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NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matem6tica editor: Leopoldo Nachbin Cohomology of Completions SAUL LUBKIN NORTH-HOLLAND 42
Transcript
Cohomology of Completions (North-Holland Mathematics Studies, 42) Cohomology of Completions
NORTH-HOLLAND MATHEMATICS STUDIES
Universidade Federal do Rio de Janeiro and University of Rochester
Cohomology of Completions
SAUL LUBKIN
Department of Mathematics University of Rochester Rochester, N. Y. 14620, U.S.A.
1980
42
Il North-Holland Publishing Company. 1980
All rights reserved. No part a/this publication may be reproduced. stored in a retrieval system, or transmilled. in any form or by any means. electronic. mechanical. photocopying. recording
or otherwise. without the prior permission a/the copyright owner.
ISBN: 0444860428
Sale distributors/or the U.S.A. and Canada
ELSEVIER NORTH-HOLLAND. INC. 52 VANDERBILT AVENUE.
NEW YORK, N.Y. 10017
Lublrln. Saul., 1939- Cohomology of completions.
(Notas de matem&tica ; 71) (Noith-Holland mathe­ matics studies ; 42)
Bibliography: p. 1. Complexes. Cochain. 2. Modul.es (Algebra)
3. Spectral sequences (Mathematics) I. Title. II. Series. QAl.N86 no. 71 cQAl69J 5l0s C5l2'.55J 80-19364 ISBN 0-444-86042-8
PRINTED IN THE NETHERLANDS
To Laure, my wife
PREFACE
In this book, we study, primarily, the cohomology of the
t-adic completion of cochain complexes of left A-modules, where
A is a ring and tEA.
To this end, in the Introduction, we develop the necessary
basic homological techniques that are used throughout this book.
The text is also filled with many examples and applications to,
e.g., algebraic geometry and algebraic topology.
For example, using the deep, basic finiteness theorem,
Chapter 5, Theorem 1, we prove the finite generation of lifted
p-adic cohomology "using the /\" of a scheme proper over a
Noetherian O-algebra A, see [P.A.C.]. This connects up with
research done by the author over the past seventeen years on
the Weil zeta function stemming from the original Weil Conjec­
tures, see e.g., [W.C.], [C.A.W.], [P.P.W.C.], [B.W.V.], [P.C.T.],
[F.G.P.R.] and [P.A.C.].
Also, the general theory of Poincare duality is studied
somewhat in Chapter 7, and this is used, in algebraic geometry,
to prove Poincare duality for the lifted p-adic cohomology of
complete non-singular algebraic varieties, see [P.A.C.]; as well
as applications to the more traditional case, in algebraic
topology, of the usual singular cohomology of compact, oriented
topological manifolds, see the Examples of Chapter 7.
And, in Chapter 8, we easily retrieve, as another applica­
tion to the field of algebraic geometry, most of the well-known
theorems on the finite generation of the cohomology of formal
schemes proper over an affine, for finitely generated ideals.
The book is written in such a way that a graduate student,
vii
with very little other than basic background, and having only
the most rudimentary knowledge of homology, should be able to
profit from various parts of it. And, in part towards this
end, there is an extensive Introduction.
The first chapter of the Introduction covers the general
theory of abelian categories and may be skipped by those readers
who are not interested in that subject. The second chapter of
the Introduction contains what is perhaps the most thorough
study ever made of spectral sequences. Yet, it is so written
that is should be accessible to a beginner in the subject.
However, many researchers, even those who are very advanced in
the subject, who have to study or construct new spectral se­
quences, at any level, may be able to profit from the general
study made.
The greater part of this book is written at the level of
left modules over a ring, rather than for the more general
abelian categories, so as to make the material more accessible
to the beginner. However, for the readers interested in the
greater generality, the first chapter of the Introduction
supplies a thorough initiation to the general theory of abelian
categories, including how to work in abelian categories.
The chapter on spectral sequences, Chapter 2 of the Intro­
duction, is written at the abstract level of abelian categories­
however, consistent with the general philosophy, it has been
written in such a fashion that the beginner can mentally substi­
tute the words "abelian group" for "object", and "the category
A of abelian groups" for "an abelian category A", throughout
that chapter, if his only interest is that concrete case.
