Foundations of Relative Homological Algebra

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA SAMUEL EILENBERG AND

J.

C. MOORE 1)

Introduction The notion of a derived functor for a fuoctor T: (1'- 93 (with suitable conditions imposed on the categories (j and

:.B

and on the (unctor T) is one of the key

notions of homological algebra. Since its original definition [4] this notion has undergone a multitude of generalizations. Some of these generalizations went in the direction of avoiding the use of projective resolutions, following an idea of Yoneda [12]. This will not be the point of view adopted here. Granting that the definition of derived functors is to use projective resolutions, the procedure breaks up neatly into two quite well separated steps. The first of these steps is the definition of projective resolutions, their existence and basic properties. The second step is the definition of the derived functors and the study of their properties. In this paper we shall not be concerned with the second step at all; this will be deferred to subsequent papers. The first step will be our main concern here. In each category, the notion of projective objects is inherent. However it has

been recognized for some time that more latitude in the choice of projective objects (or equivalently in the choice of exact sequences) should be permitted. Thus Hochschild [8] in studying the category of modules over an algebra A considered r-projective modules where r is a subalgebra of A. Heller [7] considered additive categories with a distinguished class of "proper" morphisms. Buchsbaum [2] considered abelian categories in which a class of morphisms called an "h. f." ciass was given subject to a number of natural conditions. Butler and Horrocks [3] modify and extend Buchsbaum's approach by starting with an abelian category and a distinguished class of short exact sequences. The main notion of this paper is that of a "projective class of sequences" in an arbitrary (pointed) category. Each such class carries with it its own projective objects. One can then talk about projective resolutions, and if the category is additive, all the usual properties of the resolutions hold. In particular, this will permit the development of homological algebra in some additive categories Received by the editors April 23, 1964. I)The first author was partially supported by the Office of Naval Research and the National Science Foundation, while the second author was partially supported by the Air Force Office of Scientific Research during the period while this research was in progress.

I

2

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

which are not abelian, e.g., the category of comodules over a coalgebra over an arbitrary commutative ring. With the notion of a projective class established in Chapter I, the balance -of the paper is devoted to methods of finding such projective classes and their projective objects. The main tool here is the "adjoint theorem" of Chapter 11. Chapter I begins with a brief review of categories and continues with the definition and some basic properties of projective classes. Chapter 11 begins with a review of adjoint functors and con~nues with the adjoint theorem (Chapter 11, §2) which tums out to be the main tool for constructing projective classes. Chapter III gives a varied assortment of examples of projective classes in various categories, all arrived at by the adjoint theorem. Chapter IV studies the category cet of complexes over an abelian category et and the various sub-categories of cet. In particular, it is shown that the "double resolutions" of Cartan-Eilenberg [4, Chapter XVII] are simply resolutions in cet relative to a suitable projective class. CHAPTER I. RELATIVE HOMOLOGICAL ALGEBRA 1. Review of categories

Let et be a category. We shall use the abbreviated notation et(A, B) instead of the usual notation Homet(A, B) for the set of morphisms f: A -+ B of objects

A and B in et. The dual category is denoted by et*. All categories et considered here will be assumed pointed, i.e., containing an object Ao such that et(A, Ao) and et(A o' A) consist of single elements for

every object A in [11, p. 504]. For any two such trivial objects AO and A ~

there is a unique isomorphism AO ~ A ~ in et ,and we shall write 0 for any

object Ao with the property above. -For any two objects A, A' in et we define

the trivial map A -+ A' as the composition A -+ 0 -+ A'. -The trivial map will be written as 0 A'A or again simply as O. A typical example is the category

S of sets

with basepoints with morphisms

being maps of sets preserving the basepoints. -Thus for any category et and for any objects A, A' in et, et (A, A') is any object in the category S. This yields the "representation" functor et* x et -...

S.

A family Po-: A-...Ao-' a-€ I of morphisms in a category et is called a prod-

uct if for any family of morphisms f0-: B -+ A0- in et there is a unique morphism

f: B-. A such that Po-f = fo- for all a€ I. Dually a family io-: A0--+ A, a€ I there exists a unique f: A-+ B such that fi 0- =f0- for all a€ I.

is a coproduct if for each family f0-: A0-- B, a€ I

Let et be a category and let Po-: A-+Ao-' a-€ I be a product. For each

3

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

a'€:£ there exists then a unique morphism i~: A 0--. A such that p0-0i = lA 0- ' v Prio- = 0 for r f.a.If the family lio-l is a coproduct then we say that io-

Ao- I

A,

'

a€:£

Po-

is a biproduct. A pre-additive category is a category

a such that each set a(A, A') is

given the structure of an abelian group in such a way that composition is distributive: (gI + g2)f=gd + g21, g(II +12 ) = gll + gh for 1,11' 12 : A'-.A and g, gl' g2: A-. A". It is easy to see that in a pre-additive category and for a finite set of indices there is no need to distinguish between products, coproducts and biproducts. Further one can verify that ig

A0 " .

'

A,

a= 1, 2

PO"

is a biproduct if and only if PIi l

=

A pre-additive category

1 Ar P2 i 2

=

1A2 , ilPI + i 2 P2 = lA·

a in which biproducts exist for any two objects

Al

and A2 is called additive. It can be shown that a category in which biproducts exist for any two objects Al and A2 can be converted in at most one way into an additive category [11, p. 511-512]., Given morphisms

such that rc = lA" we say that r is a retraction, c is a coretraction and that A' is a retract of A. Amorphism [:A'-.A is called an epimorphismif a([, B):a(A, B)-.a(A~B) is injective (as a mapping of sets) for every B in

a. Monomorphisms are defined

dually. Let (1.1) be morphisms in

a.

We say that (1.1) is a sequence if ji

kernel of j if ji = 0 and for any

I: B -. A

=

O. We say that i is a

such that j[ =0, [admits a unique

factorization 1= ig. It follows that i is a monomorphism. If kernel of j then there exists a unique cI>: A' ---. A' such that cl> is an isomorphism. Cokernels are defined dually.

i: A' -. A is another icl> =, i, and this cl>

The sequence (1.1) is said to be exact if i admits a factorization

A'-:'A~A such that

is an epimorphism and k is a kernel of j.If further l is a retraction

4

SAMUEL EILENBERG AND

J.

C. MOORE

then we say that (1.1) is split exact. Coaxactness and split coexactness are defined dually. The notions of "sequence", "exact sequence", etc. are carried over to longer @agrams

A2 -

••• :-+

Al -

AO -

A_I -

A_ 2 -+

•••

(terminating or non-terminating at either end) by applying them to each consecutive pair. In particular a sequence non-terminating in both directions is called a

complex. It is easy to see that in the category

S

exactness has the usual meaning and

all exact sequences split because all epimorphisms are retractions. Further in (1.1) i is a kernel of j if and only if for every 8 € (1 the sequence

0 - (1(8, A') is an exact sequence in

(1(8, A) -

(1(8, (1")

S.

Let T: (1-. $ be a functor. We shall always assume that T(O)

=

0 (i.e., T

carries trivial objects into trivial objects). If (1 and $ are pre-additive categories

h)

and T(fI + = T(f ) + T(f } then we say that T is additive. It can be shown I 2 that if (1 and $ are additive categories and the functor T: (1-. $ preserves finite biproducts then T is an additive functor. A functor T: (1 -

T(f2} imply

fl

$ is said to be faithful if fl' f 2 : A I

-

A in (1 and TU I ) =

= f 2•

Proposition 1.1. A faithful functor T: (1-. $ reflects epimorphisms and mono-

morphisms, i.e., if f: A -. A I and T(f) is a monomorphism (epimorphism) then f itself is a monomorphism (epimorphism). Proof. Let gl' g2: AII_ A be such that fgI = fg 2• Then T(f} T(gl} = T(fg 2 ) = T(f) T(g2)' If T(f) is a monomorphism then T(gI) = T(g2} and since T is faithful, gI = gr Thus

f

is a monomorphism.

Proposition 1.2. Let T: (1- $ be a faithful and kemel preserving functor.

If the category (1 has kemels then T reflects exact sequences, i.e., given

A'!.... A.!. A"

(1.2)

such that (1.3)

is an exact sequence, then (1.2) is an exact sequence. Proof. Since T (ji)

=

T (J) T (i)

There is then a factorization

=

0 and T is faithful, it follows that ji =0.

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

5

of i such that k is a kernel of j. Then T(k) is a kernel of T(j) and since (1.3) is exact, it follows that T (1) is an epimorphism. Since T is faithful, it follows that 1 is an epimorphism and (1.2) is exact. 2. dosed and projective classes

Let (1 be a category, P an object of (1 and E a sequence in (1. Then S. Given a class & of sequences of (1,

(1 (P, E) is a sequence in the category

P

let

be the class of all objects P in (1 for which (1(P, E) is exact {or every

E € &. We then write & ~ P. Similarly, given any class P of objects in (1 let & be the class of all sequences E in (1 such that (1(P, E) is exact for every

P € P. We then write P ==? &. Clearly, for any class & we have

and for any class

P we have P=9&=9P=9 &=:;;.."

,

&~ P =9 &. Similarly, a class P of objects is called closed if P = P i.e., if P =:;;. & ~ P. A class & of sequences is called closed if & = &, i.e., if

If

P =i> &

then & is closed and if & ~

The objects of

P will

P then P is

closed.

usually be called &projective, while the sequences

in & are called P-exact. The proof of the following two propositions is elementary and is left to the reader. Proposition 2.1. If retract of P, then P' € Proposition 2.2. If a €

I

P P.

is a closed class of objects in (1, P €

P

and P' is a

P is

a closed class of objects in (1 and i er : Per -. P, is a coproduct in (1, then P € P if Ol7,d only if Po- € P for every a € I.

Let & be a closed class of sequences in a category (1 and let & ~ shall say that & is projective if the following condition holds.

P.

We

For every morphism A -. A' in (1 there exists amorphism

P -.. A with P € P and with P-.. A-..A' in &.

(2.1)

All these notions may be dualized by passing to the category (1*. Thus we shall write & * =9 ~ if &* ~ ~ in the category (1*; we shall then say that the objects of ~ are &-injective. We shall say that & is an injective class in (1 if &'" is a projective class in (1"', etc. H & is a projective class of sequences in (1 and &' is an injective class of

6

SAMUEL EILENBERG AND

sequences in

a then we say that &, and &"

in &, coincide with the complexes in say that

P

J. C.

MOORE

are complementary if the complexes

&". If&, ~ P

and

&" * ~ ~

then we also

and ~ are complementary.

