Eilenberg, Moore - Foundations of Relative Homological Algebra

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FOUND TIONS OF R L TIV HOMOLOGIC L LGE R

SAMUEL EILENBERG AND J. C MOORE 1Introduction

The notion of a derived functor for a fuoctor (1 - 9 (with suitable conditions imposed on the categories j and : B and on the (unctor T is one of the keynot ions of homological algebra. Since its original definition [4] this notion hasundergone a multitude of generalizations. Some of these general izat ions went inthe direction of avoiding the use of projective resolutions, following an idea ofYoneda [12]. This will not be the point of view adopted here. Gran ting that thedefinition of derived functors is to use projective resolutions, the procedure breaksup neatly into two quite well separated steps.

The first of these steps is the definition of project ive resolutions, their existence and basic prope rtie s. The second step is the definition of the der ived functors and the s tudy of their properties. In this paper we shall no t be concerned withthe second step at all; this will be deferred to subsequent papers. The first stepwill be our main concern here.

In each category, the not ion of proj tiv objects is inherent. However it hasbeen recognized for some time that more latitude in the choice of projective objects or equivalently in the choice of exact sequences) should be permitted . ThusHochschild [8] in studying the category of modules over an algebra A consideredr-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 categoryand a distinguished class of short exact sequences.

The main notion of thi s paper is that of a projective class of sequencesin an arbitrary (pointed) category . Each such class carries with it its own projective objects. One can then talk about projective resolutions, and the categoryis additive, all the usual properties of the resolutions hold. In particular, thiswill permit the development of homological algebra in some additive categories

Received by th e editors April 23 , 1964.I)The first author was partially supported by th e Office of Naval Research and theNat iona l Sci ence Foundati on , whi le the s econd author was part ial ly supported by the Air

Force Office of Scienti fic Research during th e period while thi s research was in progress.



2 FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA

which are not abelian, e .g ., the category of comodules over a coalgebra over anarbitrary commutative ring.

With the notion of a projective class established in Chapter I, the balance ofthe 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 defini tion and some basic propert ies of projective classes. Chapter begins with areview of adjoint functors and o n ~ n u s with the adjoint theorem Chapter 11 2)which tums ou t to be the main tool for constructing projective classes. ChapterIII gives a varied assortment of examples of projective classes in various categories, a ll arrived at by the adjoint theorem. Chapter IV studies the category ce tof 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 RELATIVE HOMOLOGICAL ALGEBRA1 Review of categories

Let et be a category. We shall use the abbreviated notat ion et A, B insteadof the usual notation Homet A, for the set of morphisms f: B of objects and B in et . The dual category is denoted by et .

All categories et considered here will be assumed pointed, i.e., containingan object Ao such that et A, Ao) and et Ao A consist of single elements forevery object in [11, p. 504]. For any two such trivial objects O and there is a unique isomorphism O in et nd we shall write 0 for anyobject Ao with the property above. -For any two objects A, A in et we definethe t rivial map A A as the composition A 0 A . -The trivial map will bewritten as 0A A or again simply as O.

A typical example is the category S of sets with basepoints with morphismsbeing maps of sets preserving the basepoints. -Thus for any category et and forany objects A, A in et, et A, A is any object in the category S. This yieldsthe representation functor et x et -... S.

A family Po-: A-...Ao- a- I of morphisms in a category et is called a produc t if for any family of morphisms f : B A in et there is a unique morphismf: B A such that Po-f = fo for all a I . Dually a family io-: A0-- A, a Iis a coproduct i f for each family f : A0-- B, a I there exists a unique f: A-+ Bsuch that fi =f for all a I .

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



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRAa : there exists then a unique morphism i ~ : A A such that p i = lA v 0- 0 0 0-Prio- = 0 for r f a f th e family lio l is a coproduct then we say that

io-Ao- I A, a:Po-

3

is a biproduct.A pre-additive category is a category a s uc h th at e ac h se t a A, A ) i s

g iv en t he st ru ctu re o f a n abelian group in such a wa y that composition is distributive: gI g f=gd g21, g II 12) = gll gh for 1,11 12: A -.A an d g g lg2: A A . I t is easy to s ee t ha t in a pre-additive category an d for a finite setof indices there is no need to distinguish between products, coproducts an d biproducts. Further on e can verify that

i gA0 A , a = 1, 2PO

is a biproduct if an d only i fP Ii l = 1A r P2 i2 = A2 , i lPI i 2P2 = lA

A pre-additive category a in w hich b ip ro du cts exist for any two objects Alan d A is called additive. t can be shown that a category i n w hi ch biproductsexist for an y two objects Al and A2 can be c on ve rt ed i n at m os t o ne way into anadditive category [11, p. 511-512].,

Given morphisms

s uc h t ha t rc = lA we sa y that r is a retraction, c is a coretraction an d that Ais a retract of A.

Amorphism [ : A . A is called an epimorphismif a [, B ): a A , B - . a A ~ Bis i nj ec ti ve a s a mapping of sets) for every B in a. Monomorphisms a re d ef in eddually.

L et 1.1)

be morphisms in We say that 1.1) is a sequence if ji = O. We say that i is akernel of j i f ji = 0 an d for an y I B A s uc h t ha t j[ = 0 , [ a d m i t s a uniquefactorization 1= ig. t follows that i is a monomorphism. If i: A . A is anotherkernel of j then there exists a unique cI A A such that i l = i an d this ll is an isomorphism. Cokernels ar e defined dually.

T he s eq ue nc e 1.1) is said to be exact if i admits a factorizationA : A ~ A

s uc h t ha t is an epimorphism and k is a kernel of j f further l is a retraction



4 SAMUEL EILENBERG AND J. C. MOOREthen we say that (1.1) is split exact.

Coaxactness and split coexactness are defined dually.The not ions of sequence , exact sequence , etc. are carried over to

longer @agrams : A - Al AO - A_I A_

(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 acomplex.

It is easy to see that in the category S exactness has the usual meaning andall exact sequences spl it because all epimorphisms are retractions. Further in 1.1) i is a kernel of j if and only i f for every 8 the sequence

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

Let T 1 . be a functor. We shall always assume that T(O) = 0 (i.e., Tcarries trivial objects into trivial objects). I f and are pre-additive categoriesand T(fI + h = T(fI) + T(f2} then we say that T is additive. It can be shownthat i f and are additive categories and the functor T 1 . preservesfinite biproducts then T is an addit ive functor.

A functor T - is said to be faithful if fl f2 : A I - A in and TU I) T(f2} imply fl = f2Proposition 1.1. A faithful functor T: 1 . reflects epimorphisms and mono-

morphisms, i.e., i f f: A A I and T(f) is a monomorphism (epimorphism) then fi tself is a monomorphism (epimorphism).

Proof. Let gl g2: AII_ A be such that fgI = fg2 Then T(f} T(gl} T(fg2) = T(f) T(g2) I f T(f) is a monomorphism then T(gI) = T(g2} and since Tis faithful, gI = g r Thus f is a monomorphism.

Proposition 1.2. Let T: 1 be a faithful and kemel preserving functor.If the category has kemels then T ref lects 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) = 0 and T is faithful, it follows that ji =0.There is then a factorization



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 5of 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 followsthat 1 is an epimorphism and (1.2) is exact.

2. dosed and projective classesLet (1 be a category, P an object of (1 and E a sequence in (1. Then

(1 P E is a sequence in the category Given a class of sequences of (1,let P be the class of all objects P in (1 for which 1 P, E is exact {or everyE . We then write P. Similarly, given any class P of objects in (1 let be the class of all sequences E in (1 such tha t 1 P E is exact for everyP We then write P ==? .

Clearly, for any class wehave

and for any class P we haveP=9 =9P=9 =:;;. . ,

class of sequences is called closed if = i.e., if P= 9 .Similarly, a class P of objects is called closed i P= P i.e., i P=:;;. PP=i> then is closed and i P then P is closed.

The objects of P will usually be called projective, while the sequencesin are called P-exact.

The proof of the following two propositions is elementary and is left to thereader.

