of Cone( f ) and homotopy equivalences

5.1 Null-homotopic morphisms Cone( f ′) Cone( f )uφ

5.1.1

A degree +1 graded map Cone( f ′) Cone( f )uχ

is represented bya matrix

χ =

(χtt χts

χst −χss

). (88)

with entries being degree +1 graded maps

B′ Buχtt , B′ [1]Au

χts ,

and[1]A Bu

χst , [1]A′ [1]Au−χss .

5.1.2

Given a null-homotopic morphism Cone( f ′) Cone( f )uφ

contracted

by a homotopy χ , the identity

φ = [χ, ∂]

translates into 4 identities:

φtt = f ′ χst + [χtt, ∂], φts = f ′ χss − χtt f + [χts, ∂] (89)

andφst = [χst, ∂], φss = χst f + [χss, ∂] . (90)

5.1.3

In those 4 identities we take into account that, in terms of the inputcomplices A , B , A′ and B′ , the ‘off-diagonal’ entries of (88) have degrees0 and 2, and thus are even, while the diagonal entries have degree 1, andthus 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 inthe matrix homotopy category M(A) . A morphism (υ; ϕt, ϕs; θ) is null-homotopic precisely when the following occurs: υ is null-homotopic inCh(A) , i.e.,

υ ∼η 0 , (91)

for some map η of degree 0 from B to A′ , the primary homotopies ϕt andϕs are homotopic to f ′ η and η f , respectively,

ϕt ∼χt f ′ η and ϕs ∼χs η f , (92)

and the secondary homotopy ψ is homotopic to f ′ χs − χt f ,

ψ ∼θ f ′ χs − χt f . (93)

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

5.2.2 Homotopy split commutative squares

We shall say that a strictly commutative square

A′

f ′

Aϕs

oo

f

B′ Bϕt

oo

(94)

is homotopy split if there exists a morphism of chain complices η such thatboth triangles in

A′

f ′

Aϕs

oo

f

B′ Bϕt

oo

η

__(95)

are homotopy commutative. A homotopy splitting of the commutative squareconsists of the morphism η and the corresponding homotopies

ϕt ∼χt f ′ η and ϕs ∼χs η f .

67

5.2.3

The resulting 2 homotopies between

ϕt f = f ′ ϕs and f ′ η f

may or may not be homotopic. If they are, and

χt f ∼θ f ′ χs ,

we say that a homotopy splitting is strong and the quartet (η; χt, χs; θ) isthen referred to as a strong homotopy splitting data. If the square admits astrong 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 isstrongly homotopy split.

5.2.5

Providing a contracting homotopy for (0; ϕt, ϕs; 0) in M(A) is equivalentto supplying the splitting morphism η , a pair of primary homotopies χtand χs between ϕt and f ′ η , and between ϕs and η f and, finally, asecondary 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) , amorphism (ϕt, ϕs; θ) is null-homotopic if ϕt and ϕs are null-homotopic,

ϕt ∼χt 0 and ϕs ∼χs 0, (96)

and the homotopy ψ is homotopic to f ′ χs − χt f , exactly as in the caseof M(A) , cf. (93).

5.3.2

In UT(A) a commutative square is null-homotopic precisely when both ϕtand ϕs are null homotopic and the resulting 2 homotopies that contractϕt f = f ′ ϕs are themselves homotopic. In particular, a morphismnull-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

f

A

f

B B

(97)

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

idB− f η = [χt, ∂] , 0 = [ f , χ] + [θ, ∂] (98)

and0 = [η, ∂] , idA−η f = [χs, ∂] . (99)

where [ f , χ] denotesf χs − χt f .

5.4.2 A homotopy equivalence data in Ch(A)

A pair of morphisms of chain complices

Af

// Bg

oo (100)

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 andB . By definition, A and B appear in it on equal footing. If complices Aand 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 andhA , 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) thatcorrespond to isomorphisms in the homotopy category K(A) .

In contrast with the definition of a homotopy equivalence data wheref and g appear on equal footing, f is primary data while the homotopyinverse g and the two homotopies are secondary. One needs to be awareof 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 homotopyinverse of g while χt and χs are the corresponding homotopies hB andhA . A homotopy equivalence data with fixed f is the same as a homotopysplitting of (97) while a contraction of f in M(A) is a strong homotopysplitting.

Exercise 38 Show that the monomorphism

A⊕ B Au xιB

is a homotopy equivalence if A is a contractible complex.

5.5.4

We shall now investigate the structure of a homotopy equivalence in greaterdetail. We start by making a few observations.


[ 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 beextended to a contraction of f in M(A) precisely when [ f , h] is null-homotopic in Ch(A) .

70

Exercise 40 Prove the identities[h2

B, ∂]= [ f g, hB] and

[h2

A, ∂]= [g f , hA]. (102)

5.5.6

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

hB ˜ [ f , h] g = ( f hA − hB f ) g.

Since hB : B −→ [−1]A is a morphism of chain complices,

h′B ˜ hB + hB

is a homotopy that contracts idB− f g .

5.5.7 Calculating f hA − h′B f

In order to simplify calculations we shall omit the composition symbol ,

f hA − h′B 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

¯hB ˜ f [g, h] = f (g hB − hA g).

Since ¯hB : B −→ [−1]A is a morphism of chain complices,

h′′B ˜ hB − ¯hB

is again a homotopy that contracts idB− f g .

71

5.5.9 Calculating f hA − h′′B f : the first method

Aided by the identities established in Exercises 39, 33 and 40, we calculate

f hA − h′′B 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

= [( f hA − hB f )hA , ∂] + [h2

B f , ∂]

= [( f hA − hB f )hA + h2

B f , ∂]

= [h2

B f − hB f hA + f h2

A , ∂] .

5.5.10 Calculating f hA − h′′B f : the second method

f hA − h′′B 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 )

= [ f h2

A, ∂]− [hB( f hA − hB f ) , ∂]

= [ f h2

A − hB( f hA − hB f ) , ∂]

= [h2


A , ∂] .

5.5.11

We established the following important result.

Proposition 5.1 Given a homotopy equivalence ( f , g; hB, hA) , let

h′B ˜ hB + [ f , h] g = hB + ( f hA − hB f ) g (103)

andθ′ ˜ −( f hA − hB f ) hA (104)

and, also, let

h′′B ˜ hB − f [g, h] = hB − f (g hB − hA g) (105)

andθ′′ ˜ −h2

B f + hB f hA − f h2

A . (106)

72

Then both(g; h′B, hA; θ′) and (g; h′′B, hA; θ′′)

are contractions of f in M(A) .

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

5.5.12

We found that by modifying just one of the two homotopies in a homotopyequivalence data, one is able to produce a strong homotopy splitting ofsquare (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 differentmethods, the identity

[ f , h] + f [g, h] f = [h2


A , ∂]. (107)

By exchanging f and g we obtain a symmetric identity:

[g, h] + g[ f , h]g = [h2

Ag− hAghB + gh2

B, , ∂]. (108)

5.5.14

It follows that if0 = [g, h] + [g2, ∂],

for some degree 2 graded map g2 , then

0 = [ f , h] + [ f g2 f − h2

B f + hB f hA − f h2

A , ∂]

and vice-versa, if0 = [g, h] + [ f2, ∂],

then0 = [ f , h] + [g f2g− h2

Ag + hAghB − gh2

B , ∂].

This yields the following result.

73

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

B

g

B

g

A A

(109)

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 homo-topy 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) isnull-homotopic if and only if ν f is null-homotopic. More precisely, find aformula for a contracting homotopy of ν in terms of a given contracting homotopyof ν f and a given homotopy equivalence data ( f , g; hB, hA) .

In particular, the set of homotopy equivalence data that provide astrong homotopy splitting of square (97) is a torsor over the subgroup ofnull-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 ifevery 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

χ2n−1 =

(χtt,2n−1 χts,2n−1

χst,2n−1 −χss,2n−1

). (110)

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

∂Cone ( f ) + χ =

(∂B + hB f

g −(∂A + hA)

)(111)

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 ) + χ

)2

= idCone( f ) (114)

is equivalent to 4 equations

idB−f g =(∂B + hB)

20 = [f, ∂ + h] (115)

0 = [g, ∂ + h] idA−g f =(∂A + hA)

2. (116)

5.6.3 Homotopy n -equivalences

Each of the 4 equations in (115)–(116) has infinitely many componentequations according to their degree. The ‘diagonal’ equations have onlycomponents of even degree, the off-diagonal equations have only compo-nents of odd degree.