It should be noted that if &, is an exact ptojective class in an abelian cate-

gory

et

then the morphisms f: A -> A' such that the sequence Ker f- A-> A' ->0

is in &, form an h.t class in the sense of Buchsbaum [2]. This establishes a bijection between all the exact projective classes and all the h.f. classes which possess enough projectives. The details are left to the reader. 3. Resolutions and derived functors Let &, be a projective class in a category

et,

and let A be an object of

et.

A left complex X over A is a complex

••• --. Xn

dn

->

Xn - l

-> ...........

Xo -.. 0 -.. •••

together with amorphism f: Xo -> A such that fd l =0. The morphism f is called the augmentation and frequently is regarded as amorphism f: X -> A of complexes where A is treated as a complex with A in degree zero and zero el~ewhere. The sequence ••• --. X

n

-> ••• ..-.

Xo-.. A-.. 0 -> •••. will be denoted by X. ~

The left complex X over A is said to be &'-acyclic if X is in

&'.

It is said

to be &'-projective if each X n is &'projective.If X is both &,projective and &,acyclic then X (taken together with its augmentation f: X-.. A) is called an

&,-projective resolution of A. Proposition 3.1. If &, is a projective class in a category

ject of

a has &'.projective resolutions.

Proof. Let A €

et.

et,

then every ob·

Applying (2.1) successively we find sequences in &,

Xo ~ A -+ 0, dl

Xl

-+

X2

->

d2

Xo -.. A, Xl

...... .....

dl -+ ,.

Xo'

...

,.

such that Xn is &,projective for n = 0, 1, •••• Combining these into a single sequence yields the desired resolution. Proposition 3.2. Let f: X -+ A be an &,-projective left complex over A and let Tf: Y -+ B be an &'-acyclic left complex over B. Then for any morphism

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

f: A--... B there exists amorphism F: X --... Y of complexes such that "IF

7 =

fi.

If

further the category (f is pre-additive then any two such morphisms F are homotopic. The standard inductive proof [4, p. 76] applies without modification and. will not be repeated. Let T: (f- 93 be an additive functor defined on a pre-additive category (!

93.

and with values in an abelian category

Let & be a projective class in (f and

let A be an object in (f with an &-projective resolution a complex in

93

f:

X_ A. Then T(X) is

and its homology depends only on A. The nth derived functor of

T relative to & is defined as (L & n 1') (A ) = H n (T (X )). Since these matters will be discussed in greater generality somewhere else, we shall not enter into this any further here. If the category (f is additive, then the usual properties of resolutions of sequences 0 -+ A' way the connecting morphism

A--... A"--... 0 in & hold and we obtain in the usual

L~ T(A

IJ )

-

L~_l T(A")

with the usual properties.

Let now (f be a pre-additive category. We shall consider (f(A, B) as a func· tor (f x (f* --... M* where M is the category of abelian groups, and M* is its dual.

If & is a projective class in (f then Ext{i;(A, B) is defined as Rn «fU, B)) where X is an &projective resolution of A.If &' is an injective class in (f then Ext{i;,(A, B) is defined as Rn «(f(A, Y)) where Y is an &'-injective resolution of B. If the classes & and &' are complementary then Ext& and Ext&, coincide. This fundamental property of Ext is established in the usual way using the fact that

g is in

&' andY is in &. One then proves that the complexes

(f (X, B) and (f (A, Y) have the same homology as a suitably defined complex (f(X, Y). 4. Categories with kernels

In this section (f will denote a category with kernels. Let & be a class of sequences. Let be the class of all morphisms such that the sequence A'A_ 0 is in &.We shall write & ~

m

m.

Let m be a class of morphisms.Given a sequence

A'!..ALA"

(4.1)

we have a factorization

A':'i~A where k is a kernel of j. Let & be the class of all those sequences (4.1) for which 1 € m.We shall write m==> &•. Proposition 4.1. If m==>& ==> P then P € P if and only if (! (P, f) lS surjective for every f € If P ==> & =9 then f € if and only if (! (P, f) is

m.

m

m

8

SAMUEL EILENBERG AND

J.

C. MOORE

m

surjective for every P € P. If &, is closed then &, =9 =9 &,. The closed class &, with &, =9 P, &, =9 is projective if and only if for every A € (( there exists amorphism f: P--... A in with P € P. The class & is exact (i.e., composed of exact sequences only) if and only if is composed of epimorphisms.

m m

m

The proof follows from the observation that

0-. (1(8,

A) --...

(((8, A) --... (((8, A")

is exact for every object 8. Therefore (((8, A') --... (((8, A) --... (((8, A") is exact if and only if (1(8, l) is surjective. Proposition 4.2. Let

mo

be the class of all retractions in ((, and let o =9 &0 =9 Po' Then &0 is the class of split exact sequences and is projective, while Po is the class of all objects of ((.

m

The proof follows trivially from the observation rhat

f

is a retraction if and

only if (((8, f) is surjective for every object 8 in (1. An object P of (( is called projective if (f(P, f) is surjective for all epimorphisms

f.

Proposition 4.3. Let

:J\.

m1 be the class of all epimorphisms and let m1

='9

Then &'1 is the class of all exact sequences while PI is the class of all projective objects in ((. The class &'1 is projective if and only if for every A € (( there exists an epimorphism f: P --... A with P projective.

&1 =»

Proof. We clearly have

m1 =~ &'1 =9 m1 and &'1

is the class of all exact

sequences. Further &'1 =9 PI and PI is the class of all projective objects. Suppose now that for every A € (( there exists an epimorphism f: P --... A with P € PI' Suppose that (4.1) is in &1' Then (( (P, 1) is surjective for every P € Choose an epimorphism f: P --... A with P € PI' Then (((P, 1): (((P, A') --... (( (P,

A)

is surjective and thus

epimorphism and (4.1) is in

&1'

f=

1m for some m: P --... A'. Therefore l is an

The remaining assertions are obvious.

A category (( in which the class called projectively perfect. Clearly

&1

PI'

&'1

of all exact sequences is projective is

is the largest possible exact projective class. In the category

S of pointed sets we have &0 = &1' since every epimorphism is a retraction. Thus S has only one exact projective class. It is easy to see that the only other projective class in S is the class &, consisting of all sequences in S. The only &projective objects are the trivial objects consisting of a single point.

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

9

5. Subcategories Let & be a projective class in a category

11

and let & =9

egory (11 of (1 will be called an &-subcategory if class for (1'. If further &' =9 (1' ('\ &-subcategory of 11.

P (in

&' =

P.

A full subcat-

(t' ('\ & is a projective

(1') then we say that

11' is

a normal

If (1' is a normal &-subcategoryof (1 and if (:X'--.A' is an &'_ pro jective resolution of A' € (1' in (1' then (: X' --. A I also is an &-projective resolution of A' in (1. Thus if (1 is a pre-additive category and (i" is a pre-additive subcategory of (1 (i.e., the inclusion functor ]: (1' --. (1 is additive), then for any additive functor T: Cl' --. ~ with values in an abelian category the derived functors L &, (T j) coincide with (L &T)]. Proposition 5.1. Let & be a projective class in a category (j, let (i" be an &-subcategory of (f and let be a subcategory of which is a normal &-subcategory of Then is a normal (Ct' ('\ &)-subcategory of <1'. Proof. We have (111 ('\ &) = f\ &, and thus is a &)-subcategory of (f'. If p" in (1" is (a" f\ &)-projective then p" is &-projective and thus also (a ' ('\ &)-projective.

a.

a" a"

(a' ('\

a"

a'

a"

(a' ('\

Proposition 5.2. Let (f' be a full subcategory of (1, let & be a projective

class in (1 and let

& =9 P.

Pc (1'

If

then (1' is a normal &-subcategory of (t.

Proof. Let &'= (1' ('\ & and let &'===;;. p' (in (1'). Then PeP'. If E' is a a sequence in &' such that (1'(P', E') is exact for every P'in pi, then it follows that E' € & and therefore E' € &'. Thus the class &' is closed (in (f'). If A'1 -. A'2 is a morphism in (1 then there exists a sequence P-. A'1 -. A IZ in & with P € P. Then P € P' and condition (2.1) holds. Thus &' is a projective class in (j". Now let P' €

P'.

There exists then in & a sequence p-Lp '-.0

with P € P. Since this sequence is also in &', it follows that a'(p ', f) is surjectivp.. Thus we have fg = 1 for some g: p' -. P. Thus P' is a retract of P and and consequently P' €

P.

Thus

P'

=

P

as required.

6. Resolvents Given a category (1, we denote by (12 the category in which the objects are morphisms f: Al -. AZ in (1 and in which amorphism (CPl' CPZ) : f-. f' is given by a commutative diagram

Al CPl

f

----+

AZ

1

1

CPZ' A'1 -....... A'z

f'

10

SAMUEL EILENBERG AND

J.

C. MOORE

We define the "range" and "domain" functors

R, D: (12

--+

(1

by setting

Then

J (f) = f

R(f)

= A2,

D(f}

=

R(c/>l' c/>2)

AI'

defines amorphism

=

c/>2'

D (c/>l' c/>2) =c/>l'

J: D --+ R.

Let ff, be a projective class in (1 and let ff, functor

==9

P.

A resolvent for ff, is a

satisfying the following conditions: (R.l) Re= D

(R.2) For any morphism (R.3) For any

f,

De (f) €

P

f: A ---> A' De(f) e(f). A L A '

is a sequence in ff,. Given an object A in (1 let 0 A denote the trivial map 0 A : A --+ O.Define eo(A)

for n

=

=

e(OA)' en(A) =e(e n _ 1 (A))

1, 2, " ' . Then the sequence e2(A)

.

el(A).

• , , ~ De (A) - - . Del (A) -------. De (A) 2 o

eO(A~

A-

0

is an ff,-projective resolution of A, which is functorial. It is called the canonical resolution of A relative to the resolvent e. Proposition 6.1. A functor e: ({2 --+ (12 is a resolvent of a projective class ff, in (1 if and only if Re = D and for any morphisms f: A --+ A I and g in (1 (1(De(g), De(f}} (1(De(g),e(f». (1(De(g},A) (1(De(g),fl (1 (De (g), A')

(6.0

is an exact sequence., The ff,-projective objects of (1 are then the retracts of objects De (g) where g ranges over all morphisms of (1, Thus ff, is unique.

Proof. If e is a resolvent for ff, then De(g) is &-projective by (R.2) and thus by (R.3), (6.1) is exact. Conversely assume that (6.1) is exact. Taking g

=

f it follows that fe (f)

=

O. Let

P be

the class of all retracts of objects

De (g) where g ranges over all morphisms in (1, and let

P =9 K

Then ff, is a

closed class of sequences and De(f) e (f). A

is in ff,. Since De (f)

IS

L

~'

&-projective, it follows that the class ff, is projective

FOUNDA nONS OF RE LAnVE HOMOLOGICAL ALGEBRA

and that e is a resolvent for

11

&.