Proposition 2.1. If P is a closed class of objects in (1, P P and P is aretract of P then P

Proposition 2.2. If P is a closed class of objects in (1 and i er : er P,a I is a coproduct in (1, then P P Ol7 d only o P for every a I .

Let be aclosed class of sequences in a category (1 and let Weshall say that is projective the following condition holds.

For every morphism A A in (1 there exists amorphismP 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 weshall write = 9 if * in the category (1*; we shall then say that theobjects of are -injective. We sha ll s ay that is an injective class in (1 i' 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 J. C. MOOREsequences in a then we say that and are complementary if the complexesin coincide with the complexes in . If and then we alsosay that and are complementary.I t should be noted that if is an exact ptojective class in an abelian category et then the morphisms f: A > A such that the sequence

Ker A-> A >0is in form an h t class in the sense of Buchsbaum [2]. This establishes a bi jection between all the exact projective classes and all the h .f. classes whichpossess enough projectives. The details are left to the reader.

3. Resolutions and derived functorsLet be a projective class in a category et and let A be an objec t of et.

A left complex X over A is a complexdn . Xn > Xn - l > Xo-.. 0 -..

together with amorphism f : Xo> such that fd l =0 . The morphism f is calledthe augmentation and frequently is regarded as amorphism f : X > of complexeswhere is treated as a complex with in degree zero and zero l ~ w h r Thesequence --. X > Xo-.. 0 > will be denoted by X.nThe left complex X over A is said to be -acyclic if X is in . It is saidto be -projective i f each Xn 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. is a projective class in a category et then every object of a has .projective resolutions

Proof. Let A et. Applying (2.1) successively we find sequences in Xo 0,dlXl Xo -.. A

d2 dlX2 > Xosuch that Xn is ,projective for n = 0 1, Combining these into a singlesequence yields the desired resolution.

Proposition 3.2. Let f X A be ,-projective left complex over A andlet Tf: Y B be -acyclic left complex over B. Then for any morphism



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 7f: A B there exists amorphism F: X --... Y of complexes such that IF = fi. Iffurther the category f is pre-addit ive then any two such morphisms F are homo-topic.

The standard inductive proof [4, p. 76] applies without modification and. willno t be repeated.Let T: f be an additive functor defined on a pre-additive category

and with values in an abelian category 93 t be a projective class in f andlet A be an objec t in f with an -projective resolution f: X _ A. Then T X) i sa complex in and its homology depends only on A. The nth derived functor ofT relative to is defined as L 1 ) A ) =H T X )). Since these matters willn nbe discussed in greater generality somewhere else, we shall not enter into thisany further here. I f the category f is additive, then the usual properties of resolut ions of sequences 0 A A A --... 0 in hold and we obtain in the usualway the connecting morphism T A IJ ) ~ l T A ) with the usual properties.

Let now f be a pre-additive category. We shall consider f A B) as a functor (f x (f* --...M* where M is the category of abelian groups, and M* is its dual.I f 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.I f is an injective class in fthen Ext{i;,(A, B is defined as Rn f A, Y)) where Y is an - inject ive resolution of B. I f the classes and are complementary then Ext and Ext , coincide. This fundamental property of Ext is established in the usual way usingthe f ac t 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 complexf X Y .

4. Categories with kernels this section (f will denote a category with kernels. Let be a class of

sequences. Let mbe the class of all morphisms such that the sequence -A_ 0 is in .We shal l write

Let m be a class of morphisms.Given a sequenceA ' . .A L A

we have a factorizationA i ~ A

(4.1)

where k is a kernel o f j. Let be the class of all those sequences (4.1) forwhich 1 m.We shall write m==> .

Proposition 4.1. If m==> ==> P then P P i and only if P f) l sur-jective for every f If P mthen f mi and only i P f) is



8 SAMUEL EILENBERG AND J. C. MOOREsurjective for every P If , is closed then , m ,. The closed class, with , , mis projective and only i f for every A there existsamorphism f: P ... A in mwith P The class is exact i.e., composed ofexact sequences only and only mis composed of epimorphisms

The proof follows from the observation that0 1 8, A 8, A --... 8, A )

is exact for every object 8. Therefore(((8, A ) --... 8, A --... 8, A )

is exact i f and only i f 1 8, l) is surjective.Proposition 4.2. Let mo be the class of al l retractions in and le tmo 0 Po Then 0 is the class of split exact sequences and is projec-tive, while Po is the class of all objects of .The proof follows trivially from the observation rhat f is a retraction if and

only i f 8, f is surjective for every object 8 in 1.An object P of is called projective i f (f(P, f is surjective for all epimor-

phisms f.Proposition 4.3. Let m1 be the class of all epimorphisms and let m=1 = J\ Then 1 is the class of al l exact sequences while PI is the classof al l projective objects in . The class 1 is projective i f and only i f for every

A there exists an epimorphism f: P --... A with P projective.Proof. We clearly have m 1 m1 and 1 is the class of all exactsequences. Further 1 PI and PI is the class of all projective objects.

Suppose now that for every A there exists an epimorphism f: P --... A withP PI Suppose that 4.1 is in 1 Then P, is surjective for every P PIChoose an epimorphism f: P --... Awith P PI Then (((P, 1): (((P, A ) --... P, A is surjective and thus f = 1m for some m: P --... A . Therefore l is anepimorphism and 4.1 is in 1 The remaining assertions are obvious.

A category in which th e class 1 of all exact sequences is projective iscalled projectively perfect

Clearly 1 is th e largest poss ib le exact projective class. In the categoryS 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 o therprojective class in S is the class , consist ing of all sequences in S The onlyprojective objects are the trivial objects consisting of a single point.



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 95. Subcategories

Let be a projective class in a category and let = 9 P. A full subcategory 11 of (1 will be called an -subcategory i f '= t \ is a projectiveclass for (1 . If further = 9 (1 ( \ P (in (1 ) then we s ay that 11 is a normal-subcategory of

I f (1 is a normal -subcategoryof (1 and i f : X -- . A is an _ projective resolution of A (1 in (1 then : X -- . A I also is an -projective resolution of A in (1. Thus i f (1 is a pre-additive category and (iis a pre-addit ive subcategory of (1 (i .e. , the inclusion functor ] : (1 -- . (1is additive), then for any additive functor T: l--. with values in an abelian category th e derived functors L , Tj coincide with LT)].

Proposition 5.1. Let be a projective class in a category j , let (ibe an -subcategory of (f and let a be a subcategory of a' which is anormal -subcategory of a. Then a is a normal Ct \ )-subcategoryof



10 SAMUEL EILENBERG AND J. C. MOOREWe def ine the range and domain functors

R, D: 12 -- + 1by setting

R f) = A2 ,D f} = AI

R c/>l c >2 = c/>2D c/>l c >2 =c/>l

Then Jf = f defines amorphism J: D -- + RLet ff, be a projective class in 1 and let ==9 A resolvent for ff, is a

functor

satisfying the following conditions:(R.l) Re= D(R.2) For any morphism f De f P(R.3) For any f: A > A

De f) e f) . ALis a sequence in ff,.

Given an object A in 1 let 0 A denote the trivial map 0 A : A -- + O.Defineeo A) = e OA) en A) =e en_1 A))

for n = 1, 2, . Then the sequencee2 A el A . e O ~ , , De 2 A) Del A) -------. Deo A) A - 0

is an ff,-projective resolution of A, which is functorial. I t is called the canonicalresolution of A relative to the resolvent e.

Proposition 6.1. A functor e: {2 -- + 12 is a resolvent of a projective classff, in 1 and only 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 .0is an exact sequence., The ff,-projective objects o f 1 are t he n t he r et ra ct s of objects De g) where g ranges o ve r a ll morphisms o f 1, Thus ff, is unique.