75

5.6.4

If we retain in expansions (112)–(113) only terms of degree less or equaln , we shall say that they constitute a homotopy n -equivalence betweencomplices A and B if all the component equations not involving terms ofdegree 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 com-plices, 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. Inparticular, 0-equivalences do not extend, in general, to 1-equivalences.

5.6.8

We know that a homotopy equivalence does not extend, in general, tocontracting 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 , seeProposition 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) , afterpossibly 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 + hB f + f2

g −(∂A + hA)

)2

= idB⊕[1]A . (117)

Equation (117) is equivalent to ( f , g; hB, hA) being a homotopy equivalencedata satisfying

0 = [ f , h] + [ f2, ∂] (118)

and 4 additional equations

0 = f2g + h2

B 0 = [ f2, h] (119)

0 = [g, h] 0 = h2

A + g f2 . (120)

5.6.11

In view of Exercise 30, Cone( f ) admits an integrable contraction when fis a homotopy equivalence. In particular, any any homotopy equivalenceadmits 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

Cn P′nπn

oooo

be a family of arbitrary epimorphisms with P′n projective. It induces thecorresponding family of morphisms of complices

C Cone(id[n]P′n

)u (121)

given by...

∂n+2

...

Cn+1

∂n+1

0

oo

Cn

∂n

P′nπn

oooo

Cn−1

∂n−1

P′n

∂nπnoo

Cn−2

∂n−2

0

oo

......

whose coproduct provides an epimorphism

C Poooo

78

from the contractible complex

P ˜ än∈Z

Cone(idP′n

)(122)

with projective termsPq = P′q ⊕ P′q+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 , thecomplex Cone(id[n]Q)

· · · 0oo Qoo Q 0

oo · · ·oo

n−1 n

is a projective object in Ch(A) .

6.1.3

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

6.1.4

It also follows that any projective object in the category of chain complicesof an abelian category with sufficiently many projectives is isomorphicto a direct summand of a contractible complex with projective terms: allterms of such a complex being direct summands of projective objects areprojective.

79

6.1.5

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

, where

P′q˜ ZqP

is the object of q -cycles of P , each P′q is projective, and P is isomorphic tothe coproduct

än∈Z

Cone(id[n]P′n)

in the category of chain complices.

6.1.6

A coproduct of any family of projective objects being projective, we obtainthe following description of projective objects in several categories of chaincomplices.

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

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

(c) Exactly the same statements hold also for the categories Ch+(A) , Ch-(A) ,Chbd(A) , and Ch≤n(A) for any n.

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, thenfor any complex C in Ch≥l(A) , there exists a epimorphism

C [l]Q⊕ Poooo

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 projec-tives, then a chain complex is a projective object in Ch≥l(A) if and only if itis a contractible complex with projective terms.

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

(c) Exactly the same statements hold also for the categories Ch[l,m](A) for arbi-trary m ≥ l .

Exercise 45 Find the analog of the above characterizations of projective objects inthe 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 objectsin the corresponding categories of chain complices.

Exercise 47 For each of the chain complex categories C mentioned in Propositions6.1 and 6.2 consider the n-homology functor

Hn : C −→ A

and determine its left and right derived functors LqHn and RqHn .

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

6.2.3

Acyclic complices with projective terms may not be contractible if they arenot bounded below. For example, if A is the category of unitary modulesover the ring

k[ε] = k[t]/(t2)

of dual numbers, the following complex P of rank 1 free, and thereforeprojective, k[ε] -modules

· · · k[ε]×εoo k[ε]×ε

oo · · ·×εoo (123)

is acyclic but not contractible. Indeed, contractibility of a complex ispreserved by any additive functor. On the other hand, tensoring by k

81

over k[ε] , which is certainly an additive functor, transforms (123) into thecomplex with zero boundary operators

· · · k0oo k0

oo · · ·0oo .

6.3 Complices with projective terms

6.3.1

Consider a commutative diagram in a preadditive A with rows beingextensions and compositions in columns being 0,

Pq

∂Pq

fq

C′′q

∂′′q

Cq

∂q

πqoooo C′q

∂′q

oo

ιqoo

Pq−1

∂Pq−1

fq−1

fq−1

!!

C′′q−1

∂′′q−1

Cq−1

∂q−1

πq−1

oooo C′q−1

∂′q−1

oo

ιq−1

oo

Pq−2

fq−2

fq−2

!!

C′′q−2Cq−2πq−2

oooo C′q−2

oo

ιq−2

oo

(124)

82

6.3.2

If Pq is projective, then there exists an arrow fq making the upper trianglein

Pq

∂Pq

fq

fq

!!

C′′q

∂′′q

Cq

∂q

πqoooo C′q

∂′q

oo

ιqoo

Pq−1

fq−1

fq−1

!!


oooo C′q−1

oo

ιq−1

oo

commute.

6.3.3

Letδq ˜ fq−1 ∂P

q − ∂q fq. (125)

Sinceπq−1 δq = fq−1 ∂P

q − ∂′′q πq fq = 0 ,

arrow (125) uniquely factorizes through ιq−1 ,

δq = ιq−1 δ′q

and

ιq−2 ∂′q−1 δ′q = ∂q−1 ιq−1 δ′q = ∂q−1 δq

= ∂q−1 fq−1 ∂Pq = fq−2 ∂q−1 ∂P

q = 0

which means, in view of ιq−2 being a monomorphism, that

∂′q−1 δ′q = 0.

83

6.3.4

If the C′ column is exact at C′q−1, then by using projectivity of P′q again,

we obtain an arrow δq such that

δ′q = ∂′q δq

and we setfq = fq + ιq δq.

Exercise 49 Verify that∂q fq = fq−1 ∂P

q .

6.3.5

We established the following result.

Lemma 6.3 Consider a commutative diagram (124) with rows being extensions,compositions in columns being zero, Pq being projective and the C′ -column exactin C′q−1

. Then there exists an arrow

fq : Pq −→ Cq

making the diagram

Pq

∂Pq

fq

fq

!!

fq

!!

C′′q

∂′′q

Cq

∂q

πqoooo C′q

∂′q

oo

ιqoo

Pq−1

fq−1

fq−1

!!


oooo C′q−1

oo

ιq−1

oo

commute.

84

6.3.6 Epi quasiisomorphisms

Let us call an epimorphism π occuring in an extension of chain complices

C′′ Cπoooo C′oooo

an epi quasiisomorphism if C′ is acyclic.

6.3.7

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

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

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

6.3.8

In both situations one constructs a lift of a morphism P −→ C′′ to C ,beginning in a degree m such that either Pq , or both C′′q and Cq , are zero forall q < m , by induction on q and aided by Lemma 6.3. Proposition 6.4 canbe called with justification the Fourth Fundamental Lemma of HomologicalAlgebra.

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

Proof. Suppose that

P′ Pu uf

is an epi quasiisomorphism. The Lifting Property established in Proposition6.4 shows that there exists a right inverse f ′ of f .

For a general quasiisomorphism, consider the commutative diagram

Q⊕ P

(ε f )

||||

P′ Pf

ooaa

ιP

aa

85

where ε is an epimorphism and Q is a contractible complex with projectiveterms, e.g., the canonical projection

P′ Cone(id[−1]P′

)u u ε .

Since ιP is, according to Exercise 38, a homotopy equivalence, every quasi-isomorphism f between bounded below complices with projective termshas a homotopy right inverse f ′ ,

f f ′ ∼ idP′ .

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

idHqP′ = Hq( f f ′) = Hq f Hq f ′

combined with the fact that Hq f is an isomorphism. Let f ′′ be a homotopyright inverse of f ′ . Then

f ∼ f ( f ′ f ′′) ∼ ( f f ′) f ′′ ∼ f ′′.

In other words, a homotopy right inverse of f is also a homotopy leftinverse.

An alternative proof. A morphism f is a quasiisomorphism preciselywhen its cone Cone( f ) is acyclic. The cone of a morphism betweenbounded below complices with projective terms is a bounded below projec-tive complex with projective terms.