Now let P be &'projective.Tben taking f: P-O we have the ex~t sequence

d(p, De (f})

->

d(p, P) -

0

so that P is a retract of De (f). Let

d be a category with kernels. Choosing a kernel Dk (f) k (n, A

for any morphism f: A-. A' yields a functor

k: d 2 -. (1'2 such that Rk

=

D. Moreover, for any object B in (f the sequence 0-. I1(B, Dk(f}) _11(B, A) -. (j(B, A')

is exact. Thus by 6.1, k is a resolvent for a projective class &. As f ranges overall morphisms of 11, the objects Dk(f} and their isomorphs range through the class Po of all objects of I1.Tbus by 6.1,& =9 Po and thus from 4.2 it follows that & = &0 is the class of all split exact sequences in et. We have thus Proposition 6.2. If 11 is a category with kernels then the kernel functor k: cF _ (f2 is a resolvent for the class &0 of split exact sequences. Let d be a category with kernels, and with kernel functor k. Let & be a projective class in (1 with resolvent e.Then for any f: A - A' in (1 we have fe (f)

=

0 and therefore we have a commutative diagram

De(f) ~(~ A a(f)\

I

L

A'

k(f}

Dk(f)

for a unique morphism a(f}.If a(f)

=

e(g)wbere g: Dk.(f)-O then we say that

the resolvent e factors through the kernel. For any A €

11

let 0 A : A -

0 and

write e (0 A) as F(A)

~

A.

Then F: 11- 11 is a functor and f: F -1(1 i~ a morpbis~.With this notation, if e factors through the kernel then De(f} =F(A) where A = Dk(f) and e(f} is the composition

F(ii)

~ 1 !.Y.2..... A.

We say that the resolvent e is defined by the pair (F, f), Conditions (R.2) and (R.3) yield conditions

(R. 2 ') F (A) €

P

for any A € (j

12

SAMUEL EILENBERG AND

J.

C. MOORE

(R.3 / ) F (A) f(A ), A --. 0 is in & for any A € (f. Conversely if (f has kernels and (F, i) is a pair consisting of a functor

F: (f --.. (f and amorphism f: F--.. 1(:j', satisfying (R.2 ') and (R.3 ') then the formu· lae above define a resolvent e for &. Frequently the pair (F, i) will itself be referred as a resolvent for &. Proposition 6.2. If

& is an exact projective class in a category

(j' and if

(F, i) is a resolvent, then F is faithful. Proof. Let fl'

h : A--.. A' be such that

F([I) = F(f1 ). Since fidA) = f (A ') F (f ) for i = 1, 2, it follows that f i (A) = f i (A). Since the sequence i 1 2 F (A) ~ A -----. 0 is in & it is exact and thus f (A) is an epimorphism. Thus

f1 =

fz· CHAPTER 11. THE ADJOINT THEOREM 1. Adjoint functors We shall use the symbol

(a,

13) :.s -1 T: ((!, 53)

to designate the following situation: (f and

53

are categories, T: (f--.. 53,

5: 53 --.. (! are functors, are morphisms of functors satisfying (1.1) We shall say that the functor 5 is the coadjoint of T or that T is the ad· joing of 5. We shall sometimes use the abbreviated notation (a, 13) : 5 -1 T or even just 5 -1 T. An alternative approach may be obtained as follows. Given functors

T: (f--.. 93 and 5: 93--.. (1 and given amorphism 13: 153 --.. T5 define for A in (f and B in 53 b:(f(5(B), A)--..53(B, T(A)) by setting

b(l/J)

=

T (l/J) 13 (B) for

l/J: 5(B) -. A.

(1.2)

We have

b(at/f5(r)) = T(a)b(l/J)r for a:A-. A', r:B'-.B t3(B)

=

(1.3) (1.4)

b(I S (B»)'

Conversely, given b as above satisfying (1.3), formula (1.4) defines amorphism of functors

13: 153-' T5

and (1.2) holds. Thus the study of

13

may be reduced to

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

13

that of b and vice-versa. Proposition 1.1. Given functors T: (1-.~. S:~-+(t and given amorphism

f3: 1~ -- TS there exists a morphism a: ST -+ 1(1 such that (a, (3) : S--1 T: «(1. 93) if and only if for every A in (1 and B in ~ the mapping b: (1(S(8). A) -. ~(B. T(A» defined by (1.2) is a bijection. If this is the case then a is dnique and is given by a(A)

= b- 1 (IT(A»' The mapping a:~(B, T(A» -. (1(S(8), A)

defined by a(ep) = a (A) Seep)

(1.2*)

is the inverse of b. Proof. Given a: ST -.1(1 we define a by (1.2*) and find that conditions (1.1) imply ab = 1 and ba = 1. Conversely if b is a bijection we set a =b -1 and

a(A) = a(lT(A»' Then a is a morphism of functors and (1.1) is proved by computation. The condition that b is bijective when stated explicitly reads: Given ep: B -. T(A) there is a unique t/J: S(B) -. A such that commutativity holds in the triangle

B

f3(~)

• TS(8)

~

1

T(t/J)

T(A) Stated in this way the condition asserts that the morphism f3(B): B-. TS(B) is "universal" for all morphisms B-. T(A). Proposition 1.2. If (a, f3): S~ T :((1.~) then passing to the dual catego-

ries we have (f3*. a*): T* ~ S*: (~*. (1*). It suffices to record the duals of (1.1). Proposition 1.3. Let (a, f3): S ~ T: «(1.~) Then T preserves monomorphisms, products, and kernels while S preserves epimorphisms, coproducts and co-

kernels. Proof. We shall use the notation (1.2). Let f: A-. A' be a monomorphism and let gl' g2 : B-. T(A) be such that T(f) gl == T(n g2' Since

T(f)gj for i

= 1,

= T(f}ba(gj)

== b(fa(gj»

2, it follows that f a (gl) = fa(g2)' Thus a(gl)

= a(g2)

and gl

= g2'

14

SAMUEL EILENBERG AND

J.

C. MOORE

Let Per: A -. Aer , a€ I, be a product. Thus for each B in 93 the natural mapping (1(S(B), A) -. x (1(S(B), A ) er er is a bijection. Applying a to both sides yields that the natural mapping 93(B, T(A)) -. x 93(B, T(A )) er er is a bijection and thus T(p er): T(A)-. T(A er ), a€ I,is a product.

Now let f: A' -. A be a kernel of g: A -. A". Sincegf =0, we have T(g) T(n = T(O) = O. Suppose h: B-. T(A) is such that T(g)h = O. Then b(ga(h)) = T(g) ba(h) = T(g)h = O. Consequently ga(h) = 0 and a(h) admits a factorization a(h) = fh' with h': S(E) = A'. Then h =ba(h) = b(fh') = T(f}b(h'). Since T(f} is a monomorphism this factorization of h is unique and thus T(n is a kernel of T (g). The second half of 1.3 follows by duality. Proposition 1.4. Let S

-+ T: «(1, 93) where er and 93 cire additive categories.

Then the functors T and S are additive.

Proof. By 1. 1 and 1. 2, T is a functor which preserves products. It is known that such functors are additive. Similarly for S. Proposition 1.5. Let (a, are equivalent:

13):S-4

T:«(1,

93).

Then the following conditions

(i) T is faithful. (H) T reflects epimorphisms.

(iii) a (A) : ST (A) -. A is an epimorphism for every A in (1.

Proof. (i)

~

(ii) follows from 1. 1.

(ii) ~ (Hi). Since (Ta) (f3T) = IT it follows that Ta(A) is an epimorphism. Thus by (H) a(A) is an epimorphism. (Hi) =9 (i). Let f l , h : A -. A ~ and let T (f1 ) = T (f2). Then f l a (A) = = a(A')5T(f ) = f2a(A), so that f =1 since a(A) is an epimor2 2 l

a(A')ST(fl ) phism.

2. The adjoint theorem Let (a, f3): 5 -4 T: «(1,

93).

We shall utilize the natural isomorphisms a: 93(B, T(A) ~ (1 (5(B), A)

with b = a-I, as given in § 1. Given any class ff, of sequences in by T-lff, the class of all sequences E in (1 such that T (E) € ff,.

93

we denote

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

15

Theorem 2.1. If & is a projective class in :B then T- 1 & is a projective class in (f. The T-l&-projective objects of (f are the retracts of objects S(B) where B ranges over aU the &-projective objects of :B. We begin by proving Lemma 2.2. Under the conditions above if f: A -. A' in (f and if

B -!-.. T(A) is in

&

then

~

TCA')

(2.1)

S(B)~ALA'

(2.2)

is in T-l&. Indeed we have

fa (g) =a(TC[)g) =a(O) = 0 and thus (2.1) is a sequence. We must show that the sequence

TS(B)

~

T(A)

'0!.2. T(A')

(2.3)

is in &. Let then h: P-. TCA), where P is &-projective, be such that TCn h = O. Since (2.1) is in & we have h=gk for some k:P-.B. Since g= Ta(g){3(B),it follows that h == Ta(g)k' for some 1/: P-. TS(B).Thus

:B(P, TS(B»-. :B(P, T(A» -. :B(P, T(A"» is exact and consequendy (2.3) is in &.Consequently (2.2) is in T-l& as required. Proof of 2.1. Let & ==9 P, &' = T- 1 & ==9 P'.If P € P and E' € &' then T(E') € & and :B(P, T(E'» is exact. Therefore applying the isomorphism a we find that (f(S(P), E') is exact and thus S(P) € P'.Thus every retract of S(P) also is in P'. Next, let E' be a sequence in (f such that (f(P', E') is exact for every

P' €

P'.Then (f(S(P), E') is exact for every P €

P.

Thus applying

the isomorphism b we find that :B(P, TCE'» is exact for every P € P. Consequendy T(E') € & and thus E' € &'. This shows that P' ==9 &' and thus &' is closed. For an arbitrary f: A-.A', a sequence (2.1) in & with B € the class & is projective. Then (2.2) is in &' and S(P) € (2.1) is fulfilled and the class &' is proj ective.

P'.

P exists

since

Thus condition I,

Finally let P' € P'.Then in & we have a sequence

P.!. TCP') -. 0 with P €

P.

Using this for (2.1) we obtain the sequence

S(P)~ P'~O in

&'.

Since P' €

P', (1.(P', S(P» -. (1.(P', P')

is surjective. Thus we have

16

SAMUEL EILENBERG AND

a(g) 11. = lp' for some 11.: P' is complete.