Preface ix
background material on abelian categories, and more, if desired.)
Description of some selected results; applications.
In the Introduction, Chapter 1, the no~ion of abelian
category is thoroughly developed, and it is shown how to prove
theorems in an abstract abelian category. In section 2, the
familiar notions of subobject and quotient-object are studied,
and in section 3 abelian categories are introduced formally.
Then, the previously published paper, "Imbedding of Abelian
Categories", ([I.A.C.)) is reproduced, as section 4 of the
chapter. The reader learns, in this section, how to work, in
many ways, as freely in a general abstract abelian category,
as in the category of left modules over a ring. Also, in sec­
tion 5 of this chapter, the notion of a sUbquotient of an ob­
ject in an abelian category (or of an abelian group) is defined
formally and developed (for the first time to the best of the
author's knowledge). It is believed that this will facilitate
the understanding of the definition of the "Eoo-term" of a
spectral sequence, as in the next chapter. (Another applica­
tion to algebraic topology, of this material (not discussed in
the text) is a better understanding of the higher order coho­
mology operations, as in the Steenrod algebra.)
In Chapter 2 of the Introduction, what is probably the most
extensive study to appear so far, is made of the theory of
spectral sequences. However, this material is suitable to be
read by a graduate student who has not had too extensive a
background. To develop the theory properly, the general notion
x Preface
of graded abelian category, is introduced in section 3. Fil-
tered objects, and their completions and co-completions, are
studied in section 6. (E.g., in section 6 it is shown that,
whenever the completion of the co-completion, and the co-comple-
tion of the completion, of a filtered object both make sense,
then they are canonically isomorphic.) And the abutment, the
partial abutments, see section 7, the Eoo-term, etc., are all
studied for the spectral sequence of an exact couple (section
7), the spectral sequence of a filtered cochain complex (sec-
tion B)and the spectral sequence of a double complex (section
10) •
In sections 2 and 5, the spectral sequence of an exact
couple is studied. Among other things, one obtains a short
exact sequence, the middle term of which is Eoo' and the first
[
of V)
In section 7, the two partial abutments of the spectral
sequence of a conventional, bigraded exact couple are defined
and studied. These are the direct limit abutments 'Hn, nE 7. ,
anc the inverse limit abutments "Hn, n E 7.. These are two sets
of "would-be" abutments, and are filtered objects. The associ-
ated gradeds of each of these sets is a different subquotient
of The delicate question of whether or not these par-
tial abutments are perfect is studied (e.g., in section 7, and
in section 9, Theorem 4 and Corollary 4.1). It is equivalent
to ask whether or not the left and right defects are zero
Preface xi
(section 7).
creasing subobjects, of the
The right defect is the subobject and the left defect
(In the category of abelian groups, the right
defect-but not in general the left defect- is always zero.)
The associated graded of the direct limit abutment is canoni-
cally isomorphic to and the associated graded of
the inverse limit abutment is canonically isomorphic to
C~,q/B~,q.
For the spectral sequence of a filtered cochain complex,
we also construct the "best possible hope" for an abutment for
a filtered cochain complex, the integrated partial abutments
Hn, nE 'l'. These can be thought of as "tying together" the
direct limit and inverse limit abutmenGof the exact couple of
the filtered cochain complex- in the precise sense that, the
associated graded of the integrated partial abutments are
canonically isomorphic to see section 8. And the
integrated partial abutments can also be thought of as "tying
together" the dual partial abutments of the filtered cochain
complex (section 8, Theorem 10). In fact, there are four, in
general non-isomorphic, exact couples induced by a filtered
cochain complex, all having the same spectral sequence (but
in general different sets of partial abutments), see section
8.