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

Thus, Cone( f ) is contractible and therefore f is a homotopy equiva-lence.

6.4 Cartan-Eilenberg resolutions

6.4.1

Suppose that C is a complex in an abelian category A with sufficientlymany projectives.

86

6.4.2

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

......

...

PHp1

PZp1

PBp1

PHp0

PZp0

PBp0

Hp Zp Bp

0 0 0

u∂H

p2

u∂Z

p2

u∂B

p2

u∂H

p1

u∂Z

p1

u uπp1

u∂B

p1

u xιp1

uεH

p

uεZ

p

u uπp0

uεB

p

u xιp0

u u

u uπp

u

u xιp

87

6.4.3

By applying again the Third Fundamental Lemma, we obtain an extensionof augmented projective resolutions

......

...

PBp−1,1 PC

p1PZ

p1

PBp−1,0 PC

p0PZ

p0

B′p Cp Zp

0 0 0

u∂B

p−1,2

u∂C

p2

u∂Z

p2

u∂B

p−1,1

u∂C

p1

u uβ′p1

u∂Z

p1

u xζp1

uεB′

p

u

εCp

u uβ′p0

u

εZp

u xζp0

u u

u uβ′p

u

u xζp

where εB′p is εB

p−1composed with the canonical isomorphism of Bp with

B′p .

88

6.4.4

The First Fundamental Lemma yields morphisms of augmented resolutions

......

PZp1

PBp1

PZp0

PBp0

Zp Bp

0 0

u∂Z

p2

u∂B

p2

u∂Z

p1

u∂B

p1

u xβp1

uεZ

p

uεB

p

u xβp0

u u

u xβp

89

6.4.5

By combining constructions of Sections 6.4.2, 6.4.3 and 6.4.4, we obtain thesequence of morphisms of augmented projective resolutions

......

...

· · · PCp−1,1 PC

p1PC

p+1,1 · · ·

· · · PCp−1,0 PC

p0PC

p+1,0 · · ·

· · · Cp−1 Cp Cp+1 · · ·

0 0 0

u∂C

p−1,2

u∂C

p2

u∂C

p+1,2

u∂C

p−1,1

u∂p−1,1

u∂C

p1

u∂p1

u∂C

p+1,1

u∂p+1,1

u∂p+2,1

u

εCp−1

u∂p−1,0

u

εCp

u∂p0

u

εCp+1

u∂p+1,0

u∂p+2,0

u

u∂p−1

u

u∂p

u

u∂p+1

u∂p+2

(126)

where∂pq ˜ ζp−1,q βp−1,q β′pq .

Exercise 51 Show that∂p−1,q ∂pq = 0 .

6.4.6

If we set∂←pq = (−1)q∂pq and ∂↓pq = ∂C

pq,

then (Ppq, ∂←pq, ∂↓pq

)p,q∈Z

forms a double complex with Ppq = 0 for q < 0 and diagram (126) can beinterpreted either as a double complex P augmented by [−1](C) or as a

90

morphism of double complices

P

[0](C)u

ε (127)

where [m](C) denotes the double complex obtained by placing C in them -th row (in order not to confuse it with the shifted complex [m]C weenclosed C in parentheses).

6.4.7 Cartan-Eilenberg resolutions

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

...

Hp1P

Hp0P

HpC

0

u

u

uε

u

...

Zp1P

Zp0P

ZpC

0

u

u

uε

u

...

Bp1P

Bp0P

BpC

0

u

u

uε

u

and

...

Pp1

Pp0

Cp

0

u

u

uε

u

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

91

6.4.8

For any Cartan-Eilenberg resolution of C , the induced morphism of com-plices

C Tot Pu ε , (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 manynonzero Ppq on each diagonal p + q = n . Our construction produces aCartan-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 itis a complex with projective terms.

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

C Pqis

oooo

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

C′′ Cπoooo C′ooι

oo (129)

and an epi quasiisomorphismP

ε

C′′

92

the pullback extension

P

ε

Q

ε

πoooo C′ooι

oo

C′′ Cπ

oooo C′ooι

oo

(130)

is graded-split in view of projectivity of terms of P , hence isomorphicto the cone extension of a certain morphism f : C′ −→ [−1]P . Since thekernel of ε is isomorphic to the kernel of ε and the latter is acyclic, ε is anepi 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 C′′ , there exists anepi quasiisomorphism from a graded-split extension, cf. (130), where P denotesa bounded below complex with projective terms or, equivalently, from the coneextension of a certain morphism f from C′ to a bouded below complex withprojective terms.

6.4.11

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

P′

qis

P

qis

C′ Cf

oo

there exists a ‘lift’ of f to P and P′

P′

qis

P

qis

foo

C′ Cf

oo

(131)

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

93

Proof. LetQ

ε

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 ofthe hypothesis, Q is acyclic and therefore contractible, cf. Proposition 6.5.A morphism

K Puf

factorizes through ε , hence it is null-homotopic in view of contractibilityof Q .

6.4.12

If f ′ is another lift of to P and and P′ , cf. diagram (131) the differencef ′− f factorizes through the acyclic kernel of the left epi quasiisomorphismin (131) and thus, according to Lemma 6.8, is null-homotopic.

We arrive at the following proposition.

Proposition 6.9 Any assignment

C 7−→

P

Cuu

ε (C ∈ ObA)

of an epi quasiisomorphism from a bounded below complex with projective termsgives rise to a unique additive functor from Ch+(A) to the homotopy categoryK+(PA) of bounded complices with projective terms

Ch+(A) K+(PA)wP , C 7−→ P, (132)

and an epi quasiisomorphism of functors

J+ P

Quu

ε (133)

94

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

Ch+(A) K+(A)wQ

. (134)

6.4.13

Note that the composition of P with the functor [0] that embeds A onto afull subcategory of Ch+(A) yields a projective resolution functor from A

to K+(PA) . Moreover, any such resolution functor admits an extension toCh+(A) as we saw while constructing a Cartan-Eilenberg resolution of anarbitrary complex.

6.4.14

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

K+(A) K+(PA)wP (135)

that will be denoted by the same symbol P .

6.4.16

Epi quasiisomorphism of functors (133) from the category of chain com-plices Ch+(A) now becomes a quasiisomorphism of endofunctors of thecorresponding homotopy category K+(A)

J+ P

idK+(A)

uε . (136)

95

6.4.17

Recall that every morphism C −→ C′ in the homotopy category is repre-sented by a composite

C′ C′′ Cu uη

u xι

where η is an epimorphism in Ch+(A) , C′′ is a direct sum of C and acontractible complex, and ι is the corresponding monomorphism embed-ding of C into C′′ , cf. the first proof of Proposition 6.5. We shall rememberthat the quasiisomorphism in (136) is represented by an epimorphism inCh+(A) .

6.4.18

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

6.4.19

In particular, the composite functor

K+(PA) K+(PA)wPJ+

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 projectiveresolution functor is a retraction of the homotopy category of bounded belowchain complices onto the full subcategory of complices with projectiveterms.

6.4.21

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

96

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

idK+(PA) P J+w'η

,

and the counit being an epi quasiisomorphism,

idK+(A) J+ Pu uqis

ε.

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

P

Cuu

ε ,

the correspondence assigning to a homotopy class of a morphism

C P′uf

the homotopy class of its lift

P P′uf

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

6.4.22

We say that a functor F : C −→ C′ inverts arrows belonging to a certainsubclass S ⊆ ArrC if Fσ is an isomorphism for every σ ∈ S .

Theorem 6.11 (a) Any projective resolution functor (135) inverts all quasiiso-morphisms.

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

F J+ P

Fuu

Fε (137)

97

which is an isomorphism of functors if and only if F inverts quasiisomor-phisms.

Proof. Part (a) is a consequence of the argument preceding the statementof the theorem.

If F inverts quasiisomorphisms, then Fε is an isomorphism itself. Vice-versa, for every additive functor F , the composite functor F J+ P invertsquasiisomorphisms, hence if Fε is an isomorphism, F inverts quasiisomor-phisms.