-+

J. C. MOORE

S(P). Thus P' is a retract of S(P) and the proof

Suppose now in' 2.1, that e: ~2 -+ ~2 is a resolvent for liJ. Then for every

f: A --> A' in ({ the sequence DeT(f):!J!.2. T(A)!:!l2. T(A') is in liJ and DeT(f) is &-projective. It follows from 2.2 that

SDeT(f) a(eT(f». A

L

A'

is in T-1liJ, while 2.1 implies that SDe T(f) is T-l&-projective. There~ore

a(e T (fn yields a resolvent for T-1liJ. This yields Corollary 2.3. If e is a resolvent for & in ~ then e'(f) = a(eT(f)) for f: A --> A' in ({ is a resolvent for T-1liJ in ({. Alternatively, e' (f) is the composition

SDeT(f) SeT(f). ST(A) a(A\ A If (F,

d is a resolvent pair for

liJ in ~ then (SFT, (') where /

is the com-

position

. a reso l vent palr . Jor ( T-l (!) c.

~s

Corollary 2.4. If the category ({ has kernels, T is faithful and the class liJ is exact, then the class T-1liJ is exact. If further T preserves epimorphisms and liJ is the class of all exact sequences in ~ then T-1liJ is the class of all exact sequences in ({. Indeed, by 1.3, T preserves kernels and therefore by I, 1.2, T reflects ex· act sequences. Thus T- lliJ is exact. If further T preserves epimorphisms, then since it preserves kernels it also preserves exact sequences. This implies the second statement. Proposition 2.5. Let liJ be a projective class in a category ({ and let ({' be

a subcategory of ({. If the inclusion ftmctor I: ({' -+ ({ has a coadjoint P--1I, then ({' is an liJ-subcategory of ({. Further ({' isa normal &-subcategory of ({ if and only if P(A) is liJ-projective if A is liJ-projective. This follows from 2.1 since ({'(') liJ =r1liJ. 3. The multiple adjoint theorem A family of functors S 0-: ~ 0--> ({, family of objects Ro-E ~o'

Q

€

I,

Q

E

I,

is called cointegrable if for any

the coproductEBSo-(Ro-) exists. 0-

17

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

Theorem 3.1. Let (ao-' f30-) : Sowhere the family ISo-I, u'€

I,

-+ To-: «(f, ~o-)'

a'€ I

is cointegrable. For each a€

I,

let ff,o- be a pro-

jective class in ~o- and let ff,o- =9 Po-. Then the family ff,1t = QT~Iff,o- is a projective class in (f. The ff,1t.projective objects are the retracts of coproducts

E!10- S o-(P 0-)'

Po- € Po-' a€

I.

Proof. Let ~ be the product of the categories ~o- with projections p 0-: ~ -. ~ 0-' .and let T: (f -. ~ be the functor defined by the conditions p o-T = T 0-' a€ I. Let S: ~-. (f be the coproduct EB S Po-: ~-. (f with injections 0- 0io-:So-Po--'S. Then we have (a, f3) : S -1 T:

«1,

~)

where a(i 0- T)

= a 0-

Indeed the relations (1.1) follow by an easy computation. Let ff, be the class of sequences E in ~ such that Po-(E) € &0- for all a€::£, and let P be the class of all objects P € ~ such that p o-(P) € Po- ·for all a€ I. Then ff, is a projective class in ~ and ff, =:> P.Since ff,1t = T-Iff"

the conclusion follows from 2.1We shall omit the statement of the analogue of 2.3. Corollary. If the category (f has kernels, if the family IT 0- 1, a€

I

is col-

lectively faithful (i.e., if f I , f 2 : A_ A' in (f and T 0-(f1) = T0-(f2 ) for all a€ = f ) and if each class ff,o- is exact, then the class ff,,1t is exact. 2

I

implies f I

This follows from 2.4 since T: (f-. ~ is faithful. Taking ~o-=(f, T 0-= So- = la in 3.1 we obtain Proposition 3.3. Let ff,C7' a'€::£ be a family of projective classes in (f and let ff, o- =:> Po-. Suppose that for each family lA 0- I, Ao- € <1, a€ I the coproduct EB 0- Boexists. Then the class ff, = 0 ff, is projective and the ff,-projective objects in 0- 0(f are the retracts of coproducts E!1 Po-' Po- € Po-' a€ ::£. 04. A refinement In some rather special situations it is possible to refine the conclusions of the adjoint theorem 2.1 as concerns the structure of projective objects. Theorem 4.1. Under the conditions of Theorem 2.1 assume that a functor L: (f-. ~ is given such that (I) LS is isomorphic with 1~ (II) If y: A -. A I is a morphism in (f such that L (y) is an isomorphism and

18

SAMUEL EILENBERG AND

J.

C. MOORE

if A and A I are T- 1fJ-pro jective then y is an isomorphism. Then for any object A in (1' the following properties are equivalent: (i) A is T-1fJ-projective (ii)

L (A) is fJ-projective and A ~ SL (A)

(Hi) A ~ S(B) for some fJ-projective object B in

93.

Proof. (ii) =9 (iii) is obvious and (iii) =9 (i) follows from 2.1. Thus only (i) =9 (ii) needs to be established. Let ell: LS-+ 193 be an isomorphism of functors.From the commutative diagram

LSeIl

LSLS

• LS

j~

~LS j LS

>

193

ell we deduce that eIlLS = LScIl since ell is an isomorphism. Let A be T-1fJ-projective.By 2.1, there exist morphisms

A

!-.. 5(B) J:.... A

such that B is &projective and l/J8

=

lA' Thus the composition

L(A)~ LS(B)

L(t/J) ,

L(A)

is IL(A)'Since ell (93) : LS(B) ~ B it follows that LS(B) is &projective and thus L (A) is also fJ-projective as a retract of LS(B). Let y denote the composition

SL(A) SL( 8). SLS(B)

~ SW) ~ A

Then

L (y)

=

L (l/J)LScIl (B)LSL (8)

=

L (l/J)eIl (LS(B»LSL (8)

= L(l/J)L (o)eIl(L (A» = eIl(L(A» and thus L (y) is an isomorphism. Since both A and SL (A) are T-l&projective it follows from (11) that y: SL (A) -+ A is an isomorphism as required. As a refinement of 3.1 we have Proposition 4.2. Under the conditions of 3.1 let La: (1' -+ 93 a' O'€ ~ be a family of functors such that (la) La preserves the coproducts

E:B S/B r),

T

€ It

T

(Ib) LaSa~ 1:B

a

and LaST

= 0 for T';'0'.

(11) If y: A -+ A I is a morphism in (1' such that L CT(y) is an isomorphism

19

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

for each u€~, and if A and A' are ti/"-projective then y is an isomorphism. Then for each object A in (1 the following properties are equivalent: (i) A is &,It.projective, (ii) L(J(A) is &'(J-projective for all u€~ and A~EBS(JL(J(A), (J (iii) A ~ EB S(J(B(J) for some family IB(JI, where B(J is &'(J-projective for all u€~. (j Proof. Using the notation of the proof of 3.1 we consider the functor L: (1-.53 satisfying per L == L (J for all u€ ~. From (la) and (Ib) we deduce

PaLS == L(J(~ SrPr) == EBLaSrPr ==L(JSaPa ~ r

r

P(J

and therefore LS ~ 153. Condition (11) obviously implies condition (11) of 4.1. Since SL == EBS(JPaL =EBSaL a , the conclusions follow from 4.1. a (J

5. Generators Proposition 5.1. Let (1 be a category with coproducts and, P an object of (1. Let (P) =9 &'p=?> Pp. Then &'p is a projective class in (1 and class of retracts of coproducts of copies of P.

Pp is the

Proof. Consider tbe functor T = (1(P, ): (1-. S. We construct a coadjoint S--+T as follows. For each~ € S let S(~) = P~ = EBA(J' u€ ~ where ACT=O CT if u is the base point of ~ and A a = P if u is not the base point of ~. Let i(J: A(J-> P~ be the natural injections. For every morphism cjJ: ~-> ~' in S,denote by S(cjJ) = P 4> the unique morphism P~-.P~' satisfying P 4>i(J= i4>«J) if cjJ(u) is not the base point of ~' and P4>i CT =O if cjJ(u) is the base point of ~'. This yields a functor S: S-. (1. Now define

a(A): ST(A) = P(1(P ,A) -. A, a(A)i4> = cjJ: P -. A if cjJ ,f 0, a(A)i4> = 0:0-. A 13(~):~ -. (1(P, P~) = TS(~), 13(~)u=i(J: P f3(~)u= 0:

-4

P~

P -. P~

if cjJ = O.

if u-,f 0, if u-=O.

Relations 1.1 are easily verified so that we have

(a, 13):S--1 T: «(1,

S).

Let &'0 be the class of all (split) exact sequences in S and let &'0 ==9 PO.Then by _I, 4.2 Po consists of all the objects of S. By the adjoint theorem 2.1, &,' = T-l&,O is a projective class in (1, and if &" ==9 pi then P' is the class of retracts of objects S(~), i.e., p' is the class of retracts of coproducts of P. Since P €

Pp

and

Pp

is closed under coproducts and retractions, it follows

20

SAMUEL EILENBERG AND

J. C. MOORE

Pp.On the other hand (P) C pI implies Pp =(P) C P'. Thus Pp = pI &p = &'. This completes the proof. Let k be the kernel functor in the category S. By I, 6.2,k is a resolvent the class &0' Therefore 2.3 yields a resolvent e for the projective class &p.

that

PI

C

and therefore

for Explicidy for any f: A -+ A', e (f} is the composition

SDkT(f} SkT(f). ST(A)~ A. A resolvent pair for

&p

is even easier to describe. It is simply (ST, a).

We shall say that the object P of (1 is a generator if the functor T is faithful.

= (1(P, )

Corollary 5.2. Let (1 be a category with kemels. If P is a generator, then

&p

is an exact projective class. If further P is projective then &p =&1 is the class of all exact sequences in the category (1, which is thus proje-ctively perfect.