In section 9, Corollary 4.1, it is also shown that if the
spectral sequence is such that the cycles stabilize, then the
left defect=O, so that if we are in the cat. of abo gps., then
xii Preface
the integ. part. abts. are a true abutment. If also the
spectral sequence is semi-stable (see section 9, Remark 3 after
Proposition 5.1), then the integrated partial abutments are
also complete (section 9, Proposition 5). (And then similar
observations also hold for the set of partial abutments, and
also for the dual set of partial abutments, of the completion
of the co-completion of the filtered cochain complex, see near
the end of section 9). The spectral sequence of a double
complex is studied in section 10. And also, in section 10,
Example 3, the Adams spectral sequence is studied from a spec­
tral sequence-theoretic point of view, as an interesting ex­
ample from algebraic topology of an important spectral sequence
coming from an exact couple in which the inverse limit abut­
ments act as an abutment (in the sense of section 7, Definition
1), rather than the more common direct limit abutments.
It is shown by Examples, that most of the pathology not
excluded by the theories of Intro. Chap.2, actually do occur.
Spectral Sequences were first introduced by J. Leray.
After the Introduction, we begin the main theme of the
book. One can begin the book at this point, if one wishes,
looking back when necessary to the Introduction.
Let A be a ring (not necessarily commutative) with
identi ty, and let t ~ A.
Let C* be any cochain complex of left A-modules, in­
dexed by all the integers. Then we have the cohomology groups,
all integers n, which are left A-modules. We also have the
Preface xiii
and also
all integers n, i, with i > O. We study the connections be-
tween these groups.
In particular, as a special case of Introduction, Chapter 2,
section 5, we obtain a short exact sequence, the middle term
of which is and the first and last terms of which depend
entirely on Hn(C*) and on Hn+l(C*) respectively. (Chap-
ter 1, Theorem 2). No hypotheses are required. This essen-
tially determines the Ew-term of the Bockstein spectral sequence
in the fullest generality.
(It was, essentially, in the making of a study of these
results, that I was led to the Introduction, Chapter 2.)
In Chapter 2, we study an exact sequence of six terms,
interconnecting Hn(C*) for all integers
n,i, with i > O. I introduced this sequence over thirteen
years ago in [P.P.W.C.], where it appeared as (1.8) (and the
actual research had been done earlier, at the University of
Oxford, in 1963-64). We strengthen and improve slightly on
these previous results.
In Chapter 3, a well known short exact sequence is noted,
in the case that Cn is t-adically complete for all integers
n.
xiv Preface
In Chapter 4, we assume that Cn is complete for all in-
tegers n, and use the results established in Chapters 1,2 and
3 to obtain some new results. Most striking is Theorem 6 of
Chapter 4. See also Proposition 5, and Corollary 6.1 of Chap-
ter 4.
In Chapter 5, we combine the results of Chapter 1 (especi-
ally Proposition 6) with those of Chapter 4 (especially Propo-
sition 2 - a corollary of Theorem 1, which is a special case
of Theorem 6) to obtain the theorem (Theorem 1) that gives the
chapter its heading. We state it here.
Chapter 5, Theorem 1. Let A be a ring with identity and
let t~ A be an element in the center of A. Suppose that
A is t-adically complete. Let C* be a ~-indexed) cochain
complex of left A-modules such that c i is t-adically complete
for all integers i, and that the endomorphism "multiplication
by t":Ci+ci is injective, all integers i. Suppose that the
ring A is left Noetherian (i.e., that all left ideals are
finitely generated. This assumption can be deleted if we com-
plicate the statement of the Theorem somewhat- see the Remarks
following Theorem 1 of Chapter 5). Let n be any fixed in-
teger. Then if
for the fixed integer n,
then
This Theorem is, perhaps, a somewhat surprising result.
Preface xv
Can it be generalized? For example, can the hypothesis that
"multiplication by t:Ci -+C i is injective, for all integers . " 1,
be eliminated? Chapter 5, Theorem 4 answers this in the affir-
mative, if one assumes in addition that for the fixed integer
n,
where 0* is the subcochain complex of C* such that
Also in Chapter 5, we introduce a new concept, "the per-
cohomology groups of a cochain complex of left A-modules C*
with coefficients in a right A-module, M," Hi (M,C*), i E 'J'
which coincides with Hi (M @ C*) if C* is bdd. above and is flat A
over A, for all integers i, but which in general is a new
invariant. Then, in the above theorem, (Theorem 1 of Chapter
5), if we assume that the element tE A is a non-zero divisor,
and if we replace "Hn(C*/tC*) with "H~(A/tA,C*)'" then the
hypothesis that "mulitplication by t:ci -+ c i is injective, for
all integers . " 1, can be deleted. This is a somewhat finer
result than Theorem 1. This is Chapter 5, Proposition 2.