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

6.5 The derived categories

6.5.1 Localization of a category

Let Σ ⊆ ArrC be any subclass of the class of arrows of a category C . Wecall a functor

C C′wΛ

a localization of C with respect to or at Σ , if any functor

C DwF

that inverts arrows from Σ factorizes uniquely through Λ

C

Λ

F// D

C′F

??

6.5.2 Localization of a category

The functor Λ is referred to as the localization functor while its target, C′ , isoften 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 isomorphismof functors. In particular, its target C′ is unique up to a unique isomorphismthat is compatible with the corresponding localization functors. Genericnotation for C′ is C[Σ-1] . There is no generally agreed notation for thelocalization functor.

6.5.4

Any small category has a localization at any subset Σ ⊆ ArrC . This isdemonstrated by the construction of C[Σ-1] as the quotient of the categoryF freely generated by the disjoint union of the set of all arrows of C andthe set Σ

ArrCt Σ .

Elements of the second summand correspond to formal inverses of arrowsbelonging to Σ . Let us denote those elements by σ .

6.5.5

One then considers the congruence ∼ generated by all relations occuringbetween arrows of C and additional relations

σ σ = idc and σ σ = idc′

where c is the source and c′ is the target of σ . The latter express the factthat σ is an inverse of σ ∈ Σ .

6.5.6

The free category F has the same objects as C . The set theoretic inclusionof ArrC into ArrCt 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 ∈ ArrC, 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 introducedby Oysteyn Ore in his study of formal inverses in rings of differentialoperators in early 1930-ties.

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

Proof. Let us fix a projective resolution functor P . Let L be the categorywith the same objects as Ch+(A) and the morphisms between C and C′

declared to be the morphisms between PC and PC′ ,

HomL(C, C′) ˜ HomK+(A)(PC, PC′).

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

K+(A) LwΛ

that inverts all quasiisomorphisms. If

K+(A) BwF

is a functor inverting all quasiisomorphisms, we set F to be F on the classof objects and

F(α) ˜ F(εC′) F(α) F(εC)−1 (138)

where εC and εC′ are the corresponding epi quasiisomorphisms from PCto 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 classof all quasiisomorphisms is one of several categories called the derivedcategory of A . The above is the fastest proof of its existence but the modelof D+(A) our construction provides explicitly depends on a particularprojective resolution functor P . We shall see soon advantages of thisapproach. If it is, however, desirable to have a model of D+(A) independentof any such functor, one can represent D+(A) as the category of rightfractions with denominators being quasiisomorphisms. Verification thatthe class of quasiisomorphisms in K+(A) satisfies right Ore conditions isan easy exercise in view of the results of this Chapter. Then the existenceof the category of right fractions follows from general and rather lengthyconsiderations. The criterion of equality of morphisms in categories offractions is unfortunately highly implicit.

6.5.9

Let us represent key categories and functors by the following commutativediagram

K+(PA)

ΛJ+

P

OO

A[0]

// Ch+(A)Q

// K+(A)

P77

Λ''

D+(A)

(139)

We are not using here any specific model for D+(A) but if it is the modelconstructed 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 counitof the adjunction

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

101

being isomorphisms. In particular, P and Λ J+ provide an equivalenceof 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 thediagram commute

K+(PA)

ΛJ+

P

OO

FJ+

''

K+(A)

P77

Λ''

K+(B)

Λ

&&

D+(A)LF

// D+(B)

(140)

is denoted LF and called the left derived functor of F . It is induced by thefunctor

K+(A)ΛFJ+P

// D+(B)

that inverts quasiisomorphisms since P does. On objects LF acts by sendinga complex C to FPC .

6.5.12

In contrast with the fact that D+(A) can be constructed with no referenceto any projective resolution functors, the derived functors LF are definedexplicitly in terms of such resolution functors. This is one more reason whythe 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 arethe composites

A B

K+(A) K+(B)u

Q[0]

wLq F

wFJ+P

uHq (141)

102

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

A B

D+(A) D+(B)u

ΛQ[0]

wLqF

wLF

uHq (142)

6.5.14 The derived category D−(A) and the (total) right derived func-tor RF

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

K−(IA) K−(A)wJ− .

6.5.15

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

6.5.16

The dual of the diagram defining LF

K−(IA)

ΛJ−

I

OO

FJ−

''

K−(A)

I77

Λ''

K−(B)

Λ

&&

D−(A)RF

// D−(B)

(143)

103

defines the (total) right derived functor RF . Here I denotes an injectiveresolution functor

K−(A) K−(IA)wI .

Here I is left adjoint to J− while I and Λ J− again provide an equivalenceof categories.

6.5.17 The (classical) right derived functors RqF

If B is an abelian category, then the classical right derived functors RqFare the composites

A B

K−(A) K−(B)u

Q[0]

wRqF

wFJ−I

uH−q (144)

or, equivalently, the composites

A B

D−(A) D−(B)u

ΛQ[0]

wRqF

wRF

uH−q (145)

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

Extl−kA (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 . Thegroup on the right hand side of (146) was calculated in Exercise 9. Inparticular,

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 reso-lution of their first argument, they are right derived functors. This is sobecause HomA( , ) is a contravariant functor of the first argument. Whenviewed as a covariant functor, A is replaced by Aop , bounded below com-plices become bounded above complices and projective resolutions becomeinjective resolutions. In follows that the Ext groups as functors of the first,contravariant argument, are the right derived functors

ExtqA(M, N) = Rq HomA( , N)(M) (148)

of the functor

Aop −→ Ab, Mop 7−→ HomA(M, N).

6.6.3

The Ext groups as functors of the second, covariant argument, are the rightderived functors

ExtqA(M, N) = Rq 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

Rq HomA( , N)(M) = Rq 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 valuesLqF(M1, . . . , Mn)

of classical left derived functors of additive functors of n arguments aredefined as homology groups Hq of the totalization of the n -complexobtained by replacing each covariant argument by its projective resolutionand each contravariant argument by its injective resolution.

6.7.2 Right derived functors RqF

The valuesRqF(M1, . . . , Mn)

of classical right derived functors are defined as homology groups H−qof the totalization of the n -complex obtained by replacing each covariantargument by its injective resolution and each contravariant argument by itsprojective resolution.

6.7.3 Balanced functors of n arguments

A functor is said to be left balanced if replacing any argument by thecorresponding resolution – projective, in the case of covariant, and injective,in the case of contravariant arguments, makes it an exact functor of theremaining arguments.

6.7.4

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

6.7.5

Calculating left derived functors of left balanced functors of n argumentscan be performed by replacing any single argument by the correspondingresolution. And similarly for calculating right derived functors of rightbalanced functors.

106

6.7.6

The HomA functor is right but not left balanced.

6.7.7 TorRq groups

The functor of tensor product

mod-R, R-mod Abw⊗R

and its bimodule variants are left but not right balanced. The left derivedfunctors of ⊗R are denoted

TorRq (M, N) ˜ Lq ⊗R (M, N) . (150)

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

TorZ1(A, Z/nZ)

happens to be the n -torsion group of an abelian grup A . Indeed,

Z Zu ×n

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 multiplicationby n map

A Au ×n .

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, inparticular, that any projective abelian group is free and that any non freeabelian 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 Cone( f )π[1]Aoooo BooιB

oo Af

oo (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: everyconsecutive three morphisms in this infinite sequence of morphisms

Xn−3=[1]Xn Xn−2

fn−2

oo Xn−1

fn−1

oo Xnfn

oo

is isomorphic to the triple of the form (151)

[1]Xn Cone( fn)π[1]Xn

oooo Xn−1oo

ιXn−1

oo Xnfn

oo (152)

as follows from Exercises 15 and 16. Thus, any such triple determines thewhole infinite sequence up to an isomorphism and the corresponding shiftin n .