This follows from 2.4 and the remark that P is projective if and only if (1(P, f) is surjective for any epimorphism f.Thus P is projective if and only if the functor T preserves epimorphisms. OIAPTER III. EXAMPLES 1. Groups Let § be the category of groups and group morphisms with the usual composition.The groups consisting of the Unit element alone are ttivial. -The category

§

has products (namely, the usual cartesian products) and coproducts (namely, the free products). Every morphism f: G-+ GI has a kemel i: H -+ G, where H is the subgroup

of G composed of elements g € G with f(g) phism. Also

f

=

e', and i is the inclusion mor-

has a cokernel which is the natural morphism GI -+ G'/N where N

is the least invariant subgroup of G' containing the subgroup f(G). It should be noted that an exact sequence in § is also coexact, but a coexact sequence need not be exact. The group Z of additive integers plays a special role. The functor -+ S is the functor which to each group assigns the underlying set. From 11, .§5 we know that Thas a coadjoint S -1 T, where the functor S: S-. §

T = § (Z, ): §

assigns to each pointed set I different from the base point.

the free group generated by the elements of I

The following properties of the functor T should be noted: T is faithful and preserves and reflects monomorphisms, epimorphisms, kemels and exact sequences. All these facts are obvious except the preservation of epimorphisms. This requires proving that an epimorphism J: H-. G in § is surjective, i.e., maps H

21

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

onto G. This fact has been established in [9]. We give another proof here which has the advantage of being valid also for the full subcategory of g determined by the finite groups. Let then j: H -> G be an epimorphism in g.Replacing H by itsmage in G we may assume that H is a subgroup of G and that j is the inclusion. 'We must prove that H

G. First assume that H has index 2 in G; then H is an invariant subgroup of G. Then 7Tj = OJ where 7T, 0: G-+ G/H are the factorization morphism and the trivial morphism. Thus 7T =0, a contradiction. =

Thus we may assume that the set G/H of right cosets of H in G has at least 3 elements. Let ep be a permutation of the set G/H which leaves the coset H and only the coset H fixed. Let 7T: G-+ G/ H be the natural map rrg = Hg and let.,,: G/H -> G be such that 7T'" = lC/ H' ." (If) = e (where e is the identity element of G). Every element g of G can then be written uniquely as a(g) .,,(7T(g» with a(g) € H. Define ,\ (g)

a(g)." (ep (7T(g))).We verify that ,\ is a permutation

=

of the set G. Let P be the group of all the permutations of G and consider the morphisms k, l: G-+ P defined by

k(g)x =gx for g, x € G, l(g) =,\-lk(g)'\. The condition k(g)

= l(g) is equivalent with the condition ,\k(g) = k(g)'\, i.e.,

with the condition '\(gx) =g'\(x) for all x € G.1f g € H then 7T(gX) = rr(x) and

ga (x) = a (gx) so that ,\ (gx) = g,\ (x) and k (g) = l (g). Thus kj = lj and since j l. Thus ,\ (gx) = g'\ (x) holds for all g, x € G. Taking x =e we find '\(g) =g. Thus .,,(ep7T(g» =.,,(7T(g» and ep7T(g) =7T(g). Consequendy g € H and H = G.

is an epimorphism we have k

Since the functor T

=

=

g (Z, ) is faithful and preserves epimorphisms, it fol-

lows that Z is a projective generator for the category

§.

From 11, 5.2 and 11, 5.1

we deduce that the category g is projectively perfect and the projective objects in g are the retracts of coproducts of copies of Z.Since the coproducts of copies of Z are free groups and since a subgroup of a free group is free it follows that the projective objects in for

§

§

are precisely the free groups. A resolvent pair (F, ()

is obtained by defining F (G) to be the free group generated by the non-

trivial elements of G and taking

d G) : F (G) -+ G

to be the morphism which maps

each generator of F (G) into itself. The full subcategory §f determined by the finitely generated groups is.a projectively normal subcategory of epimorphisms in

§

it follows that

§. §f

Since the epimorphisms in

§f

are also

is projectively perfect. However , it will

follow from a result in the next section that

§f

does not have a resolvent pair.

Not only is the category § not injectively perfect, but we shall show that every injective object in § is trivial. Indeed suppose G is injective and let

22

SAMUEL EILENBERG AND

J.

C. MOORE

g € G, g t 1. Let H be a simple group and H' be an infinite cyclic subgroup of H with generator h. Let f': H' -. G be the morphism defined by [' (h) =g. Since G is injective,

f'

admits an extension [: H -. G. Since [(h) =g

t

1, the kernel

of [ is a proper invariant subgroup of H and since H is simple it follows that [ is a monomorphism. Therefore card (H) ::; card (G). Thus to conclude the proof it suffices to show that simple groups containing infinite cyclic subgroups can be constructed with arbitrarily high cardinality. Let X be an infinite set and let P be the group of permutations of X.For Tr

Tr

€ P we denote the set of

fi~ed points of

by FTr and its complement in X by MTr• By a theorem of R. Baer [I] the subset

N of P defined by the condition card (M Tr ) < card (X) is a maximal invariant proper subgroup of P, i.e., H = PIN is simple. It is clear that card(H) ~card(X) and that H contains infinite cyclic subgroups. 2. Abelian groups Let G' = ZM be the category of abelian groups with the usual morphisms and composition. This is an abelian category. This category has arbitrary products and coproducts. Indeed the products are the usual direct products (sometimes called the "unrestricted" direct sums or products) and the coproducts are the usual direct sum (sometimes called the "restricted" direct sums or products). The group 2 is a projective generator for G' and therefore

n,

5.2 shows that

G' is projectively perfect and the free abelian groups are the projective objects of G'. Also from 11,

§ 5 we

obtain a resolving pair (F, f) for G'.The functor F is not

additive and indeed we shall show that G' does not have a resolvent (F,

d

with

F additive. Indeed suppose that F were additive. Let Q be the additive group of rational numbers. Then the morphism n: Q-. Q is an isomorphism for any integer n

t

O.Thus F(n): F(Q)-.F(Q) is an isomorphism. Since F is additive we

have F(n) =n so that F(Q) is divisible. Since F(Q) is also free it follows that

F (Q)

=

0, a contradiction.

It is easy to verify that the group Q1

QI2 is a cogenerator for G' i.e., a generator for the dual category G'*.Since Q1 itself is easily shown to be injec· =

tive it follows from 11, 5.2 that the category G' is injectively perfect. The injective objects of (f are the retracts of products of copies of QI.On the other hand it is well known that the injectiveabelian groups are precisely the divisible abelian groups. Again 11, §5 yields a core solvent pair (f, G) in which the functor G is however not additive. In fact a core solvent (f, G) for (:j' with G additive does not exist. Indeed, suppose that G is additive. Let 2 n = 21n2 for some n> 1Then the morphism n: 2 n -.2 n is zero and thus G (n) : G(2 n ) -. G(2 n) is zero. Since G is additive we have G (n)

= n and since G (2 ) is divisible, n n:G(2 n )-.G(2 n) is an epimorphism. Thus G(2n ) =0, a contradiction.

23

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

er determined by the finitely generated err is a projectively normal subcategory of err itself is projectively perfect. We shall show that err has no re-

Let (11 be the full subcategory of abelian groups. It is easy to see that

er,

and that solvent pair. Indeed suppose that (F, f) is a resolvent pair for (1r For any inte-

ger n ~ 1 let An denote the image of the morphism F (n) : F (Z) -. F (Z).Given another integer m> 1 we consider the natural morphism TT: Z-. Zmn.Since

TTmn = 0 we have F (TT) F (mn) =0 and therefore F (TT) (A mn ) = O.On the other hand TTn;iO and since F is faithful (by 1,6.2) we have F(TT)F(n) = F(TTn);i 0 and thus F (TT) (An) ;io. Since A mn C An it follows that F (TT) defines a non-zero morphism AniA mn -. F(Zmn)' Since F(Zmn) is a free abelian group it follows that rank (A t / A mn )

>0

and therefore rank (An)

Thus Al

=

> rank (A mn ), if m> 1.

F(Z) has infinite rank, a contradiction.

Let j" denote the full subcategory of

er

determined by the torsion groups.

Given any abelian group A let r(A) denote the torsion subgroup :>f A. Then if A is injective (i.e., divisible), it follows that r(A) also is injective. From this we deduce easily that j" is an injectively normal subcategory of (1 and is injectively perfect. Not only does j" fail to be projectively perfect, but we shall show that the only projective objects in j" are the trivial ones. Indeed suppose that A is projective in j" and A ;i integer p

> 1.

o.

Then there exists a monomorphism i: Zp -. A for some

Consider the exact sequence p

j

0-+Zp-+Q1-+Q 1• Since

Q1

is injective there exists amorphism f: A -> Q1 such that fi

= j.

Since

A is projective and p: Q1 -+ Q1 is an epimorphism there exists amorphism g: A-+ Q1 such that gp = f. Then gp = thus j = 0, a contradiction.

f

and j = {i = gpi. However pi = 0 and

Given an abelian group A and an integer n

> 1 we have the exact sequences

o -> nA -+

A

-+

nA -+0,

o -+

A

->

An -. 0

nA

->

as well as the isomorphisms

Consider the groups ~ = Z

and let

EBEBZ , n n

IT = Q1

EB xn

Zn' n = 2, 3, .••

(~) ~ &~~ P~, (IT) * ~

&fI * _~ j"II.

24

SAMUEL EILENBERG AND

6(~, E) ~ (f(Z, E)

EEl xn a(Z n ,

C. MOORE

a and

Then by IT, 5.1 5, ~ is a projective class in A sequence E is in 5, ~ if and only if

J.

a (~, E) ~

5,11 is an injective class in

a.

E) is exact. Since

EEEl xn

n

E, n", 2, 3, ...

it follows that E is in 5,~ if and only if E and each of the sequences nE (n> 1) are exact. Similarly E is in 5,II if and only if (f(E, ll) is exact. Since

aCE, ll) ~ (1(E, Ql)

EEl

x (!(E, Zn) ~ (f(E, Ql) n

EEl xn (f(E n,

Ql)

and since Q1 is an injective cogenerator for (f, it follows that E is in 5,II if and only if the sequences E and En (n> 1) are all exact. Now suppose that the sequence E is a complex, i.e., is non-terminating in both directions. From the exactness of the sequences

o -+ n E -+ E -+ nE -+

0 0

-+

nE

-+

E -+ En

-+

0

we deduce that E is in 5,~ if and only if E is in 5,Il and if and only if nE is exact for every integer n'" 1, 2, •• '. Thus the classes 5,~ and 5,11 are complementary and the complexes in 5,~ (or 5,11) are the pure exact sequences. The objects of Po!. are retracts of coproducts of copies of ~. Since ~ is a coproduct of cyclic groups every coproduct of copies of !. is again a coproduct of cyclic groups. Since a subgroup of such a group is again such a group [5, p. 46], it follows that the objects of P~ are precisely the coproducts of cyclic groups. The objects of j'II are retracts of products of copies of copies of

n

n.