( If the element tE A is not a non-zero divisor, then that
assumption can still be deleted if we replace "Hn{C*/tC*)"
with "H;[T] {C*, .?'[T] / (T • .?' [T]) ) ") .
As an application of the finiteness theorems of Chapter 5
(particularly Chapter 5, Theorem I), we prove, in the notations
of [P.A.C.]'
xvi Preface
valuation ring of mixed characteristic, let A be an O-algebra
and let A = A ®l< (0). Suppose that A is Noetherian. Let X - 0
be a scheme simple and proper over A and embeddable over A
(for this latter, it suffices that X be either quasiprojec-
tive over A or liftable over ~, see [P.~C.l), and let
Hh (X,~"), h ~ 0, be the lifted p-adic cohomology using the
"I\n, as defined in [P.A.CII!]. Then Hh (X,!2.") is finitely
generated as ~"-module, all integers h.
This is proved in Examples 1 and 2 at the end of Chapter
5. In fact, it was in the course of proving Theorem A" above
that we were led to prove the very general Chapter 5, Theorem
1, (which will probably also have applications to other branches
of mathematics) .
I also state Chapter 4, Theorem 6:
Chapter 4, Theorem 6. Let A be a ring with identity and
let tE A be any element. Let C* be any (~indexed) cochain
complex of left A-modules such that Cn is t-adically complete
for all integers n. Then for every integer n,
(1) Hn(C*)!(t-divisible elements) is t-adically complete.
(2) Hn(C*) has no non-zero infinitely t-divisib1e e1e-
ments.
(3') For every integer n, we have an epimorphism from
(*)The hypothesis "unramified" can be replaced weaker condition "ordO (p) ~p - 1, where p is the istic of the residue class field of 0".
by the character-
Preface xvii
the group of equation (3) onto lim [Hn(C*/tic*)]. i>O
(4) Also, for every integer n, there are induced
natural isomorphisms of abelian groups (or of left
A-modules if t is in the center of A):
(t-divisible part of Hn(C*»~
(5) Also, for every integer n, the subgroup of
Hn(C*), the t-divisible part of Hn(C*), is con-
tained in the subgroup
of Hn(C*).
And Note 2 of Theorem 6 states,
Note 2. If for any fixed integer n, we have that
Im(dn:Cn ~ c n+ l ) has no non-zero t-torsion elements, then for
that integer n the epimorphism of equation (3') is an iso­
morphism, and then also the two subgroups of Hn(C*) discussed
in conclusion (5) coincide.
A special case of Chapter 4, Theorem 6 is that in which
(multiplication by t): Cn~Cn is injective, for all inte-
gers n. Then Note 2 above applies for all integers n. That
special case is Chapter 4, Theorem 1 and Theorem 1'.
Under the more general hypotheses of Theorem 6, one might
wonder whether one can obtain the stronger conclusions of
Note 2 above, whether or not (multiplication by t): Cn ~ Cn
is injective. The answer is "NO", as is shown in Remark 2
xviii Preface
following Corollary 6.1 ("Example 2"). However in Chapter 4,
Theorem 6' (in Remark 4 following Corollary 6.1), it is shown
that, if the hypotheses are as in Chapter 4, Theorem 6 and if
in addition the element tE A is a non-zero divisor in the
center of the ring A, and if we work with the appropriate
percohomology groups as in Chapter 5, then we obtain all the
conclusions of Note 2, where in conclusion (3'), respectively:
(5), we must replace "Hn(C*/tic*)" by ttH~(A/tiA,C*)'" re­
spectively: replace "Hn-l(C*/tiC*)" by "H~-l(A/tiA,C*)'"
(In Chapter 4, Remark 7 following Corollary 6.1, it is shown
further that the hypotheses that Itt is a non-zero divisor"
and that Itt is in the center of the ring A" can be elimi-
nated if one uses the percohomology groups
and instead,
Perhaps also relevant are Chapter 5, Corollaries 1.1 and
1.2. In Chpater 5, Corollary 1.1, it is shown, e.g., that,
under the hypotheses of Chapter 5, Theorem I, we have that
(l) and
(2)
Does this result hold if we have the more general hypotheses
of Chapter 5, Proposition 2? (I.e., when we delete the hypo-
thesis that "multiplication by t:Cn -+ Cn is injective, for
all integers n"?) The answer is "no", as is shown by an
example (Chapter 5, Remark 4 following Theorem 4). (However,
if one uses the appropriate percohomology groups instead,
Preface xix
laries 1.1' and 1.1".)