Proof. I f e is a resolvent for ff, then De g) is -projectiveby (R.2 ) andthus by (R.3), (6.1) is exact. Conversely assume that (6.1) is exact. Takingg = f it follows that fe f = O. Let P be the class of all retracts of objectsDe g where g ranges over all morphisms in 1, and let P K Then ff, is aclosed class of sequences and

De f) e (f). A Lis in ff,. Since De f IS -projective,it follows that the class ff, is projective



FOUNDAnONS OF RELAnVE HOMOLOGICAL ALGEBRA 11

F A) A.

and that e is a resolvent for Now let P be projective.Tben taking f: P -O we have the x ~ sequence

d p, De f} ) d p, P - 0so that P is a retract of De (f).

Let d be a category with kernels. Choosing a kernelDk f k n, A

for any morphism f: A- . A yields a functork: d2 (1 2

such that Rk = Moreover, for any object B in (f the sequence0 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 rangesoverall morphisms of 11 the objects Dk f} and their isomorphs range throughthe class of all objec ts of I1.Tbus by 6.1, = 9 and thus from 4.2 it follows that = 0 is the class of al l split exact sequences in et We have thus

Proposition 6.2. I f 11 is a category with k erne ls t he n the kernel functork: cF _ (f2 is a resolvent for th e class 0 o f split exact sequences.

Let d be a category with kernels, and with kernel functor k Let be aprojective class in 1 with resolvent e.Then for any f: A - A in 1 we havefe f = 0 and therefore we have a commutative diagram

De f A L Aa f \ I k f}

Dk f)for a unique morphism a f} . I f a f = e(g)wbere g: Dk. f -O then we say thatthe resolvent e factors through the kernel. For any A 11 let 0A : A - 0 andwrite e 0A as

Then F: 11- 11 is a functor and f : F1 1 i ~ a m o r p b i s ~ i t h this notation, ife factors through the kerne l then De f} = F A ) where A = Dk f) and e f} isthe composition

F ii 1 Y 2 AWe say that the resolvent e is defined by the pai r F, f , Conditions (R.2) andR.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 i f has kernels and F, is a pair consist ing of a functor

F: f --.. f and amorphism F--.. 1 :j , satisfying R.2 ) and R .3 then the formulae above define a resolvent e for .

Frequently the pair F, will itself be referred as a resolvent for .Proposition 6.2. If is an exact projective class in a category (j and i f

F, is a resolvent, then F is faithful.Proof. Let fl h :A--.. A be such that F [I) = F f1. Since f idA) =

f A ) F fi) for i = 1 2 it follows that f1 i A) = f2 i A). Since the sequenceF A) A -----. 0 is in it is exact and thus f A) is an epimorphism. Thusf1 = fz

CHAPTER 11. THE ADJOINT THEOREM1. Adjoint functors

We shall use the symbola , 13 :.s -1 T: , 53

to designate the following situation: f and 53 are categories, T: f--.. 535: 53 --.. are functors,are morphisms of functors satisfying

1.1)We sha ll s ay 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 oreven just 5 -1 T

An alternative approach may be obtained as fol lows. Given functorsT: f--.. 93 and 5: 93--.. 1 and given amorphism 13: 153--.. T5 define for A in fand B in 53

b: f 5 B), A)--..53 B, T Aby setting

We havebl/J = T l/J 3 B) for l/J: 5 B)

b at/f5 r)) = T a)b l/J)r for a : A - . A , r : B - .Bt3 B) = b I S B)

1.2)

1.3)1.4)

Conversely, given b as above satisfying 1.3), formula 1.4) defines amorphismof functors 13: 153- T5 and 1.2) holds. Thu s the study of 3 may be reduced to



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRAthat o f b and vice-versa.

13

Proposition 1.1. Given functors T: 1 - . ~ . S : ~ - t and given amorphismf3: 1 ~ TS there exists a morphism a: ST -+ 1 1 such that a, 3) : S--1 T: 1. 9i f and only i f for every A in 1 and B in ~ the mapping

b: 1 S 8). A) ~ B . T Adefined by 1.2) is a bijection. If this is the case then a is dnique and is givenby a A) = b- IT A The mapping

a : ~ B , T A 1 S 8), A)defined by

a ep) a A) Seep 1.2*)is th e inverse of b.

Proof. Given a: ST - .1 1 we define a by 1.2*) and f ind that conditions1.1) imply ab = 1 and ba = 1. Conversely if b is a bijection we set a = b -1 anda A) = a lT A Then a is a morphism of functors and 1.1) is proved by computation.

The condition that is bijective when stated explicitly reads:Given ep: B T A) there is a unique t/J: S B) A

such that commutativity holds the triangle

B f 3 ~ TS 8) T t/J)T A)

Stated in this way the condition asserts that th e morphism f3 B): B TS B)is universal for all morphisms B T A).

Proposition 1.2. If a , f3 : T : 1 . ~ then passing to th e dual categories 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 monomorph-

isms, products, and kernels while S preserves epimorphisms, coproducts and co-kernels.

Proof. We shall use th e notation 1.2). Let f: A -. A be a monomorphismand let gl g2 : B T A) be such tha t T f) gl T n g2 Since

T f)gj = T f}ba gj) b fa gjfor i = 1, 2, it follows that fa gl) fa g2) Thus a gl) = a g2) and gl = g2



14 SAMUEL EILENBERG AND J. C. MOORELet er A .A , a I be a product. Thus for each B in the naturalmapping

1 S B), A) . x 1 S B), A is a bijection. Applying a to both sides yields that the natural mapping

93 B, T A)) . x 93 B, T A is a bijection and thus T p er): T A ) - . T A ), a I is a product.

Now le t f: A . A be a kernel of g: A .A . Sincegf =0, we haveT g) T n = T O) = O. Suppose h: B - . T A) is such tha t T g)h = O. Thenb ga h)) = T g) ba h) = T g)h = O. Consequently ga h) = 0 and a h) admits afactorization 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 nis 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 cire additive categories.

Then the functor s T and S are additive.Proof. By 1. 1 and 1.2, T is a functor which preserves products. I t is known

that such functors are additive. Similarly for S.Proposition 1.5. Let a , 13 :S-4 T: 1, 93 Then th e following conditions

are equivalent: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 fol lows from 1.1.i i Hi . Since Ta) f3T) = IT it follows that Ta A) i s an epimorphism.

Thus by H a A) is an epimorphism.Hi = i . Let f l , h A .A and let T f1 T f2). Then f l a A) =a A ) S T fl a A ) 5 T f2) f2a A), so that f l = 12 since a A) is an epimorphism.

2. The adjoint theoremLet

a f3 : 5 -4 T 1, 93We 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 we denoteby T-lff the class of all sequences E in 1 such that T E ff



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 15

(2.1)

Theorem 2.1. If is a projective class in B then T- 1 is a projectiveclass in (f. The T-l&-projective objects of are the retracts of objects S(B)where B ranges over aU the -projective objects of :B.

We begin by provingLemma 2.2. Under the conditions above i f f: A A' in and i f

B .. T(A) TCA )is in thenis in T-l&.

Indeed we have

S B ) ~ A L A

fa (g) =a(TC[)g) =a(O) = 0

(2.2)

and thus 2.1) is a sequence. We must show that the sequenceTS(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 = OSince 2.1) is in we have h=gk for some k : P - . B . Since g= Ta(g){3(B),itfollows that h Ta(g)k' for some 1/: P - . TS(B).Thus

:B(P, T S 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 thenT(E') and :B(P, T(E' exact. Therefore applying the isomorphism a wefind 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 such that (f(P', E') is exactfor every P .Then (f(S(P), E') is exact for every P Thus applyingthe isomorphism b we find that :B(P, TCE is exact for every 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 P exists sincethe class is projective. Then (2.2) is in and S(P) P . Thus condition I,2.1) is fulfilled and the class is proj ective.