A product of

has the form A EEl B where A is a product of copies of Q while B 1

is a product of cyclic groups. Since every morphism A -> B is trivial, it follows (by a general argument valid in any abelian category) that a retract of A EEl B has the form A' EB B / where A' is a retract of A and B' is a retract of B. Thus any object in &11 has the form A EB B where A is an injective (i.e., a divisible) group while B is a retract of a product of cyclic groups. Another characterization of the objects in j'II as the retracrs of abelian groups capable of carrying a compact topology was given by Los [10]. 3. Banach spaces Let K denote the field of real numbers or the field of complex numbers. We shall consider the category j3 whose objects are Banach spaces over K and whose morphisms are continuous linear transformations I: B -+ B /. Each such transformation I has a norm If 1 =' SUPlxlq II(x) I and with this norm :B(B, B') is again a Banach space. If g: B I -+ 8" then jgll S Igl Ill. This implies that j3([, C):j3(B', C)-+j3(B, C) and j3(C, f):j3(C, B)-.j3(C, B') are again morphisms and 1j3(f, C)

I Sill,

1j3(C, f}

I sI·

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

25

For each Banach space B we denote by B' the unit ball of B, i.e., the set of all points x € B with Ixl ~ 1. We denote by ~. the sub category of ~ obtained by replacing ~(B, B ' ) by ~. (B, B ' ) = ~(B, B ' )'. Thus in ~. morphisms have norm

~ 1. Isomorphisms in ~. have norm 1 and are isometries. Given a family

IB (J"J,

a €~, of Banach spaces we consider the Banach space

x B (J" whose elements are families IX(J"J, x(J" € B (J"' a € ~ with IX(J"I bounded and with Ilx(J"JI defined as sUPlx(J"l. The natural projections Pr:x B(J"-4B r are then morphisms in ~. and are a product. Indeed, given any family [(J": C-4 B (J" of morphisms in ~. and given y € C we have

I[(J"(Y) I :s I[(J"I Iyl

:s Iyl

and thus fey) =

I{(J"(Y) J is an element of x B (J". This defines f: C -4 x B (J" in ~. and p of = [(J" for a €~. Clearly f is unique. We also define the Banach space

EB B (J"

whose elements are families IX(J"J,

x(J"€ B(J"' a € ~ with ~ IX(J"I < 00. We define Ilx(J"ll = ~ IX(J"I. The natural injections i r ; Br -4 Q7 ffi Bare morphisms in ~. and are a coproduct. Indeed let (J" [(J": B (J"-4 C, a €~, be a family of morphisms in ~ .• Then for each x = Ix (J"l €

Et1 B (J"

we have ~ If(J"(x(J")

I :s ~ If(J"I

IX(J"I

:s ~ IX(J"I

< 00.

Thus [(x) = ~ f (J"(x (J") is a well defined element of C and I[(x) I :S Ixl. Thus f: Et1 B (J" -4 C is a morphism in ~. and fi (J" = [(J" for all a €~. Clearly [ is unique. Each element of morphism

Et1 B (J"

is also an element of x B (J" and there is a canonical

(): EBB(J"-4 x B(J" in ~ .. If the family ~ is finite then () is bijective and Et1 B (J" and X B (J" are identical as topological vector spaces but have different norms. Thus Et1 B (J" and x B (J" are isomorphic in the category ~ but not in the category ~ •• Let [:B 1 -+B 2 be any morphism in ~ .. Let B,=/l(O) and B"=[(B 1). The~ the inclusion B I -+ B1 is a kernel of [ and the natural projection B2-> B2 / B"

is a cokernel of [. Further [ admits a factorization in ~.

B 1 ~B/B'~ B"~B2 where

TT

is the natural projection, a is an inclusion and l is a linear isomorphism,

but not necessarily an isometry. If 1 is an isometry, then the morphism [ is called normal. It is easy to see that in ~: the notions of exactness and coexactness coincide. Consider (K)

=9

&'K =9

PK'

(K)

* =9 &,K * =9 ~K.

By 11, 5.1, &, K is a projective class in ~., &,K is an injective class in ~., PK consists of retracts of coproducts of copies of K while ~K consists of retracts

26

SAMUEL EILENBERG AND

J. C.

MOORE

of products of copies of K. A sequence

B'~ B1... B"

(3.1)

is in [hK if and only if

B'·-+ B· -+ B"· is exact in

S.

It is easy to verify that this holds if and only if i (B')

= r I (0)

and i is normal. The sequence (3.1) is in [hK if and only if ~(B", K). -+ ~(B,

K)"

-+

~(B', K).

is exact. It is an elementary exercise involving the Hahn-Banach theorem to see that this holds if and oply if HB')

=

r

1 (0) and j is normal. Thus as far as un-

limited sequences are concerned both [hK and [hK consist of exact sequences in which all morphisms are normal. Thus {f,K and [hK are complementary.

4. Rings and modules Let A be a ring and let AM be the category of left A-modules. This is an abelian category if AM (A, A') is regarded as an abelian group in the usual fashion. Given a ring morphism <1>: I

-+

A we consider the functor

T: AM

-+

IM

which assigns to each A-module A the I-module obtained from A be setting

a a = (a )a. Clearly T is faithful and exact. We also consider the functors

5: I M -+ AM, 5' :IM -+ AM defined for B and fin IM by

5(B)

=

A®IB, S(f)

5' (B) = I(A, B), 5'

=

A®If,

(n = I(A,

f)

where I( , B) stands for IM ( , B). In defining 5 we regard A as a bimodule AA I while in defining 5' we regard A as a bimodule IAk The usual isomorphisms A(A® I B, A) ~ I(B, A ® A A) = I(B, A),

A(A, I(A, B)) yield adjointness relations 5

---i

(a,

~ I(A ® A

A, B)

=

I(A, B)

T and T ---i 5'. Specifically we have

m: 5 ---i T: (AM, IM),

(fJ': a'): T ---i 5': (IM, AM)

(4.1)

(4.2)

27

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

where

a: ST {3: (3': TS'

-+

-+ 1,

1-+

a(A): A 0 I A -+ A,

A

0

a -+ Aa,

TS, {3(B):B-+A0IB, b-+ 10b,

1, (3'(B) :I(A, B)

-+

B, cp

-+

4>(1)

for cp € I(A, B),

a': 1 -+ S'T, a'(A): A -+ I(A, A), a -+ cp, cp(A) Applying il, 2.1 we find in AM a projective class &1}>

=

=

Aa.

T- 1&0 where &0 is

the class of split-exact sequences in IM. Similarly if &0 is the class of cosplitting exact sequences in IM, then &11>

=

T-l&O is an injective class. For

unlimited sequences, splitting and co-splitting exact sequences coincide and thus &1}> and &1}> are complementary classes. The &I}>-projective objects are the retracts of A 0 ! B while the &I}>-injective objects are the retracts of I(A, B) for any B € IM. 5. Coalgebras, comodules and contramodules Let K be a commutative ring. A K-coalgebra is a K-module A together with morphisms (A : A

-+

K,

cp A : A

-+

A0

such that the diagrams

A

A~A0A

l A0cp

4>1

A 0A----. A 0A 0 A d>t
1\.

A left comodule A over A is a K-module A together with a K-morphism

'P: A--. A 0

A such that the diagrams

A

,A0A

'Pj

lA0'P

A0A~A0A0A

4>®A

are commutative. Morphisms of A-comodules are defined in the usual way. There results a category AM of left comodules over A. This is an additive category with cokernels. If A is K-flat, then AM is an abelian category. We have the obvious functor T: AM

KM which to each comodule A assigns A regarded as a K-module. We also have the functor 5': KM -+ AM which to each -+

28

SAMUEL EILENBERG AND

J.

K-module B assigns the comodule A 0 B with There results an adjointness relation

C. MDDRE

ep 0

B: A 0 B -+ A 0 A

0

B.

T -i 5': (AM, KM).

t/J: A -+ A 0 A defines f3 (A) : A -+ 5' T(A). 0 B : A 0 B _ B defines a(B): T5' (B) -+ B.

Indeed, for every A € AM, the morphism For every B € KM the morphism

f

If in KM we consider the class

&0

lP

of all exact sequences (resp. the class

of all split exact sequences) then in AM we obtain the injective class

& == T-I&I

of all sequences which are exact as sequences of K-modules (resp.

the injective class

& == T-I&O

of all sequences which are split exact as sequences

of K-modules). The &-injective comodules are the retracts of comodules A 0 B with B injective K-module (resp. an arbitrary K-module). It should be noted that

& == T- 1&1 contains only coexact sequences. However, since T need not preserve monomorphisms & == T-I &1 need not contain all the coexact sequences of AM.

since T is faithful

A contramodule A over A is a K-module A together with a K-morphism

t/J: CA,A)-A

(where (A, A) stands for the K-module of K-morphisms A_A),

such that the diagrams

A

(" Alj

CA, CA, A»

~A

CA, A)

-

f

(A.ofrl

A

I

==

(A

0

CA, A)

A, A)

~ CA,

I

A)

ofr

)A

~

are commutative. There results an additive category AM# with kernels. The functor T: AM#_ KM which to each contramodule assigns the underlying K-module has a coadjoint 5 ---i T with 5 == CA, ), where for every K-module B, CA, B) is regarded as a A-contramodule using (A, CA, B»

==

CA

0

A, B)

(~,B) ---+

(A, B).

Thus the adjoint theorem yields projective classes in AM#.

If A is K-projective then the category AM# is abelian. If A is K-projective and fini;ely generated then the categories AM, AM# and AM are isomorphic where A

==

CA, K) has the structure of K-algebra induced by the coalgebra struc-

rure of A. The details are left to the reader.

CHAPTER IV. COMPLEXES IN AN ABELIAN CATEGORY 1. The category Let

cf

ccf

be an abelian category. A complex A in

of obj ects of

cf

and a family of morphisms dn : An -

cf

is a family

I An I, n €

Z,

An -I such that dndn+1 == 0

29

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

for all n € Z. Amorphism f: A --> A' of complexes is a family of morphisms

f n : An --> A'n

in (f' such that d'nf n = f n- 1 dn for all n € Z. With composition defined in the usual way we obtain an abelian category cl] of complexes over Cf. Given a complex A, the morphisms dn : An --> An -1 lead to exact sequences

such that d n

=

Un

Zn(A) ~ An - - > B n- 1 (A)

o~

B (A) ~ A ~ Z (A) ~ 0 n n n

Vn

Xn

v n -1 un' The relation dd

o~ such that wni n

W n

o~

=

B n (A)~ Zn (A)

v n' inxn

=

~

0,

(1.1)

,

(1.2)

0 then yields an exact sequence

=

~ Z'n (A)~

Bn- leA)

---+

0

un' There result exact sequences kn

in

0---. B n (A)---. Zn(A) ---.H n (A) ---. 0 in Z'n (A) ---. jn 0---. Hn (A) ~ B n - l ( A)

'(1.3)

---+

0

(1.4)

Note that a complex A is an exact sequence if and only if the homology objects H n (A) are zero for all n.

f: A--

A' is a morphism in c(l and if Bn(f) and Hn (f) are isomorphisms for all n € Z then f is an isomorphism. Proposition 1.1. If

Proof. From the conmutative diagram 0 __ B (A) n

~

1B (f) o __

B

n

Z (A) ---. H (A) n

n

n

1 Zn (f)

(A')~

Z

~

0

1H (n n

(A')~

n

H (A')_ 0 n

with exact rows, we deduce that Zn(f) is an isomorphism for all n € the commutative diagram

0 _ Zn (A) - - An ---. B n - l (A)

1Zn(f)

1f n

----+

Z.