Under the hypotheses of Chapter 5, Theorem 1, if E~(ti)
denotes the abutment of the Bockstein spectral sequence of the
cochain complex C* with respect to the endomorphism, "multi­
plication by ti", as defined in Chapter 1, then in Chapter
5, Corollary 1.2, we show that for the fixed integer n of
Chapter 5, Theorem I, we have that
(1) [lj,m E~(ti) 1 ",Hn(C*)/(t-torsion), and i>O
(2)
This Corollary generalizes, Corollary 1.2' (respectively: Corol-
lary 1.2"), to the situation of Chapter 5, Theorem 6' (respec-
tively: Theorem 6), if the Bockstein spectral sequences are
defined suitably (in essentially the only way that is possible,
see Chapter 1).
Under the more general hypotheses of Chapter 4, Theorem 1,
for all integers n we have that
(1) [lj,m E~(ti) 1'" Hn(C*)/(topological t-torsion), i>O
and (2) [lj,ml E~(ti)l'" ~iml (the kernel of the endomorphism i>O 1>0
induced by "multiplication by ti" on
{t-divisible elements in Hn+l(C*)}).
This is Chapter 4, Corollary 1.1. The conclusio~of Chapter
4, Corollary 1.1 hold equally well in the more general situa-
tion of the hypotheses of Chapter 4, Theorem 6, see Chapter 4,
xx Preface
Remark 7 following Corollary 6.1. (See also Chapter 4, Corol-
lary 1.1' in Remark 5 following Corollary 6.1.) (Perhaps also
of interest is Chapter 4, Proposition 3, that under the hypo-
theses of Chapter 2, Corollary 1.2, if n is a fixed integer
such that
(Chapter 4, Remark 6 following Corollary 6.1, is also of
some relevance to Chapter 4, Theorem 6.)
A somewhat amusing set of side results in Chapter 4 are
several lemmas and theorems that give information about t-
divisible elements, etc., in left A-modules M over a ring
with identity A, where t is an element of A. These in-
clude Lemma 1.1.1 (which asserts that if "left mUltiplication
by t": M-+M is injective, then the same is true of MAt,
the t-adic completion of M), Theorem 4, Proposition 5, Corol-
lary 5.1 and Corollary 5.2 of Chapter 4. E.g., in Chapter 4,
Proposition 5, it is shown that if every t-divisible element of
M is infinitely t-divisible (e.g., this is the case if, either
M has no non-zero t-divisible elements (as for example if M
is t-adically complete), or if (left multiplication by t):
M -+ M is inj ecti ve), then
liml (kernel of the endomorphism, "multiplication by i>O
till, of M)
" ,
where "MAt/Mil denotes the cokernel of the natural mapping
from M into its t-adic completion MAt (whether or not that
mapping is injective).
Chapters 1-5 constitute Part I of the book, and is the
main emphasis of the book. Part II consists of Chapters6 and 7.
In Chapters 6 and 7, we are concerned with the highest non-
vanishing cohomology group, and with Poincare duality, respec-
tively. Some of the results are of particular use in applica-
tions to algebraic geometry, and in particular to p-adic coho-
mology (e.g., see [P.A.C.]), and also to more traditional re-
suI ts in algebraic topology, as we show in some examples.