Finally let P P .Then in we have a sequenceP. . TCP') 0

with P Using this for (2.1) we obtain the sequenceS P ) ~ P ~ O

in . Since P' P (1.(P , S(P (1.(P , P') is sur ject ive. Thus we have



16 SAMUEL EILENBERG AND J. C. MOOREa(g) = lp for some : P -+ S(P). Thus P is a retract of S(P) and the proofis complete.

Suppose now in 2.1, that e: ~ -+ ~ is a resolvent for liJ Then for everyf: A--> A in { the sequence

DeT(f): J .2. T(A) : l2. T(A )is in liJ and DeT(f) is -projective. I t follows from 2 .2 that

SDeT( f) a(eT( f . A Ais in T-1liJ, while 2.1 implies that SDeT(f) is T-l -projective. T h e r e ~ o r ea(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)) forf A -- > A in { is a resolvent for T-1liJ in { Alternatively, e (f) is the composition

SDeT(f) SeT(f). ST(A) a A \ AIf (F, d is a resolvent pair for liJ in ~ then (SFT, ( ) where / is the com-

position

l . l c.a reso vent palr Jor )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 andliJ is the class of all exact sequences in ~ then T-1liJ is the class of al l exactsequences in {

Indeed, by 1 .3 , T preserves kernels and therefore by I , 1.2, T ref lects exact sequences. Thus lliJ is exact. I f further T preserves epimorphisms, thensince it preserves kernels it also preserves exact sequences . This implies thesecond statement.

Proposition 2.5. Let liJ be a projective class in a category { and le t { bea subcategory of { If the inclusion ftmctor I: { -+ { has a coadjo in t P--1I,then { is an liJ-subcategory of {. Further { i sa normal -subcategory of {if and only i f P(A) is liJ-projective i f A is liJ-projective.

This follows from 2.1 since { ) liJ =r1liJ.3. The multiple adjoint theorem

A family of functors S0- : ~ 0 > { Q E I , i s called cointegrable if for a nyfamily of objects Ro E ~ o Q I , the coproductEBSo- Ro-) exists.0-



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 17Theorem 3.1. Let

ao- f30- : So- To-: f ~ o - a Iwhere the family ISo-I, u I , is cointegrable. For each a I , let ff o be a projec tive class in ~ o - and let ff o =9 Po-. Then the family ff,1t = Q ~ I f f , o - is a projec tive class in f. The ff,1t.projective objects are the retracts of coproductsE So- P 0- Po- Po- a I.0

Proof. Let ~ be the product of the categories ~ o - with projectionsp 0-: ~ ~ 0- .and let T: f ~ be the functor defined by the condi tions p o-T = T 0-a I . Let S: ~ f be the coproduct EB S Po-: ~ f with injections0 0io-:So-Po-- S. Then we have

a, f3 : S 1 T: 1, ~where

a(i T = a0 0Indeed the relations 1.1 follow by an easy computation.

Let ff be the class of sequences E in ~ such that Po- E) 0- for alla::, and let P be the class of all objects P ~ such that p o- P) Po- foral l a I . Then ff is a projective class in ~ and ff =:> P.Since ff,1t = T-Iffthe conclusion follows from 2.1-

We shall omit the statement of the analogue of 2.3.Corollary. f the category f has kernels , i f the family IT 0 1 I is col

lectively faithful (i.e., i f fI , f2 : _ A in f and T0- f1) = T0- f2 ) for all a Iimplies f I = f2) and i f each class ff o is exact, then the class ff,,1t is exact.This follows from 2.4 since T: f . ~ is faithful.Taking ~ o - = f , 0-= So- = in 3.1 we obtainProposition 3.3. Let ff,C7 a :: be a family of projective classes in f and

let ff =:> Po-. Suppose that for each family lA I, Ao-



18 SAMUEL EILENBERG AND J. C. MOORE A and A I are T - 1fJ-pro jective then y is an isomorphism.

Then for any object A in 1 the fol lowing propert ies are equivalent:i) A is T-1fJ-projectiveii) L A is fJ-projective and A ~ SL AHi A ~ S B for some fJ-projective object B in 93.Proof. ii ) iii) is obvious and iii) i) follows from 2.1. Thus only

i ii ) needs to be established.Let ell: LS-+ 193 be an isomorphism of functors.From the commutative diagram

LSeIlLSLS LS

~ j j LS > 193ell

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: Asuch that B is projective and l/J8 = lA Thus the composition

L A ~ LS B L t/J) , L A)is I L A ) S i n c e ell 93) : LS B ~ B it follows that LS B is projective andthus L A is also fJ-projective as a retract of LS B .

Let y denote the compositionSL A SL 8). SLS B SW) A

Then L y) = L l/J LScIl B LSL 8 = L l/J)eIl LS BLSL 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 projectiveit follows from 11 that y: SL A A is an isomorphism as required.

As a refinement of 3.1 we haveProposition 4.2. Under the conditions of 3.1 le t La: 1 9 a O ~ be a

family of functors such thatla ) La preserves the coproducts E:B S /B r, T TIb a a ~ 1:B and LaST = 0 for T; 0 .a11 If y: A A I is a morphism in 1 such that L CT y is an isomorphism



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 19for each u ~ , and i f 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,i i L(J(A) is (J-projective for all ~ and A ~ E B S J L J A ) ,(Jiii A ~ EB S(J(B(J) for some family IB(JI, where B(J is (J-projective for

all u ~ . Proof . Using the notation of the proof of 3.1 we consider the functor L: (1-.53

satisfying p L == L for al l ~ .er (JFrom la and (Ib) we deduce

PaLS == L J ~ SrPr) == EBLaSrPr ==L(JSaPa ~ P Jr rand 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. GeneratorsProposition 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 Pp is theclass of retracts of coproducts of copies of P.

Proof. Consider tbe functor T = (1(P, 1-. S We construct a coadjointS--+T as follows. For e a c h ~ S let S ~ ) = P ~ = EBA(J ~ where ACT=Oi f u is the base point of ~ and Aa P i f u is not the base point of ~ . L eti(J: A(J-> P ~ be the natural injections. For every morphism cjJ: ~ - > ~ in Sd e -note by S(cjJ) P 4> the unique morphism P ~ - . P ~ satisfying P4>i(J= i4>J) i fcjJ(u) is not the base point of and P i =O i f 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 . Aa(A)i4> = 0 : 0 . A

1 3 ~ ) : ~ . (1(P, P ~ ) T S ~ ) , 1 3 ~ ) u = i J : P P ~f 3 ~ ) u = 0: P . ~

Relations 1.1 are easily verified so that we have

i f cj f 0,i f cj O.

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

a, 13):S--1 T: (1, S).Let 0 be the class of all spl it exact sequences in S and let 0 ==9 PO.Thenby _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 i f ==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. MOOREthat PI C Pp.On the o ther hand P C pI implies Pp = P) C P . Thus Pp =pIand therefore p = . This completes the proof.

Let k be the kernel functor in the category By I, 6.2 k is a resolventfor th e class 0 Therefore 2.3 yields a resolvent e for the projective class p.Explicidy for any f: A A , e f} is the composition

SDkT f} SkT f . S T 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 i f the functor T = 1 P,

is faithful.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 = 1 is theclass 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 ifthe functor T preserves epimorphisms.

OIAPTER III. EXAMPLES1. 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 subgroupof G composed of elements g G with f g) = e , and i is the inclusion morphism. Also f has a cokernel which is the natural morphism GI G N where Nis the least invariant subgroup of G containing the subgroup f G). It should benoted that an exact sequence in is also coexact, but a coexact sequence needno t be exact.

The group Z of additive integers plays a special role. The functorT = Z, : 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-. assigns to each pointed set I the free group generated by the elements of Idifferent from the base point.

The following properties of th e functor T should be noted: T is fai thful andpreserves 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



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 21onto G. This fact has been established in [9]. We g iv e a no th er p ro of here whichhas th e advantage o f b ein g v alid also for th e full subcategory of g determined byth e finite groups.

L et then j H G be an epimorphism in g . R e p l a c i n g H by i t s m a g e in Gwe may assume that H is a subgroup of G and that j is th e inclusion. We mustprove that H = G. First assume that H has index 2 in G; then H is a n i nv ar ia nts ubgr oup of G. Then 7Tj = OJ where T 0 : G-+ G/H ar e t he fact ori zat ion morphisman d th e trivial morphism. T h u s T = 0 , a contradiction.