7.1.3 Exact triangles

A composable triple of chain complices

[1]A Choo B

goo A

foo (153)

isomorphic in K(A) to the cone triple (151) will be referred to as anexact triangle. The terminology reflects the fact that such a triple can be

108

represented by the triangle diagram

B

g

Af

oo

C

h−1

DD(154)

with −1 marking reflecting the fact that h is a morphism from C to A ofdegree −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]B [1]A~1 f

oo Choo B

goo (155)

and [−1]E as

C Bg

oo Af

oo [−1]C~−1h

oo (156)

7.1.5 Morphisms between exact triangles

Morphisms between exact triangles E and E′ are naturally defined astriples of morphisms between the corresponding components that makethe following diagram commute

[1]A

[1]φA

C

φC

hoo B

φB

goo A

φA

foo

[1]A′ C′h′

oo B′g′

oo A′f ′

oo

(157)

7.1.6

A morphism between exact triangles extends to the corresponding mor-phism between infinite sequences obtained by replication on both sidesof shifted morphisms. Any such ‘3-periodic’ sequence of morphisms isuniquely determined, of course, by its restriction to any 3 consecutivearrows, i.e., to any of its embedded exact triangles.

109

7.1.7

Providing only φA and φB allows completion to a morphism between Eand E′ , in view of (157) being up to an isomorphism essentially a morphismbetween the cone extensions of f and f ′ . In other words, any commutativesquare

B

φB

A

φA

foo

B′ A′f ′

oo

extends to a morphism of the corresponding exact triangles.

7.1.8

By appying this observation to [1]E and [1]E′ , we deduce that also anycommutative square

C

φC

B

φB

goo

C′ B′g′

oo

extends to a morphism between E and E′ .

Exercise 55 Show that any commutative square

[1]A

[1]φA

C

φC

hoo

[1]A′ C′h′

oo


7.1.9

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

110

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

[1]A

[1]φA

C

φC

hoo

[1]A′ C′h′

oo


7.1.10

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

7.1.11

For any composable pair of morphisms

C Bg

oo Af

oo ,

the three cone triangles associated with f , g and g f , form a diagramthat can be completed to a diagram, see (53), with rows and columns beingexact triangles. The left edge of that diagram is a shift of the cone triangleof f , the top and and the bottom edges are ‘trivial in the sense that theyare the cone triangles associated with − idA and id[1]A . This leaves fourcone 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 atriangle involving the cones of f , g and g f ,

[1]Cone( f ) Cone(g)oo Cone(g f )oo Cone( f )oo .

This is the second row from the bottom in the 4× 4 diagram (53) withCone() 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

g f

f

C

ww

B

goo

[1]A′ A′

ff

B′

C′

oo

[1]A

(158)

admits a completion to a commutative diagram involving 4 exact triangles

A

g f

f

C

ww

B

goo

[1]A′ A′

ff

xx

B′

gg

C′

oo

[1]A

(159)

112

7.1.13

If we modify the above diagrams by replacing [1]A and [1]A′ by A andA′ , respectively, and considering the morphisms with those targets as mor-phisms of degree −1 , we can folding the opposite commutative trianglesso that A and A′ form opposite lying vertices of an octahedron each halfcontaining 2 exact and 2 commutative triangles. This is why the describedproperty 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

TA Choo B

goo A

foo , (160)

a triangle. Triangles admit a natural notion of a morphism,

TA

TφA

C

φC

hoo B

φB

goo A

φA

foo

TA′ C′h′

oo B′g′

oo A′f ′

oo

7.2.2

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

(Tr 1) the triangle0 Aoo A 0

oo

is distinguished, any arrow f of A occurs in some distinguishedtriangle, and any triangle (160) isomorphic to a distinguished triangleis distinguished;

113

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

TB TA−T f

oo Choo B

goo

is distinguished;

(Tr 3) a commutative diagram

TA

TφA

Choo B

φB

goo A

φA

foo

TA′ C′h′

oo B′g′

oo A′f ′

oo

with distinguished rows admits completion to a morphism of trian-gles;

(Tr 4) any commutative diagram (158) involving 3 distinguished trianglesadmits completion to a commutative diagram (159) involving 4 dis-tinguished triangles.

7.2.3

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

7.2.4

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

7.2.5

Since the axioms are modelled on the class of exact triangles in the homo-topy category of chain complices K(A) of an abelian category, the latterprovides our first example of a triangulated category.

7.2.6

Various shift-invariant subcategories of K(A) , like K+(A) , K−(A) andKb(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, namelyas isomorphic in the derived category to the cone triangles of morphisms ofchain complices in Ch(A) .

7.2.8

In contrast with the homotopy category, every extension of bounded belowor bounded above chain complices is, according to Corollary 6.7, isomor-phic in the apprpriate derived category to the cone extension of somemorphism of chain complices. Thus, every extension of bounded belowchain complices gives rise to an exact triangle in D+(A) , and every exten-sion of bounded above chain complices gives rise to an exact triangle inD−(A) .

7.2.9

There are other examples of triangulated categories besides various homo-topy categories of chain complices and their localizations like the derivedcategories. The triangulated categories K+(A) and K−(A) fit into a generalscheme how triangulated structures arise in projective-injective homotopytheory.

7.3 Two homotopy theories associated with an abelian category

7.3.1 The projective homotopy category of an abelian category

The projective homotopy category category KprA of an abelian category A is

defined as the quotient of A by the subcategory of projective objects PA .The morphisms M −→ N in K

prA are, by definition, equivalence classes of

morphisms in A modulo the abelian subgroup of those morphisms thatfactorize through a projective object. The latter are ‘null-homotopic’, bydefinition, in projective homotopy theory in A .

7.3.2

Let us assume that A has sufficiently many projectives and let

N′′ Noooo N′oo (161)

115

be any extension in A .

Exercise 57 Show that any morphism f : M −→ N′′ in A extends to a mor-phism of extensions in A

M

f

P

f

oooo M1

f1

oooo

N Noooo N′oooo

(162)

where the top row is an extension with projective P.

Exercise 58 Show that if

M

f

P

f ′

oooo M1

f ′1

oooo

N Noooo N′oooo

.

is another morphism of those extensions, then f1 − f ′1

factorizes through P.

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

7.3.3

It follows that assignment to f ∈ HomA(M, N′′) of the projective homotopyequivalence class of f1 defines a homomorphism of abelian groups

HomKprA(M, N′′) HomK

prA(M1, N′)w

and any assignment to each object M of a 1-st syzygy extension (28) inducesan endofunctor

KprA K

prA

wΩ , M 7−→ M1, [ f ] 7−→ [ f1], (163)

where [ f1] is the equivalence class of the morphism betweenn the 1-stsyzygy 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 uniqueisomorphism of the corresponding functors compatible with the respective1-st syzygy extensions.

7.3.5 The ‘loop’ functor Ω

From now on, we shall fix one such assignment and will denote the corre-sponding 1-st syzygy extension by

M PMoooo ΩMoooo . (164)

This is an abelian analogue of what in homotopy theory of spaces is knownas the path fibration associated to a topological space with a distinguishedpoint.

7.3.6

The loop endofunctor, in general, is not an equivalence, its iterationsprovide one-sided translation functors on K

prA .

7.3.7 The Ω -sequence associated with an extension in A

Any extension in A induces a half-infinite sequence of morphisms in KprA

N′′ Nπoo N′ι

oo ΩN′′∂oo ΩNΩπ

oo · · ·Ωιoo (165)

where ∂ is the equivalence class of f1 associated with f = idM .


N N′ιoo ΩN′′∂

oo

andN′ ΩN′′∂oo ΩNΩπ

oo

are both isomorphic in KprA to some extensions of N and, respectively, N′ .

117

7.3.8 Distinguished triangles in KprA

If we define ‘distinguished triangles’ in KprA as composable triples of

morphismsA Coo Boo ΩAoo

admitting an isomorphism in KprA

A

φA '

C

φC '

oo B

φB '

oo ΩA

ΩφA '

oo

N′′ Nπ

oo N′ι

oo ΩN′′∂

oo

with the initial triple of the Ω -sequence of some extension on A , thenthe class of distinguished triangles possesses, as we already saw, someproperties required by the definition of a triangulated category.

7.3.9

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

7.3.10 The injective homotopy category of an abelian category

If we express the projective homotopy category of the opposite categoryabelian category Aop as well as the related notions, in terms of the originalcategory, we obtain the injective homotopy category Kin

A of A and the socalled, mapping cone cofibration,

ΣM IMoooo Moooo . (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.

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

· · · NΣπoo ΣN′Σι

oo N′′δoo Nπ

oooo N′ιoo (167)

associated with an extension (161) in A .

118

Exercise 63 State the injective analog of Exercise 60.