E.g., in Chapter 6, suppose that we have an integer n,
such that
Hi (C* /tC*) = 0 for i> n, and such that
Then what can we say about Hn(C*) (and about Hi(C*) for
i ::.. n)? The most general such question is answered in Lemma 1.
The case in which Hn(C*/tC*) is finitely generated is covered
in Proposition 2. The later theorems and propositions of Chap-
ter 6 deal with the case in which Hn(C*/tC*) is a free
(A/tA)-module of rank one. Chpater 6, Theorem 4, is particu-
larly useful in [P.A.C.].
In Chapter 7 we study Poincare duality. The most notable
theorem is Theorem 3 and Corollary 3.1, which study the
xxii Preface
problem: If C* is a differential graded algebra over the cmv.
ring A and if A and ci are t-adically complete, for all
integers i, and if the graded (A/tA)-algebra Hi (C*) , i E 7/,
obeys Poincare duality over the quotient ring (A/tA), then
when can one conclude that the graded A-algebra,
Hi(C*)/(topological t-torsion), iE7/,
obeys Poincare duality over the ring A? Basically, one needs
that: Hn(C*)/(topological t-torsion) has annihilator ideal
zero; that Hi(C*) has no non-zero t-divisible, t-torsion ele-
ments, for all integers i; and that the ring A/tA is injec­
tive as left (A/tA)-module. The first two of these conditions
are reasonable, but the third is very restrictive. However,
this latter condition holds in the case that A is a discrete
valuation ring and t"F 0 (the resulting statement is Corol­
lary 3.1). This result, and Chapter 6, Theorem 4, are used
to prove that:
If X is a complete, non- singular (not necessarily lift­
able) embeddable ([P.A.C.]) algebraic variety over a field k
of characteristic p, and if 0 is an unramified, complete
discrete valuation ring of mixed characteristic having k for
residue class field, then if Hi(X,OA), iE7/, is the author's
lifted p-adic cohomology using the "A", as defined in [P.A.C.],
then, for Hi(X,OA)/(t-torsion), iE7, we have Poincare
duality over the ring O. See [P.A.C.]. This is the principle
application of Chapter 7 to our general study of the Weil zeta
function [W.C.]. {Chapter 6, Theorem 4 also has application
to algebraic families; i.e., in the notation of [P.A.C.], to
Preface xxiii
applications to algebraic topology.
In Chapters 1-7, we have dealt with completions with re­
spect to an ideal that is generated by a single element t. In
Part III, we make a study of the analogous situation, for comple­
tions with respect to finitely generated ideals that are not
necessarily simply generated. Part III consists entirely of
Chapter 8.
In Chapter 8, we return to more fundamental considerations
as in Chapters 4 and 5, but generalized as follows. Let A be
a ring (not necessarily commutative) and let I be a left
finitely generated two-sided ideal in A such that A is
I-adically complete. Let C* be a ~-indexed) cochain complex
of left A-modules such that Ci is I-adically complete for all
integers i. Then we state a generalization of Chapter 5, Theo-
rem 1. (Chapter 8, Theorem 1. The ideal I is required to be
generated by an "r-sequ~nce for Ci ", all integers i, see
Chapter 8, Definition 2.)
If the ideal I admits a set of generators contained in
the center of the ring A, then one can state a generalization
of Chapter 5, Theorem 4, namely
Chapter 8, Corollary 1.3. Let A be a left Noetherian ring
with identity and let I be an ideal in the ring A such that
A is I-adically complete; and such that we have an integer
r> 0 and a finite sequence tl, ... ,tr of generators of the
ideal I contained in the center of the ring A. Let
¢:7[Tl, .•. ,Trl ~A be the homomorphism of rings with identity
xxiv Preface
, 1 < i < r. Let C* be a ':1'-indexed)
cochain complex of left A-modules such that ci is I-adically
complete, all integers i. Let
j 7.I'[T l ,···,T I( [ lilT T) Cj ) D. = Tor. r 1'Tl ,···,T 1' ... ' , , l l r r
i, j E 1', i > O. Let n be any fixed integer. Then if
(1) Hn+i (D~) l
is finitely generated as left (A/I)-module,
for 0 ~ i ~ r, then
(2) Hn(C*) is finitely generated as left A-module.