Thus we may assume t ha t t he s e t G/H o f r ig ht cosets of H in G has atleast 3 elements. L e t ep be a permutation of the set G/H which leaves th e cosetH an d only th e coset H fixed. et 7T: G G/ H be th e natural map rrg = Hg andlet . , , : G/H G be such that 7T = l C / H . If = e where e is t he i de nt it y element of G . Every element g of G can then be written uniquely as a g) .,, 7T gwith a g) H. Define ,\ g) = a g). ep 7T g .We verify that ,\ is a permutationof th e se t G. L e t P be the group of all th e permutations of G an d consider th emorphisms k, l: G P defined by

k g)x = gx for g, x G, l g) =,\-lk g) \ .Th e condition k g) = l g) i s equivalent wit h th e condition ,\k g) = k g) \, i.e.,with th e condition \ gx) =g \ x) for all x G . 1 f g H then 7T gX) = rr x) an dga x) = a gx) so that ,\ gx) = g,\ x) and k g) = l g). Thus kj = lj an d since jis an epimorphism we have k = l. Thus ,\ gx) = g \ x) holds for all g, x G.Taking x = e we find \ g) =g. Thus .,, ep7T g =.,, 7T g an d ep7T g) =7T g). Consequendy g H and H = G.

S in ce th e functor T = g Z , i s f ai th fu l an d preserves epimorphisms, it fol-lows that Z is a projective generator for th e category . From 11, 5. 2 and 11, 5. 1we deduce that th e category g is projectively perfect and the projective objectsin g ar e th e retracts of coproducts of copies of Z.Since th e coproducts of copiesof Z ar e free groups an d since a subgroup of a free group is free it follows thatth e projective objects in ar e precisely th e free groups. A resolvent pair F, for is obtained by defining F G to be th e free group generated by th e non-trivial elements of G a nd t ak in g d G : F G G to be the morphism which mapseach generator of F G into itself.

Th e full subcategory f determined by th e finitely generated groups is.aprojectively normal subcategory of . Since th e e pimorphism s in f ar e alsoepimorphisms in i t follows that f is projectively perfect. However , i t willfollow from a result in th e next section that f does n ot h av e a resolvent pair.

No t only is th e category not injectively p er fe ct , b ut we shall show thatevery injective object in is trivial. Indeed suppose G is injective an d le t



22 SAMUEL EILENBERG AND J. C. MOOREg G, g t 1. Let H be a simple group and H be an infinite cyclic subgroup ofH with generator h. Let f : H G be the morphism defined by [ h = g . SinceG is injective, f admits an extension [: H G. Since [ h) gt 1, t he kerne lof [ 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 itsuffices to show that simple groups containing infinite cyclic subgroups can beconstructed with arbit ra ri ly high cardinality. Let X be an infinite set and let Pbe the group of permutations of X.For Tr P we denote the set of f i ~ points ofTr by FTr and its complement in X by MTr By a theorem of R. Baer [I] the subsetN of P defined by the condition card M Tr < card X) i s a maximal invariant propersubgroup of i .e., H = PIN is simple. It is clear that card(H) ~ c a r d X andthat H contains infinite cyclic subgroups.

2. Abelian groupsLet G = ZM be the category of abelian groups with the usual morphisms and

composition. This is an abelian category. This category has arbitrary productsand coproducts. Indeed th e products are the usual direct products (sometimescalled the unrestricted direct sums or products) and the coproducts are theusual direct su m (sometimes cal led the restricted direct sums or products).

The group 2 is a projective generator for G and therefore n 5.2 shows thatG is projectively perfect and the free abelian groups are the projective objects ofG . Also from 11 5 we obtain a resolving pair F, f for G .The functor F is notadditive and indeed we shall show that G does not have a resolvent F, d withF additive. Indeed suppose that F were additive. Let Q be the addit ive groupof rational numbers. Then the morphism n Q -. Q is an isomorphism for any inte-ger n t O.Thus F n): F Q ) - .F Q ) is an isomorphism. Since F is additive wehave F n) =n so that F Q) i s divisible. Since F Q) is also free it follows thatF Q = 0, a contradiction.

I t is easy to verify that the group Q1 = QI2 is a cogenerator for G i .e., agenerator for the dual category G *.Since Q1 itself is easily shown to be injective it follows from 11 5.2 that the category G is injectively perfec t. The injec-tive objects of f are the retracts of p roducts o f copies of QI.On the other handit is well known that the injectiveabelian groups are precisely the divisible abel-ian groups. Again 11 5 yields a core solvent pair f, G in which the functor Gis however not additive. In fact a coresolvent (f , G for :j with G additive doesnot exist. Indeed, suppose that G is additive. Let 2 = 21n2 for some n> 1-nThen the morphism n: 2 is zero and thus Gn : G G is zero.n n n nSince G is additive we have G n = n and since G i s divisible,nn:G 2n- .G 2n is an epimorphism. Thus G 2n =0, a contradiction.



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 23Let 11 be the full subcategory of er determined by the finitely generated

abelian groups. It is easy to see that err is a projectively normal subcategory ofer and that err itself is projectively perfect. We shall show that err has no re solvent pair. Indeed suppose that F f i s a resolvent pair for 1r For any integer n 1 let An denote the image of the morphism F n : F Z F Z .Givenanother integer m> 1 we consider the natural morphism TT: Z Zmn.Since n = 0 we have F TT F mn =0 and therefore F TT A = O.On the othermnhand TTn;iO and since F is faithful by 1,6.2 we have F TT F n = F TTn ;i 0and thus F TT An ;io. Since Amn C An it follows that F TT defines a non-zeromorphism AniAmn F Zmn) Since F Zmn) is a free abelian group it followsthat rank A t/ Amn > 0 and therefore

rank An > rank Amn , i f m> 1.Thus Al = 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 i f Ais injective i.e., divisible , i t follows that r A) also is injective. From this wededuce easily that j is an injectively normal subcategory of 1 and is injectivelyperfect.

Not only does j fail to be projectively perfect, but we shall show that theonly projective objects in j are the trivial ones. Indeed suppose that A is projective in j and A ;i o Then there exists a monomorphism i: Zp A for someinteger p > 1. Consider the exact sequence

j p0 - + Z p - + Q 1 - + Q 1Since Q1 is injective there exists amorphism f: A -> Q1 such that fi = j. SinceA is projective and p: Q1 - Q1 is an epimorphism there exists amorphismg: A-+ Q1 such that gp = Then gp = f and j = {i = gpi. However pi = 0 andthus j = 0, a contradiction.

Given an abelian group A and an integer n > 1 we have the exact sequenceso nA - A - nA -+0,o - nA -> A -> A n

as well as the isomorphisms

Consider the groups

and let = Z EBEBZn, IT = Q1 x Zn n = 2, 3, .n n

~ P ~ , IT) fI j II.



24 SAMUEL EILENBERG AND J. C. MOOREThen by IT, 5.1 5, is a projective class in a and 5,11 is an injective class in

A sequence E is in 5, i f and only i f a ~ E is exact. Since6 ~ E (f(Z, x a(Z E x E, n , 2, 3, n n n nit follows that E is in 5 ~ i f and only i f E and each of the sequences nE (n> 1

are exact. Similarly E is in 5 II i f and only i f (f(E, l l is exact. SinceaCE, l l 1 E, l x E, Zn f E, l x (f(En, ln n

and since Q1 is an injective cogenerator for (f, it follows that E is in 5 II i fand only i f the sequences E and n (n> 1 are a ll exact .

Now suppose that the sequence E is a complex, i.e., is non-terminating inboth directions. From the exactness of the sequences

o E E nE 0 0 nE E E 0n nwe deduce that E is in 5 ~ i f and only i f E is in 5 Il and i f and only if nE isexact for every integer n 1 2 Thus the classes 5 ~ and 5,11 are comple-mentary 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 acoproduct of cyclic groups every coproduct of copies of . is again a coproductof cyclic groups. Since a subgroup of such a group is again such a group [5 p. 46]it follows that the objects of are precisely the coproducts of cyclic groups.The objects of j II are retracts of products of copies of n. A product ofcopies of n has the form Bwhere is a product of copies of Q1 while Bis a product of cyclic groups. Since every morphism > B is trivial, it follows(by a general argument valid in any abelian category) that a retract of A Bhas the form A B / where A is a retract of A and B is a retract of B. Thusany object in 11 has the form Bwhere 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 ret racrs of abe liangroups capable of carrying a compact topology was given by Los [10].