From

0

IBn-l(f)

0----+ Zn(A')-- A'n----+ Bn_l(A')

----+

0

with exact rows, it then follows that fn is an isomorphism for all n € Z. Thus is an isomorphism.

In addition to the functors

Bn , Zn' Z~, Hn : ccf we consider also the functors

cf,

n € Z,

f

30

SAMUEL EILENBERG AND

Cn :CCf--+(1,

J. C.

MOORE

nEZ,

given by C n (A) = An' Cn (f) = fn . Next we introduce the functors

Rn : (1 -

cl1,

n E Z,

by defining Rn (D) for D € 11 to be the complex with D in position n and 0 in all other positions. We note that Z n Rn ship

=

1'1' We assert the adjointness relationU (1.5)

where w n : RnZ n --+ 1e l1 is given by the morphism w n : Zn (A) --+ An and in: 111--+ ZnRn = 111 is the identity morphism. Relations 11, (1.1) are trivially satisfied. Similarly we have

(in' x n): Z'n Finally we introduce the functors

---i

Rn: (Cf, cl1)

Qn : 11--+ by defining Qn (D) for

(1.6)

cl1

D E (1 to be the complex ID

···--+O----+D--+D--+O-··· with D in positions n and n - 1 and 0 elsewhere. We note that CnQ n assert the adjoinrness relationship

=

111' We

(1. 7) where in: 1(1- CnQ n is given by the diagram

=

1(1 is the identity morphism and an (A): QnC n (A)

--+

A

The relations 11, (1.1) are easily verified. Similarly we have the adjoint relationship (1.8) Since each of the functors Cn have both a left and a right adjoint it follows from 11, 2.2 that Cn preserves kernel and cokernels, i.e., Cn preserves exact sequences. A family A (T' a € ~ of objects in nE

Z, the objects Cn (A

cll

is called locally finite if for each

) are trivial for all but a finite number of indices a € ~. (T

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

Proposition 1.2. For any locally finite family Aa' a € there exists a biproduct ia

31

I of objects in C(1

Aa;:::::::! A.

(1.9)

Pa

Further, if T

n

denotes any of the /'{unctors Cn, Zn, Z'n, B n, Hn, Tn
Tn (A a) ,

'T n (A)

T n(p a)

is a finite biproduct (i.e., Tn(A a )

t- 0

I).

only for a finite number of indices a €

Proof. For each n € Z, the family A a,n ,a € I has only a finite number of non trivial objects and therefore we have a biproduct ia n

A o-,n~ ~ An'

( 1.10)

Pa,n

We define dn : An - > An - 1 by setting

dn -- lai a,n-l d a,n p a,n'

a€I

where the summation is only superficially infinite since only a finite number of the summands is nontrivial. Then d n-i d n

0 so that (A n, dn) defines a complex A. The morphisms in (1.10) define the morphisms in (1.9) and the verification that =

(1.9) is indeed a biproduct is straightforward. To prove the second part of 1.2, replace A a and A by the complexes A'a and A' which coincide with Aa and A for the indices n + 1, n, n - 1 but are trivial for other indices. Then (1.9) is still valid for A'a and A', and (1.9) is then essentially finite. Since Tn (A'a)

=

T n (A a ) and n T (A')n = T (A) the con-

clusion follows from the fact that Tn is additive. In the sections that follow we shall discuss various projectively perfect

classes of epimorphisms in C(1, and in various subcategories of c(1. Results for injectively perfect classes of monomorphisms can then be obtained by duality. In carrying out this dualization it is important to a priori distinguish between the categories C«(1*) and (Ca)*. An object in C«(1*) is a complex in (1 but with a differentiation of degree + 1. If we agree to write such a "co-complex" as

an

••• __ An-I ••• __

n +1 An~

An+l __ •••

and adopt the sign change rule A_n = An, d_ n = dn +1 then C (6 *) and (C(1)

*

become identified and we may write C(1 * without fear of ambiguity. 2. A general theorem Theorem 2.1. Let (f be an abelian category and ~ a full subcategory of C(1. Let M and N be subsets of Z such that

32

SAMUEL EILENBERG AND

J. C.

MOORE

(I) If A € et, m € M and N € N, then Qm(A) and Rn(A) are in (11) If IACTl is a locally finite family of objects in Cet is in :D. Then for any projective class &, in et the class &,M,N

is a projective class in

= [

:D

then their biproduct in

n &,M.N where

nm C-m 1 &,] n [ nn Z-n I&,]

:D.

Further, for any object A in (i) A is

:D

:D

:D.

:D

the following properties are equivalent:

n &,M.N -projective.

(H) B m - 1 (A) and Hn(A) are &'.projective for all m € M, n € Nand

A ~ [E!:) QmBm-l (A)] E!:)[E!:)RnHn(A]]. m

n

(Hi) There exist &'.projective objects Em ,F n € (j, m € M, n € N such that

Proof. Since the functors Q ,m € M and R , n € N have values in since

:D

m

n

:D

and

is a full subcategory of C(f, it follows that the adjointness relations

(1.7) and (1.5) imply

Qm --j Rn

Cm

--j Zn

:(:D, et), :(:D, (1),

m €M,

n € N.

Conditions (I) and (II) show that the functors Qm , Rn , m € M, n € N are cointegrable. Thus we may apply 11, 3.1. This proves that &11: = :D n &,M.N is a projective class in

:D,

aDd that the &,It·projective objects of

:D

are the retracts of

obj ects satisfying (Hi). Since for A satisfying (iii) we have B j - 1 (A) = 0 if i e M,

Hj(A) = 0 if j eN

it follows that (2.1) holds for any &,It·projective object A in

(2.1)

:D.

To prove the equivalence of (i), (ii) and (Hi) we consider the functors Bm_1::D--..d', H n :5)-et,

m€M, n€N

and show that the conditions (la), (Ib) and (11) of 11, 4.2 are satisfied. Condition (la) is satisfied because of 1.2. Condition (Ib) is satisfied because B m - 1 Qm=let, Bm-1Qj=O for i,f,m,Bm_1Rn=O, Hn R n =I(f, HnRj=O for j,f,n,HnQj=O.

To prove condition (11) let y: A -> A I be a morphism in et such that A and A I are &,It·projective and Bm - 1 (y) and Hn (y) are isomorphisms for m-I € M and

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

33

for n € N. Since A and A' satisfy (2.1) it follows that B /y) and H/y) are isomorphisms for all i € Z. Thus by 1.1, Y is an isomorphism. The equivalence of (i), (ii) and (iii) now follows from 11, 4.2 and the proof is complete.

Proposition 2.2. Let :0, M, Nand :0', M', N' be two systems satisfying the conditions of 2.1 and assume that :0 C :0', M cM', Ne N'. If for some projective class {i; in (1 we have :0 () {i;M,N C tb M', N'

then

:0

is a normal

:0' n

{i;M',N' -subcategory of

:0'.

Proof. Since {i;M',N' C &M,N it follows that

:0'

From 2.1 (iii) we deduce that every

:0 ('\

n {i;M',N' -projective object also is

{i;M,N projective. This yields the desired conclusion.

3. Projective classes h;l c(1 Let {i; be a projective class in an abelian category

a.

Taking

:0 = cll

in

2.1, we find two natural choices for the sets M and N. The first one is M = Z and N = O. This yields the projective class c{i;

= {i;2,0 =

n C~I {i;,

n € Z,

n

in c(1. The second choice is M = N = Z. This yields the projective class

C& = &2,2

nn (C-n I &

=

n Z-I&), n

n €

Z

in c(1. Comparing condition (ii) of 2.1 for these two classes we find Proposition 3.1. An object A of c(1 is C&-projective if and only if it is

C&-projective and Hn (A)

=

0 for all n € Z.

a is projectively perfect and let

Now assume that the category

&1

be the

class of all exact sequences in (1. Since the functors C n are collectively faithful, it follows from 11, 3.2 that C&1 is exact. Since further the functors Cn are exact, it follows that C&1 is the class of all the exact sequences in Ca. Thus we find Theorem 3.2. If the abelian category (1 is projectively perfect, then so is the

category C(:f. The projective objects of

crr

are biproducts

Efl Qm (Em)

for pro-

jective objects Em of (1. The class

&s

=

C&1

.

=

nn (C2&1

is of special importance. The sequences of

&s

n Z~I&I) will be called strongly exact.

34

J.

SAMUEL EILENBERG AND

C. MOORE

A sequence E is strongly exact if and only if the sequences Cn (E) and Zn (E) are exact for every n € Z. Corollary 3.3. If A in c(f is projective tlien H(A)

=

O.

Proposition 3.4. If E is a non-terminating sequence in then E is strongly exact if and only if

Ca

(i.e., if E € CC(f)

Cn (E), Zn (E), Z ~ (E), Bn (E), Hn (E) are exact for all n € A. Proof. The exact sequence

o ->

Zn (E)

->

Cn (E)

->

B n (E)

->

0

implies that Bn (E) is exact for every n € Z. The exact sequences

o---.

B n (E)

->

Zn (E)

->

Hn (E)

->

0,

0-+ Bn(E)

->

Cn(E)

->

Z~(E)

->

0

now yield the same conclusion for Hand Z'11 . n Theorem 3.5. If the abelian category

a is projectively perfect then the strong Ca.

exact sequences form a projective class fi,s in

Further, for any object A in

C(f the following properties are equivalent. (i) A is fi, s -projective, (ii) C n (A), Zn (A), Z'n (A), B n (A) and Hn (A) are proJ'ective for every n € Z. (iii) Bn (A) and Hn (A) are projective for every n € Z. Proof. Only the last part requires a proof. (i) ==> (ii). If A is fi, s -projective, then by (iii) of 2.1 we have

A ~[EBQm(Em)] EB[EBRn(F n)] with projective objects Em' F n € (f. This implies (ii). (ii) ==> (iii) is obvious. (iii) ==> (i). Since B 11 (A) and H 11 (A) are pro)' ective, the exact sequences (1.1) and (1.3) split, so that we have biproducts W

n

Z (A)= n

w' n

==

k'n

in

B n (A) .,

111

There results a biproduct

Z (A) - - H (A). n

k

n

n

35

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

Pn

Qn Bn-l (A)

qn

----+ A ----+'

Rn Hn (A)

P'n

==: A I

qn

where the morphisms Pn' p~, qn' q~ are defined using the diagrams

1

Bn - 1 (A)

1u~ •••

-----to

1 dn

An

Hn (A)

• Bn - 1 (A) W n -l

.A n- 1

1

1

i n-l

W

- - - + •••

1·'

••• - - - +

Bn -1 (A)

n

~

...