More general theorems are demonstrated, if merely I ad-
mits a finite set of generators that mutually commute, and a
less restrictive set of hypotheses apply, assuming that one
uses the percohomology~: "H~(A/I,C*)" in lieu of
"Hn(C*/IC*) ", see Corollary 1.1 and Remark 2 following Propo-
sition 2. Also, see Chapter 8, Corollaries 1.2 and 1.3'. (As
a fairly straightforward application of the general theory de-
veloped in Chapter 8 to algebraic geometry, in Examples 1-4 of
Chapter 8 we deduce by a new method almost all of the well-
known finiteness theorems about cohomology of coherent formal
sheaves over proper formal schemes.)
Theorem 6 and its Corollaries in Chapter 8 generalize
Corollary 1.1 of Chapter 5. (They concern the inverse limit
and the liml of the "percohomology groups mod In".) Other
results in Chapters 4 and 5 (particularly Chapter 5) are
generalized to the situation of Chapter 8. E.g., Chapter 5,
Proposition 3 is so generalized, see Chapter 8, Proposition 4.
Concerning the level of generality of the various chapters:
Preface xxv
Most of Chapter 1 generalizes to virtually any abelian category,
and this is proved. Chapter 2 requires an abelian category such
that denumerable direct products exist, and such that denumerable
direct product is an exact functor, see Introduction, Chapter
I, section 7, and this generalization is noted in Remarks.
Also, in Chapter I, only the cohomology sequence and not neces­
sarily the cochains, is needed; and in Chapter 2, any cohomology
theory, (not necessarily the cohomology of cochain complexes)
will do. Chapter 3 again requires an abelian category such
that denumerable direct products exist and such that the func­
tor, "denumerable direct product" is exact, (but now the coho­
mology theory must be the cohomology of cochain complexes). The
same is true for all of Chapter 4, as is noted in the text
But Chapter 5 requires that we be in the category of (7-indexed)
cochain complexes of left A-modules, where A is a ring, and
that tEA be in the center of A (or at least, be such that
tl\ CAt). Part of Chapter 6 is at the same level of generality
as Chapter 4; but the rest of Chapter 6, and Chapters 7 and 8,
again require a ring A and the cohomology of cochain complexes.
In keeping with the general philosophy to make the material
accessible to beginners, the main text of the book, Chapters
1-8, has been written for the most part at the level of gener­
ality of left modules over a ring, even when the more general
situation of abelian categories was possible; and the generali­
zations to abelian categories have been kept in Remarks, that
may be ignored by the reader who is not interested in, or cog­
nizant with, these generalizations.
Some of the results of this book are somewhat surprising.
xxvi Preface
These include Chapter 5, Theorem 1; Chapter 4, Theoremsl and
6 and Corollary 6.1. Perhaps also interesting are Chapter 4,
Proposition 5; Chapter 6, Theorem 4; and Chapter 7, Corollary
3.1; and the theorems of Chapter 8. We give examples to show
that most of the pathology that would not be excluded by these
theorems actually occurs - a very partial list is Chapter 4,
Examples 1 and 2, and Example 2 in Remark 2 following Corollary
6.1: and Chapter 7, the Example following Theorem 3 (e.g., the
latter example shows the difficulties in attempting to improve
Chapter 7, Theorem 3).
My special thanks to Mrs. Marion Lind and Mrs. Sandi
Agostinelli for their excellent work in typing up this manu-
script; and especially to Mrs. Sandi Agostinelli, for putting
in so much overtime work, and more than usual care and patience.
Terminology. Ingeneral, we use conventional terminology. E.g.,
if A is a ring and tEA and if M is a left A-module, then
an element uEM is t-divisible iff for every integer i > ° , --
there exists v. EM such that t i • v. = u. The element u is 1 1
infinitely t-divisible iff there exists a sequence vi' i> 0,
of elements of M such that t· v i + l =vi ' all integers i.::O,
and such that v ° = u. It is easy to give examples of t-di visible
elements that are not infinitely t-divisible.