3. Banach spacesLet K denote the field of rea l numbers or the f ie ld of complex numbers. We

shall consider the category j3 whose objects are Banach spaces over K andwhose morphisms are continuous linear transformations I: B B /. Each suchtransformation I has a norm If 1 = SUPlxlq II x I and wi th this norm :B B, B )is again a Banach space. If g B I 8 then jgll S gl Ill This implies thatj3 [ C):j3(B , C)-+j3(B, C and j3 C f):j3(C, B)- .j3 (C , B ) are again mor-phisms and 1j3 f, C ISill, 1j3 C f} I sI



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 25For each Banach space B we denote by B the unit ball of B i.e. the set of

a ll points x B with Ixl ~ 1. We denote by ~ . the subcategory of ~ obtained byreplacing ~ 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 andwith Ilx J JI defined as sUPlx J l. The natural projections Pr:x B J -4B r are thenmorphisms in ~ . and are a product . Indeed given any family [ J : C4 B J of mor-phisms in and given y C we have I[(J (Y) I I[(J I Iyl 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 pof= [(J fora ~ Clearly f is unique.

We also def ine 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 injec-tions i ; B 4 re morphisms in ~ . and are a coproduct. Indeed letr r Q7 J[ J : BJ -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 ~ If(J I IX(J I ~ IX(J I < 00Thus [ x) = ~ f (J (x (J ) is a well defined element of C and I[ x) I :S Ixl. Thusf: Et1 BJ 4 C is a morphism in and fi J = [(J for all a ~ . Clearly [ is unique.

Each element of Et1 B is also an element of x B and there is a canonicalJ Jmorphism

): EBB J -4 x B(Jin ~ .. If the family ~ is finite then ) is bijective and Et1 B J and X B J are iden-tical as topological vector spaces but have different norms. Thus Et1 B and x BJ Jare isomorphic in the category ~ but not in the category ~

Let [:B 1 -+B2 be any morphism in ~ .. Let B,=/ l O) and B =[ B1 .T h e ~ the inclusion B I B1 is a kernel of [ and the natural projection B2>B2/ Bis a cokernel of [. Further [ admits a factorization in ~ .B1 ~ B / B ~ B ~ B 2

where TT is the natural projection a is an inclusion and l is a linear isomorphism,but no t necessarily an isometry. I f 1 is an isometry then the morphism [ iscalled normal. It is easy to see that in ~ : the notions of exactness and coexact-ness coincide.

ConsiderK = K = 9 PK K * =9 K * = 9 ~ K .

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



6 SAMUEL EILENBERG AND J. C. MOOREo f p ro du ct s of copies of K.

A sequence

is in [hK i f a nd o nl y ifB ~ B1... B

B -+ B B 3.1)

is exact in S I t is easy to verify that this holds i f an d only i f i B ) = r 0an d i is normal.

Th e sequence 3.1) is in [hK i f an d only if~ ( B , K . ~ ( B , K ~ B , K .

is exact. It is an elementary exercise involving th e Hahn-Banach theorem to seethat this holds if an d oply if H B ) r 1 0 an d j is normal. Thus as far as -limited sequences ar e concerned both hK an d [hK consist of exact sequences inwhich all morphisms ar e normal. Thus {f,K an d [hK are complementary.

4 Rings a nd m odule sLet A be a rin g and let AM be the category o f l ef t A-modules. This is an

abelian category i f AM A, A ) is regarded as an abelian group in th e usual fashion.Given a ring morphism < >: I A we consider t he f un ct or

T: AM IMwhich assigns to each A-module A th e I-module obtained from A be settinga a I (a )a. Clearly T is faithful an d exact. We also consider th e functors

5: I M AM, 5 M AMdefined for B an d f in IM by

5(B) AIB S f) A I f ,5 B = I A , B), 5 = I A , f)

where I , B stands for IM , B . In defining 5 we regard A as a bimoduleAA I while in defining 5 we regard A as a bimodule IA k The usual isomorphisms

A A I B, A I B, A A A I B, A),A(A, I A , B I A A A, B I A, B

yi el d adj oi ntness relations 5 i T an d T i 5 . Specifically we have a, m 5 i T: AM, IM), 4.1fJ : a ): T i 5 : (IM, AM 4.2



whereFOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 27

a: ST a(A):A0 I A A, A 0 a Aa,{3: 1 TS, {3(B):B-+A0IB, b-+ 1 0 b

(3 : TS 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 = Aa.

Applying il 2.1 we find in AM a projective class 1}> = T 10 where isthe class of spl it -exact sequences in IM. Similarly if is the class of cosplitting exact sequences in IM, then 11> = T-l O is an injective class. Forunlimited sequences splitting and co-splitting exact sequences coincide andthus 1}> and 1}> are complementary classes.

The I}>-projective objects are the ret racts of A B while the I}>-injective objects are the ret racts of I A B for any B IM.

5. Coalgebras comodules and contramodulesLet be a commutative ring. A K-coalgebra is a K-module A together with

morphismsA : A K, cpA : A A 0 A

such that the diagrams ~> l A0cpA 0A A A0 A

d>t< Aare commutative. Here and in th e sequel a ll tensor products are taken over 1\

A left comodule A over A is a K-module A together with a K-morphismP: A 0 such that the diagramsA A 0 A

Pj l A 0 P ~4>Aare commutative.

Morphisms of A-comodules are defined in the usual way. There results acategory AM of left comodules over A. This is an additive category with cokernels. I f A is K-flat then AM is an abelian category.

We have the obvious functor T: AM KM which to each comodule A assignsA regarded as a K-module. We also have the functor 5 : KM AM which to each



28 SAMUEL EILENBERG AND J. C. MDDREK-module B assigns the comodule A 0 B with 0 B: A 0 B A 0 A 0 B.There results an adjointness relation

T -i 5 : AM, KM).Indeed, for every A AM, the morphism t J A A 0 A defines (A) : A 5 T(A).For every B KM the morphism f 0 B : A 0 B_ B defines a B): T5 (B) B.

I f in KM we consider the class lP of a ll exact sequences (r esp. the class 0 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 a re split exact as sequencesof K-modules). The -injec tive comodules are the retracts of comodules A 0 with injective K-module (resp. an arbitrary K-module). It should be noted thatsince T is faithful == T- 1 1 contains only coexact sequences. However, sinceT need not preserve monomorphisms == T-I need no t contain al l the coexactsequences of AM.

A contramodule over A is a K-module together with a K -mo rp hi smt J CA,A)-A (where A, A) stands for the K-module of K-morphisms A_A),such tha t the diagrams

A C C A == A0 A, A) C A)( lj (A.ofrl I I rC A) - A CA, A) Af ~

are commutative. There results an addit ive category AM with kernels. The functo r T: AM KM which to each contramodule assigns the underlying K-modulehas a coadjoint 5 ---i T with 5 == C ), where for every K-module B, C B) isregarded as a A-contramodule using

~ , BA, C == C 0 A, B) (A, B).Thus the adjoint theorem yields projective classes in AM .

I f A is K-projective then the category AM is abelian. A is K-projectiveand fini;ely generated then the categories AM, AM and AM are isomorphicwhere A == C K has the s truc tu re of K-algebra induced by the coalgebra strucrure of A. The details are left to the reader.