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

7.3.11

The class of distinguished triangles in KinA possesses again some properties

required by the definition of a triangulated category. In certain abeliancategories, e.g., in the categories of modules over Frobenius algebras,the classes of projective and injective objects coincide. In this case, bothhomotopy categories coincide and if we fix a realization of the ‘loop’ and‘suspension’ functors so that they are inverse to each other, then theyprovide a translation automorphisms on the homotopy category, and weindeed obtain a traingulated category structure on K

prA = Kin

A .

7.3.12

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

7.4 Satellites

7.4.1

The iterated ‘loop’ and ‘suspension’ functors provide an alternative ap-proach, avoiding use of chain complices altogether, in homological studyof 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 satellitesSqF and of right satellites SqF .

7.4.2 Left satellites

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

119

asS1F(M) ˜ Ker( FPM FΩMoo ) (168)

and the higher left satellites are defined by iteration:

SqF˜ S1(Sq−1F), S0F˜ F. (169)

Exercise 65 Show thatSqF = S1F Ωq−1.

7.4.3 Right satellites

The 1-st right satellite of an additive functor F with values in any abeliancategory or, more generally, in an additive category with cokernels, isdefined as

S1F(M) ˜ Coker( FΣM FIMoo ) (170)

and the higher right satellites are defined by iteration:

SqF˜ S1(Sq−1F), S0F˜ F. (171)

Exercise 66 Show thatSqF = 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 triangleis 0 . (Hint: Consider a morphism from a distinguished triangle containing the identitymorphism.)

Exercise 68 Show that a distinguished triangle with one of the morphisms being0 is isomorphic with a shift of the direct sum triangle

[1]A C0oo C⊕ A

πCoo A

ιAoo . (172)

Exercise 69 Show that in a triangulated category every epi and monomorphism issplit. More precisely, every epimorphism is isomorhic to a projection π of certaindirect sum, and every monomorphism is isomorphic to an injection ι of certaindirect 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 issemisimple 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 allof little use in a triagulated category. This perhaps surprising conclusion isfurther strengthened by the following observations.

Exercise 70 Show that any morphism in an abelian category with sufficientlymany projectives is projective-homotopic to an epimorphism. Dually, show thatany 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, thenotion of an exact functor acquires a new meaning: a functor betweentriangulated categories is declared to be exact, if it respects distinguishedtriangles. This does not contradict anything we did before, since a triangu-lated category is not an additive category with certain properties, it is, likean exact category, an additive category enriched by a structure of a newkind.

7.5.5

Recall that the correspondence

A 7−→ [0]A (A ∈ ObA)

embeds an abelian category A onto a full subcategory of various categoriesof chain complices, their homotopy category quotients and the correspond-ing derived categories. This full embedding is exact if the target is eitherthe abelian category of complices or the derived category, if by ‘exact’ weunderstand that any extension on A becomes an distinguished triangle inD+(A) or D−(A) .

121

Considered to be embedding of A into either of the homotopy cate-gories, this embedding is not exact, in general, since non-split extensionsof 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 discoveredand investigated between categories that are not equivalent as abeliancategories. Various abelian categories of D-modules and constrible sheavesare involved in such an equivalence.

This led to studying abelian subcategories cA of a given triangulatedcategory T for which ⊗ is up to equivalence its derived category. Theoryof 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 ⊆ ArrC be a class of arrows in a category C that satisfies thefollowing conditions

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

(O2) any diagram in C

·

·

Σ·

(173)

admits completion to a commutative square

Σ

·

·

Σ·

(174)

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

8.1.2

Denote by Σ-1Σ the class of arrows

Σ-1Σ ˜ α ∈ ArrC | σα ∈ Σ for some σ ∈ Σ. (175)

8.1.3 Notation

We adopt the following convention: member arrows of Σ in diagramswill be marked with symbol Σ . Similarly, member arrows of Σ-1Σ will bemarked 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·


Σ

·

·

Σ-1Σ·

Exercise 71 Prove Lemma 8.1.

8.1.5 Summits

Diagrams·

Σ

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

a b

124

is above a summit φ′ = (α′, σ′) ,

·α′

σ′

a b

or, equivalently, that φ′ is below φ , if the diagram

·

α

σ

·

α′

σ′

a b

admits completion to a commutative diagram

·

α

σ

·

α′

σ′

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 ofσ ∈ Σ .

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

8.1.7 Preordered classes

A class S equipped with a reflexive and transitive relation ≺ is said to bepreordered (by ≺ ).

125

8.1.8 Filtered preordered classes

A preordered class with any two members φ and φ′ admitting anothermember ψ such that

φ′ ≺ 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 S′ ⊆ S if for any φ and φ′ in S , there exists ψ ∈ S′

that satisfies (176).

8.1.10

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

any triple of summits

φ′ φ

χ

admits completion to a quadruple

ψ

φ′ φ

χ

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

126

Exercise 73 Use Lemma 8.1 to demonstrate Lemma 8.2.

Corollary 8.3 The relation

φ′ ∼ φ if φ and φ′ 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

Σ

·β

σ

Σ

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 abovethem.


Lemma 8.5 If φ′ is above φ , then above every span of φ and ψ there exists aspan of φ′ and ψ . Similarly, if ψ′ is above ψ , then above every span of φ and ψthere exists a span of φ and ψ′ .


Lemma 8.6 If φ′ ∼ φ and ψ′ ∼ ψ , then every span of φ and ψ is equivalent toevery span of φ′ and ψ′ .


127

8.2 The category of right fractions

8.2.1

Given a class Σ ⊆ ArrC of arrows satisfying the right Ore conditions(O1)–(O3), the category of right fractions C[Σ-1] is defined as follows. Ithas the same objects as C . Morphisms from b to a are equivalence classesof 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 theidentity morphisms given by [

(ida, ida)]. (178)

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

8.2.2

The category C[Σ-1] introduced above exists provided the equivalenceclasses 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 ,

a bwσ ,

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 thelocalization 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 repre-sentative of the equivalence class of summits.

Exercise 80 Show that two functors

C[Σ-1]F′

//

F′′// D

are equal if and only ifF′ Λ = F′′ 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 uniquelyfactorizes through the localization functor (179),

CF

//

Λ!!

D

C[Σ-1]

F

==

8.2.6 Left Ore conditions

The right Ore conditions for the opposite category Cop when expressed interms 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 bygoing step by step and dualizing all concepts and arguments from Section 8.1.2 upto 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 anobject a .

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

8.2.8 Terminal objects

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

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

8.2.9 Binary products

Exercise 85 Show that a pair of morphisms φ and φ′ in C[Σ-1] with a commonsource 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

c

πa

πb

a b

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

c[(πa,idc)]

[(πb,idc)]

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 preservesbinary coproducts, equalizers and coequalizers. Each case is a worthyexercise. By combining these partial results we establish the followingimportant proposition.

Proposition 8.8 The localization functor is exact, i.e., it preserves all finite directand 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 categoryA . We shall demonstrate that A[Σ-1] is canonically equipped with a pread-ditive 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 arepresentation 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 toequivalence 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

·

ρ′

ρ′

·

σ′′

σ′

·

ρΣ

·

σΣ

·

(182)

forming a commutative diagram with the composite arrows being membersof Σ .

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

·ρ

σ

·

ρ′

ρ′

·

σ′′

σ′

·

ρ

·

σ

·with all composites being members of Σ .

8.3.4

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

σ ≺ σ′ if σ′ factorizes through σ .

132

8.3.5

Thus, above any pair of summits (α, ρ) and (β, σ) from an object b to anobject a , there is a pair of summits (α′, τ) and (β′, τ) with a common to bothτ ∈ Σ . Furthermore, for any two such pairs with common ‘denominator’

(α, σ) and (β, σ) or (α′, σ′) and (β′, σ′),

representing morphisms φ and ψ , respectively, there is a pair

(α′′, σ′′) and (β′′, σ′′)

with common denominator which is above either of these pairs.

8.3.6

It follows that the summit

(α′′ + β′′, σ′′)

is above both(α + β, σ) and (α′ + β′, σ′).