1k w'

,

£n-1 W n-l

Un

A

n k'n

n

n

H n (A)

• B n-l (A)

Note that the first of these diagrams is commutative because dn

= W

n- l i n- lu. n

Condition (ii) of 2.1 is thus fulfilled and A is (;;s-projective. Proposition 3.6. Let E:

X-

Cf

be a projectively perfect abelian category and let

A be a left complex over the object A of

projective resolution of A in

cCf

cCf.

Then

E:

X-

A is a strong

if and only if for everyone of the functors

" " Proof. Consider the complex X. By 3.4 X is strongly exact if and only if

Tn(X) is exact for all the functors T n• By 3.5. Xi is (;;s·projective if and only if Tn (Xi) is projective for every n € Z. This yields the conclusion. We thus find that the so·called "double resolutions" of a complex considered in [4. Chapter XVII] are precisely the strongly projective resolutions. 4. Subcategories of Let

Cf

be an abelian category. Given -

00

cCf $ p and q $

00

we consider the

full subcategory Cq(j p of C(f determined by the complexes A with An n < p and for n > q. We usually omit the symbol p if P = -

00

and the symbol q if q =

=

0 for

00 •

Complexes A in which the differentiation dn : An - An - 1 is zero for all n € Z, determine a full subcategory cCf of cCf. An object of cCf is thus a family

{AnI, n € Z, of nbjects in

Cf.

We also define

CZCf = cCf

n cZa.

Starting with a projective class (;; in (f and applying 2.1 to the various subcategories listed above, with suitable choices of the sets M and N, we shall

36

SAMUEL EILENBERG AND

J.

C. MOORE

obtain projective classes in these subcategories. We shall then be able to apply 2.3 to recognize which of these are normal subcategories of Ca.

cqa for p p

= -

00.

If we set

M = {n In € Z, n:S gl,

N = 0,

then conditions (I) and (11) of 2.1 are fulfilled and we obtain the projective class

cq&; Since Cq&; = cq(f n C&;, it follows from 2.3 that Cq(j is a normal e.- in cq(j. p p p p C(9-subcategory of Ca. If we take

M = N = {nln € Z, n:S g I then again 2.1 applies and we obtain a projective class cq&; in cqa. Since A

Cq&; P

=

c q(1 P

A

n C&;

PAP

it follows from 2.3 that cq(f is a normal C&subcategory of Ca. P

cZa for p finite. If we set M = {nln € Z, p < n:S g\,

N = {pI,

then 2.1 yields a projective class CZ&;. We still have CZ&; = cZa

n C&;

(because

Cp = Zp on CzcO so that Cq(j is a C&;-subcategoryof C(j. However 2.3 no longer applies since for Cq(f we have N = {pI while for Ca (in defining C&;) we p

.

took N = O. In fact it follows from 2.1 (iii) that the cq &proj ective objects of p

cza are the objects

A = Rp (E p )

EB Qp+l CE p +1 ) EB QP+2 CE p +2 ) EB···

where En' p ~ n ~ g, are &projective objects of (1'. Since Hp{A) that A is not C&projective unless Ep

=

=

Ep it follows

O. Thus unless the category (j is trivial,

CZ&; is not a normal C&;-subcategoryof C&;. However, CZ&; is a normal Cp&subcategory of cpa. If we set

M = {nln € Z, p < n ~ g\, N = {nln € Z, p ~ n ~ gl then we find again that the conditions of 2.1 are again satisfied and 2.1 yields the projective class CZ&;. Again we verify that A

A

that CZ&; is a normal C&;-subcategory of

cCf.

CPq&; =

cq(j P

n C&;

and 2.2 implies

cZ(j for all values of p and g. We set

M = 0, N = {nln € Z, p ~ n ~ gl. Applying 2.1, we obtain a projective class CZ&;. A sequence E in CZC1 is in

CZ&; if and only if CnCE) = ZnCE) is in &; for all n € Z. Thus CZ&; = C~C1 n C&; and cza is a C&subcategory of Ca, however (unless a is triv~al) it is not a normal C&subcategoryof Ca. We also find that cq&; = Cqa n C&; and 2.2 imp P plies that cqa is a normal C&;-subcategory of Ca. p A

We summarize the above results in the following

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

37

Proposition 4.1. Let &, be a projective class in an abelian category following are normal e&'-subcategories of

ca;

ez for p

= -

a.

The

00

The following are normal C&'-subcategories of

ca

eqa cq(j p' p for all values of p and q.

5. An application Let

a be an abelian category,

full subcategory of

e_1 (f

&, a projective class in

&'.

determined by the complexes in

M = Inln €

Z, n;:: 01, N

=

a and let ~ be the We set

O.

The conditions of 2.1 are then fulfilled and we obtain a proj ective class &,1t in ~.

A sequence E is in &,1t if and only if projective objects in ~ are of the form tive objects of

a.

en (E)

is in &, for every n;:: O. The fj,tt.

EBQ (E ), n n n

n;:: 0 where En are fj,·projec-

Let

f

~

X: .. ·--.X.--.X. l--.···--.Xo--.A--.O--. ...

r

I

be an object in ~ and let ..........

W:

...

7]

A

",--,Yi--,Yi-1--",,--,Y o --' X--.O--., ..

be a &,1t·projective resolution of

1

.. · - Y L.. ,I

J

7)

j

Yi,j-l

~ yJ

1

1

1

" ' - X 'I

1 0

-

1

YO,j-l

--!... B i

1

J

••• -----+

1

0

--1

Yo,o 7) 0

--+ ••• ---.

Xo I

1 0

0

1 f

~

--- 0

f i-1 Yi-1,0- B i - 1 -

1

7) j-1l

- Xj -1

1

f.

- ' " --. Yi,o

i-l,l-l - - . ••• -

1

1

There results a double complex

1 -

... - Y ·t-1 ,I· ~

.. · - Y0,1 .

X.

O

-

1 Bo

-0

1

1

A

-0

1 0

We denote the entire double complex by W, the double complex with the bottom row replaced by zeros by V, the double complex with the extreme right column replaced by zeros by U, and the double complex with the bottom row and the right column replaced by zeros by Y. We use the same letters for the single

38

SAMUEL EILENBERG AND

J.

C. MDDRE

complexes associated in the usual fashion with these double complexes. We note the following facts: Each column of U is in &.

(5.1)

Each row of V is {j;i't··projective and therefore is split exact

(5.2)

and Y . . and B; are &projective. •

~,J

We shall show that

Bis in

&

A

where B is the extreme right column of W. Indeed, let P be any &·projective object in (f. Consider the functor

T=:DCP,). We must show that HT(B) =0. Since W/B~U it suffices to show = 0 and HTCU) = O. Since each column of U is in & we have HT = 0 on each column of U. Thus by a standard filtration argument HTCU) = O. Since X is in & and each row of V is split exact, it follows that HT is Zero on each row of W. Thus again HTCW} = O. This proves (5.3). that HT(W)

Now let T:

:B.

et ----+:B

be an additive functor with values in any abelian category

We consider the double complex T CY), the usual two filtrations of this double

complex and the associated spectral sequences.

p,q = T(Y p,q } and Elp,q = HqTCYp }' Since the row Yp is split exact, we have HqTCYp } = 0 for q> 0 and HoTCYp } ~ TCB p }. Therefore the spectral sequence collapses and HT(Y}~HT(B}. First consider the horizontal filtration. We have EO A

Since B is in & and Bi is &projective, r: B -+ A is an &projective resolution of A. Thus HT(B} =L{i;Y(A}. Consequently

HT(V) ~ L& T(A}.

(5.4)

Now consider the vertical filtration. The terms EO are given by the columns of TCV). Since the pth column of V is an &·projective resolution of Xp we have

E~,q = L~ TCXp }'

Thus

(5.5) We thus obtain a spectral sequence

H p CL&n q

CX) =~ L&TCA}.

We consider two special cases. If L & TCX.}

q

=

(5.6)

0 for q> 0 and L &0 T CX.} =

{i; ~ TCX), then the spectral sequence collapses and yields HTCX) ~ L T(A} just as if X were an &·projective resolution of A.

~

~

For the second special case consider a projective class ~ in C & and let

X be an ~-projective resolution of A. Since

in & so that (5.6) applies and yields

Xis

in

et

such that

~ it also is

39

FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

L~ (L ~ n (A) ~ L &T{A).

(5.7)

This spectral sequence is the one given by Burler and Horrocks [3, p. 171]. BIBLIOGRAPHY

[1] R. Baer, Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich, Studia Math. 5(1934), 15-17. [2] D. A. Buchsbaum, A note on homology in categories, Ann. of Math. 69(1959), 66-74. [3] M. R. C. Burler and G. Horrocks, Classes of extensions and resolutions, Phi!. Trans. Royal Soc. London, Ser. A, 254(1961), 155-222. [4] H. Cartan and S. Eilenberg, Homological Algebra, Princeton 1956. [ 5] L. Fuchs, Abelian Groups, Budapest 1958. [6] D. K. Harrison, Infinite abelian groups and homological methods, Ann. of Math. 69 (1959), 366-391[7] A. HelIer, Homological algebra in abelian categories, Ann. of Math. 68 (1958), 484-525. [8] G. Hochschild, Relative homological algebra, Trans. A.M.S. 82 (1956), 246269· [9] A. G. Kurosh, A. Kh. Livsliz and E. G. Schulheifer, The foundation of the theory of categories, Uspehi Mat. Naut. XV (6), 3-58. [10] J. Los, Abelian groups that are direct summands of every abelian group which contains them as pure subgroups, Fund. Math. 44 (1957), 84-90. [ll] S. MacLane, Duality for groups, Bull. A.M.S. 56(1950), 485-516. [12] N. Yoneda, On the homology theory of modules, Jour. Fac. Sci. Univ. Tokyo, sec. 1(1954), 193-227.