CHAPTER IV. COMPL EXE S IN N ABELIAN CATEGORY1. The category ccf

Let cf be an abelian category. A complex A in cf is a family IAn I n Z,of obj ects of cf and a family of morphisms dn : An - An such that dndn+1 == 0



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 29for al l n Z. Amorphism f: A -- > A of complexes is a family of morphismsf :A -- > A in f such that d f = f 1d for al l n Z. With composit ion de-n n n n n n- nfined in the usual way we obtain an abelian category cl] of complexes over Cf

Given a complex A, the morphisms dn : n--> n-1 lead to exact sequencesW n Uno Z A A B 1 A 0n n n-Vn Xn ,o B A A Z A 0n n n

such that dn = vn -1 un The relat ion dd = 0 then yields an exact sequenceo ~ B A ~ Z A Z A ~ B leA) 0n n n n-

such that wnin = vn inxn = un There result exact sequencesin k n0- - - . Bn A)---. Zn A) .Hn A) --- . 0n Z jn 0 Hn A n A . Bn - l

1.1)

1.2)

1.3)

1.4)

n Z

Note that a complex A is an exact sequence i f and only i f the homology objects Hn A are zero for all n.

Proposition 1.1. If f: A is a morphism in c l and i f Bn f) and Hn f)are isomorphisms for all n Z then f is n isomorphism.Proof. From the conmutative diagram

0 _ _ B A Z A . H An n n1Bn f) 1Zn f) 1Hn o __ B A ) ~ Z A ) ~ H A )_ n n n

with exact rows we deduce that Zn f) is an isomorphism for al l n Z Fromthe commutative diagram

Zn A) An Bn- l A) ----+ 1Zn f) 1fn IBn-l f )0----+ Z n A )- - A n----+ Bn_l A ) ----+

with exact rows it then follows that fn is an isomorphism for all n Z. Thus fis an isomorphism.In addition to the functors

Bn , Zn ~ Hn : ccf cfwe consider also the functors



30 SAMUEL EILENBERG AND J. C. MOORECn :CCf--+ 1, nEZ ,

given by Cn A) = An Cn f = fn.Next we introduce th e functors

Rn : 1 cl n E Zby defining Rn D for D to be the complex with D in position n and 0 ina ll o th er positions. We note that Z R = 1 1 We assert th e adjointness relation-n n Uship

1.5)where wn : RnZn -- 1e is given by the morphism wn : Zn A) -- An an din : 111--+ ZnRn = is the identity morphism. Relations 1 .1) a re triviallysatisfied. Similarly we have

in x n): Z n ---i Rn : Cf cl1Finally we i nt ro du ce t he f un ct or s

1.6)

Qn : 11--+ clby defining Qn D for D E 1 to be th e complex

ID - - +O - - - -+D - -+D- -+O- with D in positions n an d n - 1 an d 0 elsewhere. We note that CnQn = Weassert the adjoinrness relationship

1.7)where in : 1 1- CnQn = 1 1 is th e identity morphism an d an A): QnCn A) -- Ais given by the diagram

The relations 1.1 ) are easily verified. Similarly we have t he a dj oi nt r el at io n -ship

1.8)Since each of th e functors Cn have both a left an d a r ight a djoint followsfrom 2.2 that Cn preserves kernel and cokernels, i.e., Cn preserves exact se-quences.A family A T a of objects in cll is called locally finite if for each

Z th e objects C a re t ri vi al for all but a finite number of indices a n T




1.9)i aAa;::::::: A.Pa

denotes any of the {unctors C , Z , Z , B , H ,/ n n n n nurther, Tn

Proposition 1.2. For any locally finite family Aa a I of objects in C 1there exists a biproduct

Tn



32 SAMUEL EILENBERG AND J. C. MOOREI) If A et, m M and N N, then Qm(A) and Rn(A) are in :D.11 If IACTl is a locally f inite family of objects in :D then their biproduct in

Cet is in :D.Then for any projective class , in et the class :D n M.N where M N = [ nC- 1 ,] n[nZ- I ,]m m n n

is a projective class in :D.Further, for any object A in :D the following properties are equivalent:i A is :D n M.N -projective.H Bm- 1 (A) and Hn(A) are .projective for all m M, n Nand

A [E :) QmBm-l A)] E :)[E :)RnHn A]].m nHi) There exist .projective objects E F j, m M, n N such tha tm n

Proof. Since the functors Q m M and R n N have values in :D andm nsince :D is a full subcategory of C f it follows that th e adjointness relations1.7 and 1 .5 imply

Qm --j Cm : :D, et ,Rn --j Zn : :D, 1),

m M,n N.

Conditions I and II show that the functors Q , R m M n N are coin-m nt egrable. Thus we may apply 11, 3.1. This proves that : = :D n M.N is a pro-jective class in :D, aDd that the ,Itprojective objects of :D are the retracts ofobj ects satisfying Hi).

Since for A satisfying iii we haveB j- 1 (A) = if i e M, Hj(A) = if j 2.1

it follows that 2.1 holds for any ,Itprojective object A in :D.To prove the equivalence of i) , i i and Hi) we consider the functors

Bm_1::D--. .d , Hn :5 -et mM n Nand show t ha t the conditions la , Ib) and 11) of 11, 4.2 are satisfied. Conditionl a is sati sf ied because of 1.2. Condition Ib) is satisfied because

Bm- 1Qm=let , Bm-1Qj=O for i , f ,m,Bm_1Rn=O,HnRn =I f HnRj=O for j , f ,n,HnQj=O.

To prove condition 11) le t y: A A I be a morphism in et such that A and A Iare ,Itprojective and Bm - 1 y and Hn y are isomorphisms for m -I M and




The class

for n N. Since A and A satisfy 2.1 it follows that By and Hy are isomorphisms for all i Z Thus by 1 1 Y is an isomorphism. The equivalence of i) , ii) and i i i) now follows from 11 4 2 and th e proof is complete.

Proposition 2.2. Let :0 M Nand :0 M , N be two systems satisfying theconditions of 2 1 and assume that :0 C :0 M cM , N . If for some projectiveclass {i; in 1 we have :0 ) {i;M N C M, Nthen :0 is a normal :0 n {i;M ,N -subcategory of :0 .

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

From 2 1 (iii) we deduce tha t every :0 n {i;M ,N -projective object also is:0 \ {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 Taking :0 = cll in2 1 we find two natural choices for the sets M and N. The f ir st one is M = Zand N = O. This yie lds th e projective class

c{i; = {i;2,0 = n ~ {i; n Znin c(1. The second choice is M = N = Z. This yields the projective class

C = 2,2 = n C I n Z-I ), n Zn n nin c(1. Comparing condition i i of 2 1 for these two classes we find

Proposition 3.1. An object A of c 1 is C -projective i f and only i f it isC -projective and Hn (A) = 0 for all n Z.

Now assume that the category a is projectively perfect and let be theclass of all exact sequences in 1. Since the functors C are collectively faith-nful, it follows from 11 3 2 that C 1 is exact. Since further th e functors Cn areexact, it follows that C 1 is th e class of all the exact sequences in Ca. Thuswe find

Theorem 3 2 If the abelian category 1 is projectively perfect, then so is thecategory C :f. The projective objects of crr are biproducts Efl Qm(Em) for pro-ject ive objects Em of (1.

s = C 1 = n (C2 1 n Z ~ I I. nis of special importance. The sequences of s will be called strongly exact.