8.3.7

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

8.3.8

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

Exercise 88 Show that the class of (0ab, idb) is a neutral element of so definedaddition 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 summitsfrom b to a is an ‘abelian group’ class without necessarily being a set. Inthe case, however, when all such classes are sets, we obtain the categoryof fractions A[Σ-1] equipped with the canonical structure of a preadditivecategory.

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 doesA[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 quasi-isomorphisms in the category of chain complices ChI,J(A) , cf. Section1.3.4.

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 thethird right Ore condition while the class of quasiisomorphisms in K−(A) satisfiesthe third left Ore condition.

8.4.2 The existence of the category of fractions K+(A)[Qis−1]

LetC P

qoooo (183)

be a quasiisomorphism of bounded below complices.

134

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

C′′

f

~~

g

C′ C

,

with g being an epi quasiisomorphism, is below

( f , q)

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

8.4.3

Since every morphism of chain complices factorizes as a homotopy equiva-lence followed by an epimorphism, cf. Section 6.4.17, we deduce that everyequivalence class of summits for Σ = Qis ⊂ ArrK+(A) is represented bya summit

( f , q)

with a fixed quasiisomorphism q , as in (183). This not only shows existenceof the category of right fractions K+(A)[Qis−1] but also demonstrates that

HomK+(A)(P, C′) −→ HomK+(A)[Qis−1](C, C′), f 7−→ [( 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-gory of right fractions K+(A)[Qis−1] . Dually, one obtains a constructionof the derived category D−(A) as the category of left fractions

[Qis−1]K−(A).

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

135

9 Categories of filtered objects

9.1 Subobjects

9.1.1

The class of monomorphisms with target b ∈ ObC is preordered by therelation

µ ≺ µ′ “µ factorizes through µ′”,

i.e., there exists ν ∈ ArrC such that

·µ

ν

b

·ddµ′

dd

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


µ ≺ µ′ and µ′ ≺ µ (185)

implies that the above ν is an isomorphism.

9.1.2

In other words, (185) holds if and only if monomorphisms µand µ′ areisomorphic by an isomorphism

·

µ′

·'ν

oo

µ

b b

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

136

9.1.3

The class Sub b of subobjects of b may be a set even if C is not a smallcategory. For example, subobjects of a set S in the category of sets are inbijective correspondence with subsets of S . The latter form a set accordingto one of the axioms of Set Theory.

9.1.4

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

9.1.5

Given subobjects M and M′ of b , if

if µ ≺ µ′ for some µ ∈ M and µ′ ∈ M′ ,

thenµ ≺ µ′ for all µ ∈ M and µ′ ∈ M′ .

We shall write in this caseM ⊆ M′

and say that M is contained in M′ , or will write

M′ ⊇ M

and say that M′ contains M . The relation of being contained is a partialorder on Sub c .

9.1.6

The largest member of Sub b is given by the isomorphism class of theidentity 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 arereferred to as, respectively, the intersection and the union of the family.

137

9.1.8 Notation

We employ the notation⋂i∈I

Mi˜ infMi | i ∈ I and⋃i∈I

Mi˜ supMi | 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 subobjectsof b that contain every Mi . Since C ∈ U , this set is nonempty and itsinfimum exists in view of the hypothesis. This infimum is automaticallythe supremum of Mi | i ∈ I .

9.1.9

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

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 , thena pullback λ of µ by φ is a monomorphism,

·

µ

·oo

λ

b aφ

oo

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 ofa 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 forma single equivalence class of monomorphisms with target a . The corre-sponding subobject of a will be denoted Ker φ and will be referred to asthe 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 epimor-phisms with source b . Morphisms that serve as a cokernel of a morphismφ form a single isomorphism class of epimorphisms whose source is b ,the target of φ . The corresponding quotient object of b will be denotedCoker φ and referred to as the cokernel of φ .

9.1.13

Any morphism in the category of arrows ArrC between monomorphismsµ and µ′

·

µ

·

λ

φsoo

b aφt

oo

(186)

is uniquely determined by its target component φt . In particular, HomArrC(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

·

µ

·

λ

b aφ

oo

(188)


·

µ

·

λ

oo

b aφ

oo

139


HomC(a, b; λ′, µ′) = HomC(a, b; λ, µ)

if λ′ is isomorphic to λ and µ′ is isomorphic to µ .

9.1.14 Morphisms between subobjects

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

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

We shall say in this case that φ preserves the corresponding morphisms andexpress this by writing

φ(L) ⊆ M.

9.1.15

Equipped with this definition of a morphism, subobojects of arbitraryobjects of C form a category that will be denoted SubC .

9.1.16

Naturally, there is a functor

C SubCwI , c 7−→ [idc], (189)

and a functorC SubCu T (190)

assigning object c to its subobject M . Note that T I = idC , i.e., C is aretract of SubC .

Exercise 98 Show that T is left adjoint to I .

140

9.1.17

There is also an obvious functor from the full subcategory MonoC of ArrCconsisting of monomorphisms in C to SubC ,

SubC MonoCu[ ]

, (191)

that assigns to a monomorphism µ its equivalence class [µ] .

Exercise 99 Show that any assignment to each M ∈ SubC of a monomorphismµ ∈ M gives rise to a unique functor

SubC MonoCwS . (192)

Exercise 100 Show that S is both left and right adjoint to [ ] .

9.1.18

By design [ ] S = idSubC , i.e., SubC is a retract of MonoC , and theisomorphism

idMonoC ' S [ ]

is implemented as follows: given a monomorphism µ in C , if µ′ = S([µ])is the assigned representative of the class [µ] , then there is a unique isomor-phism

·

µ′

·'oo

µ

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

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

9.1.19

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

MonoC Cws

141

produces a ‘source’ functor

SubC Cw

that assigns to a subobject M the source of the monomorphism-representativeassigned 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 ofsimilar equivalence classes associated with a .

Let us consider a composable sequence of n monomorphisms

ι : · // ι1// · · · // ιn−1

// · // ιn// a

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

· // ι1// · · · // ιn−1

// · // ιn// a

· //ι′1

// · · · //ι′n−1

// · //ι′n

// a

to a diagram

· // ι1//

ν1

· · · // ιn−1// · // ιn

//

νn

a

· //ι′1

// · · · //ι′n−1

// · //ι′n

// a

If this is so, we writeι ≺ ι′.

As before, if ι ≺ ι′ and ι′ ≺ ι , then ν gives rise to an isomorphism ofn -sequences. The equivalence classes of n -sequences terminating at a arecalled n -step filtrations of a . Subobjects of a are precisely 1-step filtrationsof a .

142

9.2.2

Filtrations of a given object can be equivalently described in terms ofequivalence 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 inte-gers, 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 filtrationsform 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 objectsof C . So, Fltr1 C = SubC and, for n = 0 , it is natural to consider Fltr0 C tobe C itself.

9.2.5

All of these categories of ‘filtered objects’ are related to each other by anumber of obvious functors that either discard certain terms from filtration,or repeat some terms. The simplest examples we already considered forn = 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 filtrationsare sequences of subobjects

. . . ⊇ Fp−1 ⊇ Fp ⊇ Fp+1 ⊇ . . . .

The difference between increasing and decreasing filtrations is reflected innotation: in the latter the indices are superscripts. Reindexing

Fp˜ F−p

transforms a decreasing filtration into an increasing one (and vice-versa).

9.2.8 Complete Z-filtrations

If ⋃p∈Z

Fp = [ida],

we say that the filtration is complete of a .

9.2.9 Separable Z-filtrations

If ⋂p∈Z

Fp

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

b ·φ

oo

λ

a

(194)

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) ≺ (φ′, λ′) “(φ, λ) is produced from (φ′, λ′)”,

i.e., there exists ν ∈ ArrC such that the diagram

·′φ′

uuν

λ

b ·φ

oo

λ

a

commutes. Such an arrow is a monomorphism and is unique when itexists.

9.3.3

Like before for subobjects, if (φ, λ) ≺ (φ′, λ′) and vice-versa, then ν is anisomorphism. Equivalence classes of partial morphisms from a to b willbe called submorphisms.

9.3.4 Composition of submorphisms

Given a composable pair of partial morphisms

c ·ψ

oo

µ

b ·φ

oo

λ

a

(195)

145

the partial morphism (ψ φ, λ µ) is said to be their composite if thecommutative square in

c ·ψ

oo

µ

·φ

oo

µ

b ·φ

oo

λ

a

(196)

is Cartesian.

Exercise 102 If (φ, λ) is equivalent to (φ′, λ′) and (ψ, µ) is equivalent to(ψ′, µ′) , then a composite of (φ, λ) and (ψ, µ) is equivalent to a compositeof (φ′, λ′) and (ψ′, µ′) .

9.3.5

If (φ, λ) represents a submorphism Φ from a to b and (ψ, µ) representsa submorphism Ψ from b to c , then the equivalence class of their com-posites which, according to Exercise 102 is independent of the particularrepresentatives, will be denoted Ψ Φ .

9.3.6

Note that subobjects of a become identified with submorphisms repre-sented 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

b ·φ

oo

fm

λ

d

ft

@@

·

µ

ψoo a

fs

c

commute is naturally regarded as a morphism from (φ, λ) to (ψ, µ) . Notethat the middle component fm is uniquely determined by the sourcecomponent fs . Supplying the target component ft provides an additionalconstraint 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 monomor-phisms in question, i.e.,

fs([λ])⊆ [µ],

and whose restriction to their respective sources forms a commutativesquare with ft . Denote this subset of (197) by

Hom((φ, λ), (ψ, µ)

).


Hom((φ, λ), (ψ, µ)

)= Hom

((φ′, λ′), (ψ′, µ′)

)if

(φ, λ) ∼ (φ′, λ′) and (ψ, µ) ∼ (ψ′, µ′).

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 SubarrC and refer to its objects assubmorphisms or, informally, ‘subarrows’ of C .

147

9.3.9 The domain functor

Assigning to a submorphism Φ = [(φ, λ)] the subobject [λ] defines thedomain functor

SubarrC SubCwDom , Φ 7−→ Dom Φ˜ [λ]. (198)

The submorphisms whose domains ‘coincide’ with their source, i.e.,

Dom Φ = [ida],

form a subcategory in SubarrC that is naturally identified with the categoryof arrows ArrC .

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 , wedefine the preimage of M under Φ as

Φ−1M ˜ Dom(M Φ) (199)

where we identify M with the corresponding submorphism [(µ, µ)] .

Exercise 105 Show that if M ⊆ M′ , then

Φ−1(M) ⊆ Φ−1(M′).

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 , thepreimages of the terms of that filtration form, according to Exercise 105,

. . . ⊆ Φ−1(Fp−1) ⊆ Φ−1(Fp) ⊆ Φ−1(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 anysubmorphism Φ with target b , the preimage of the zero subobject 0 ⊆ bexists. 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

SubarrC SubCw .

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 isprudent to introduce it in successive steps, each step adding one more layerof 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

(Er, dr), dr dr = 0, (r ≥ r0),

connected to each other by a sequence of identifications of the homologyof term Er with Er+1 ,

Er+1 H(Er, dr)φr+1

oo .

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 sequencesof morphisms Er −→ ′Er , one for each term, that commute with thedifferentials and are compatible with the φ -identifications.

10.1.5

For each r we assume that a kernel and an image of dr

Er Zroooo and Er Broooo

exist. Either we fix both or each r (or we employ a kernel and imagefunctors, 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 ofmonomorphisms

Er Zroooo Broooo

and an extension

Er+1 Zrϕr+1

oooo Broooo

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 equippedwith a 4-step filtration

Zr0 ⊇ Zr1 ⊇ Br1 ⊇ Br0. (200)

10.1.8

Let us consider

Er+1 Zrϕr+1

oo

Er

as a partial morphism from Er to Er+1 . We shall denote the correspondingsubmorphism 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

Zrl =(Φr+1

)−1Zr+1,l−1 and Brl =(Φr+1

)−1Br+1,l−1 (1 ≤ l ≤ 2).

151

10.1.10

Thus, every term Er is equipped with a 6-step filtration and taking itspreimage under Φr+1 produces this time an 8-step filtration

Zr0 ⊇ Zr1 ⊇ Zr2 ⊇ Zr3 ⊇ Br3 ⊇ Br2 ⊇ Br1 ⊇ Br0 (202)

where

Zrl =(Φr+1


)−1Br+1,l−1 (1 ≤ l ≤ 3).

10.1.11

If we apply this argument m times, we obtain a (2m + 4) -step filtration oneach term Er

Zr0 ⊇ Zr1 ⊇ Zr2 ⊇ · · · ⊇ Zrm ⊇ Brm ⊇ · · · ⊇ Br2 ⊇ Br1 ⊇ Br0 (203)

where

Zrl =(Φr+1


)−1Br+1,l−1 (1 ≤ l ≤ m).

10.1.12

By noting that

Zrl =(Φr+l · · · Φr+1

)−1Zr+1,0 =(Φr+l · · · Φr+1

)−1Er+l (204)

and, similarly,

Brl =(Φr+l · · · Φr+1

)−1Br+1,0 =(Φr+l · · · Φr+1

)−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 submor-phism Φ of the largest and the smallest subbjects of the term l levels upfrom Er .

10.1.13

Note that each Zrl is by definition the domain of the l times iterated submor-phism 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, wehave a double sequence ofextensions

Er+l ZrlΦr+l···Φr+1

oooo Brloooo (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∞˜

⋃l≥0

Brl (207)

is contained inZr∞

˜⋂l≥0

Zrl . (208)

10.1.16

One can think of ‘elements’ of Zr∞ as those ‘elements’ of Er that are in thedomain of every iterated Φ -submorphim, and of the ‘elements’ of Br∞ asthose ‘elements’ of Er that are eventually annihilated by some iteration ofΦ .

10.1.17

By making one more step in the considerations of the previous chapter, wecould introduce the category of subquotients. This would allow us to seethat Φr+1 induces a morphism from the subquotient Zr∞/Br∞ of Er to thesubquotient 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 abovemorphisms 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,∞ // · · ·

andBr0∞ // · · · // Br∞ // Br+1,∞ // · · ·

induced by Φr+1 .

10.1.20

All of the above structures are functorial in the sense that they are respectedby 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 otherwords, a spectral sequence is a triple

E =(

Erp, Er

pdr

p// Er

p−r , Hp(Er, dr)φr+1

p

'// Er+1

p

)(r ≥ r0, p ∈ Z)

(209)where

drp dr

p+r = 0 for all r ≥ r0 and p ∈ Z .

10.2.2

All the considerations of the ungraded case give can be retraced payingattention to the presence of grading. For example, each term Er

p is equippedwith a double filtration

Erp = Zr0

p ⊇ Zr1

p ⊇ Zr2

p ⊇ · · · ⊇ Br2

p ⊇ Br1

p ⊇ Br0

p = 0 (210)

154

10.2.3

The reason for a perhaps surprising requirement that the degrees −r ofconsective differentials follow descending pattern is due to the fact that themain source of graded spectral sequences, namely the spectral sequencesassociated with filtered differential objects of A ,

(A, F, ∂),

i.e., objects A of A equipped with an increasing Z -filtration (193) and asquare-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 abigraded-morphism of bidegree (−r, r− 1) , and the φ -identifactions havebidegree (0, 0) . In other words, a spectral sequence is a triple

E =(

Erpq, Er

pqdr

pq// Er

p−r,q+r−1, Hpq(Er, dr)

φr+1

pq

'// Er+1

pq

)(r ≥ r0, p ∈ Z)

(211)where

drpq dr

p+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 payingattention to the presence of bigrading. For example, each term Er

pq isequipped with a double filtration

Erpq = Zr0

pq ⊇ Zr1

pq ⊇ Zr2

pq ⊇ · · · ⊇ Br2

pq ⊇ Br1

pq ⊇ Br0

pq = 0 (212)

10.3.3

The reason for the requirement that the bidegrees (−r, r− 1) of consectivedifferentials dr

pq follow that pattern is due to the fact that the main source

155

of bigraded spectral sequences, namely the spectral sequences associatedwith filtered chain complices

(C, F, ∂),

i.e., Z -filtered objects of the category Ch(A) of chain complices, exhibitthis pattern.

156

notes on homological algebra - ucb mathematicswodzicki/253.f16/ha.pdf · 1.1.5 note that a cokernel...

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