34 SAMUEL EILENBERG AND J. C. MOOREA sequence E is strongly exact if a nd o nly the sequences Cn E) an d n E)ar e exact for every n Z

Corollary 3.3. If A in c f is projective tlien H A) OProposition 3.4. If E is a non-terminating sequence in a i.e., i f E CC f)

then E is strongly exact i f an d only i fCn E), Zn E) , Z E), Bn E ), Hn E)

are exact for al l n A.Proof. The exact sequence

o Z E) C E) B E) 0n n nimplies that Bn E) is exact for every n Z The exact sequenceso . B E) Z E) H E) 0n n n

0 Bn E) Cn E) Z ~ Enow yield the same conclusion for n d Z .n 11

Theorem 3.5. If the abelian category a is projectively perfect then the s tron gexact sequences form a projective class fi s in a Further, for any object A inC f the fol lowing properties are equivalent.

i) A is fi s-projective, ii) C A) , Z A) , Z A), B A) and H A) are proJ ective for every n Z.n n n n n iii) Bn A) and Hn A) are projective for every n Z.Proof. Only th e las t p a rt r e qu ire s a proof. i) ii). A is fi s-projective, then by i ii) of 2.1 we have

A ~ [ E B Q m E m ] EB[EBRn Fn]with projective objects Em Fn f. This implies ii).

ii ) iii) is obvious. iii) i) . Since B A an d H A ar e pro) ective, the exact sequences11 11 1.1) an d 1.3) split, so that we h av e b ip ro du c ts

There results a biproduct

W nZ A =n w n

inB A .,

111

k nZ A H A .n k nn



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 5PnQ B A An n l ----Pn

qnR H A An n Iq n

where the morphisms Pn qn are defined using the diagramsBn - 1 A 1 Bn - 1 A H An1 1 W n i n l 1 kn n

d n t A 1 A n n n

1 n 1 , 1k wn-1 W n l n nBn 1 A B n l A Hn A

Note that the first of these diagrams is commutative because d = W l i un n n nCondition ( ii ) of 2.1 is thus fulfi lled and A is (;;s-projective.

Proposition 3.6. Let Cf be a projectively perfect abelian category and let X - A be a left compl ex over th e object A of c f Then X - A is a strongprojective resolution of A in c f i and only i for everyone of the functors

Proof. Consider the complex X. By 3.4 X is strongly exact i f and only ifTn X) is exact for all the functors Tn By 3.5. Xi is (;;sprojective if and onlyi f Tn Xi) is projective for every n Z. This yields the conclusion.

We thus find that the socalled double resolutions of a complex consideredin [4. Chapter XVII] are precisely the strongly projective resolutions.

4. Subcategories of c fLet Cf be an abelian category. Given - p and q we consider the

full subcategory Cq j of C f determined by the complexes A with A = 0 forp nn < p and for n > qWe usually omi t the symbol p if P = - and the symbol q i f q = Complexes A in which the differentiation dn : An An - 1 is zero for alln Z determine a full subcategory c f of c f An object of c f is thus a family

{AnI, n Z, of nbjec ts in f We also define Z f= c f n c a

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



36 SAMUEL EILENBERG AND J. C. MOOREobtain projective classes in these subcategories. We shall then be able to apply2.3 to recognize which of these are normal subcategories of Ca

cqa for p = we setpM = {n In Z, n:S gl N = 0,

then conditions I) and (11) of 2.1 are fulfilled and we obtain the projective classcq ; in cq j. Since Cq; = cq f n C ;, it follows from 2.3 that Cq j is a normale. p p p pC 9-subcategory of Ca we take

M= N = {nln Z n:S g Ithen again 2.1 applies and we obtain a projective class cq ; in cqa SinceA A Cq; = c q n C ; i t follows from 2.3 that cq f is a normal C subcategory of Ca.P P PcZa for p finite. we set

M= {nln Z, p < n:S g\, N = {pIthen 2.1 yields a projective class CZ ;. We still have CZ ; = cZa n C ; becauseCp = Zp on CzcO so that Cq j is a C ;-subcategoryof C j. However 2.3 nolonger appli es s ince for Cq f we have N = {pI while for Ca n defining C ; weptook N = O. In fact it follows from 2.1 iii) that the cqproj ective objects ofpcza are th e objects

A = Rp Ep p l CE p 1 QP 2CEp 2EBwhere En p n g, are projective objects of (1 . Since Hp{A) = Ep it followsthat A is not C projective unless Ep = O. Thus unless the category j is trivialCZ ; is no t a normal C ;-subcategoryof C ;. However, CZ ; is a normal Cp subcategory of cpa

we setM= {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 yieldsthe projective class CZ ;. Again we verify that Cq; = cq j n C ; and 2.2 implies

A A P Pthat CZ ; is a normal C ;-subcategory of c fcZ 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 CZC is inCZ ; if and only if CnCE = ZnCE is in ; for al l n Z. Thus CZ ; = C ~ C n C ;and cza is a C subcategory of Ca, however unless a is t r i v ~ a l i t is not anormal C subcategoryof Ca. We also find that cq ; = Cqa n C ; and 2.2 im-A p Pplies that cqa is a normal C ;-subcategory of Cap

We summarize the above results in th e following



FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA 37Proposition 4.1. Let be a projective class in an abelian category a. The

following are normal e -subcategories of ca;ez for p = 00

The following are normal C -subcategories of caeqa cq jp p

for all values of p and q.5 An application

Let a be an abe li an category a projective class in a and let be thefull subcategory of e_1 determined by the complexes in . We set

M = Inln Z n;:: 01 N = OThe conditions of 2.1 are then fulfilled and we obtain a proj ective class 1t in A sequence E is in 1t i and only if en E is in for every n;:: O The fj,tt.projective objects in are of the form EBQ (E ), n;:: 0 where En are fj,projec-n n ntive objects of a.

Let fX . . - - . X . - - . X . l - - . - - . Xo - - . A - - .O - - . . . .I rbe an object in and le t

1. . -Y . .LJ 1 1

be a.......... 7] AW , - - , Y i - - , Y i - 1 - - , , - - ,Y o - - X- - .O - - . , . . ,1tprojectiveresolution of There results a double complex1 1 f. 1- Yi,j-l - Yi,o -- ... Bi

y J 1 f i 1 J i l l l . - Yi 1 0 B i - 1 - 01 1 11 1 f O 1. . -Y . - YO j-l ----- Yo,o Bo 0,1 7) j1 7) j 1 l 7)0 1 11

-X -X j -1 . Xo A I1 1 I 10 0 0 0We denote the entire double complex by W the double complex with the bottomrow replaced by zeros by V the double complex with the extreme right columnreplaced by zeros by U and the double complex with the bottom row and theright column replaced by zeros by Y We use the same letters for the single



38 SAMUEL EILENBERG AND J C. MDDREcomplexes associated in the usual fashion with these double complexes.

We note the following facts:Each column of U is in .

Each row of V is {j;i tprojective and therefore is spli t exactand Y . and are projective.~ J

We shall show that Bis in

(5.1)(5.2)

Awhere B is the extreme right column of Indeed, let P be any projective object in f Consider the functor

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

Now let T et :B be an additive functor with values in any abelian categoryB We consider the double complex TCY), the usual two fi lt rat ions of this doublecomplex and the associated spectral sequences.

First consider the horizontal filtration. We have EO = T(Y } and El =A p q p q p qHqTCYp } Since the row Yp is split exact, we have HqTCYp } = for q andHoTCYp} TCBp} Therefore the spectral sequence collapses and H T Y } ~ H T B } .

Since B is in and Bi is projective, B A is an projective resolutionof A. Thus HT(B} =L{i;Y A}. Consequently

HT(V) L T(A}. 5.4)Now consider the vertical f il trat ion. The terms EO are given by the columns

of TCV . Since the pth column of V is an projec tive resolution of Xp we haveE ~ q = L ~ TCXp} Thus (5.5)We thus obtain a spectral sequence

H CL n X L TCA}. (5.6)p qWe consider two special cases. L TCX.} = 0 for q 0 and L TCX.} =q {iTCX), then the spectral sequence collapses and yields HTCX ~ L T(A} just as

if X were an projective resolut ion of A.For the second special case consider a projective class ~ in et such that

C and let X be an ~ p r o j e c t i v e resolution of A. Since Xis in it also isin so that 5.6) applies and yields




L A L T{A). 5.7This spectral sequence is the one given by Burler and Horrocks [3, p. 171].

BIBLIOGRAPHY[1] R. Baer, Die Kompositionsreihe der Gruppe aller eineindeutigen bbildungen

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66-74.[3] M R. C. Burler and G Horrocks, lassesof extensions and resolutions

Phi . Trans . Royal Soc. London, Ser. A 254 1961 , 155-222.[4] H. Cartan and S. Eilenberg, Homological lgebra Princeton 1956.[ 5] L. Fuchs, belian 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 , 246

269[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, belian groups that are direct summandsof every abelian group

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Eilenberg, Moore - Foundations of Relative Homological Algebra