Similarly, if M is a left A-module, where A is a ring,
and if t is an element of A, then, following the usual
terminology, an element uE M is a t-torsion element iff there
exists an integer i > ° such that i t ·u=O. We will call an
element u EM a precise t-torsion element iff t· u = 0. Thus,
Preface xxvii
if i> 1, then an element uE M is a precise ti-torsion ele-
ment if and only if ti. u = O. Thus, "u a precise t-torsion
element" implies "u a precise ti-torsion element" implies "u
. i+l . 1 ". l' a preclse t -torslon e ement lmp les "u is at-torsion
element", for all integers i > 1. The set of all precise
ti-torsion elements in M is the precise ti-torsion part of
M. (Thus, for each integer i::.l, the precisE'! ti-torsion part
of M is the kernel of the endomorphism "multiplication by
tin of the abelian group M).
However, we do deviate slightly from the most commonly
accepted notational convention in our manner of indexing a
bigraded exact couple. E.g., what would be written as
in the original reference [E.C.], is denoted in this text as
That is, we have as usual emphasized the filtration degree p,
but we have also chosen to emphasize the complimentary degree
q rather than the more traditionally emphasized total degree
n = p + q. We believe that our indexing notation helps make
comprehension overall a bit easier, perhaps because of the even
more rigidly entrenched indexing notational convention, "E~,q".
The research for the main text of this book was done
mostly at the Pennsylvania State University in 1973-74, except
for virtually all of Chapter 2 which was done at the Univer-
sity of Oxford, England in 1963-64, and was partially supported
by NSF Research Contracts, respectively an NSF Postdoctoral
xxviii Preface
Fellowship. An early version of this manuscript was pre­
pared in 1975, and is substantially the same as Chapters 1-7 of
this ms., and contains the most important parts (Theorem 1 and
its Corollaries) of Chapter 8. Some extensive "touching up" of
Chapter 8 was done at the university of Rochester in the aca­
demic year 1977-78.
done at Columbia College, in the academic year 1958-59, and
was supported by a Pulitzer Scholarship and a New York State
Regents Scholarship for Education in Engineering, Chemistry,
Physics and Mathematics. (The Exact Imbedding Theorem was
also proved, roughly simultaneously, by a somewhat different
method, in [P.F.]). Introduction, Chapter 1, section 5 was
done at the University of Rochester in Spring, 1979. Intro­
duction, Chapter 2, sections 1-7, were mostly worked out at
the University of California at Berkeley in the 1960's, and
owe a debt to an early version of [O.A.L.l, and were partly
supported by NSF grants. They were written up at the University
of Rochester in spring to fall, 1979, partly supported by an
NSF grant. Introduction, Chapter 2, sections 8-10 were com­
pleted at the University of Rochester in summer and fall, 1979,
and were partially supported by an NSF grant.
DEDICATION
PREFACE
Section 3 Abelian Categories
Section 5 Subquotients
Section 7 Denumerable Direct Product and Denumerable Inverse Limit
Theory of Spectral Sequences
The Spectral Sequence of an Exact Couple, Ungraded Case
Graded Categories
The Spectral Sequence of an Exact Couple, Graded Case
Filtered Objects
The Partial Abutments of the Spectral Sequence of an Exact Couple
The Spectral Sequence of a Filtered Cochain Complex
Convergence
xxix
v
vii
1
1
4
11
44
56
66
80
97
97
111
137
185
208
235
278
339
400
441
465
493
xxx
Cohomology of Cochain Complexes of t-Adically Complete Left A-Modules
Finite Generation of the Cohomology of Cochain Complexes of t-Adically Complete Left A-Modules
PART II
Poincare Duality
PART III
Finite Generation of the Cohomology of Cochain Complexes of I-Adically Complete Left A-Modules for a Finitely Generated Ideal I
BIBLIOGRAPHY
531
539
607
659
687
737
801
Categories
Most of the material of Parts I, II, and III of this book,
is about cochain complexes of left modules over a ring. It is
noted, in the various chapters, that some material generalizes
to suitable abelian categories. Therefore, we write this brief
introductory chapter to acquaint the reader unfamiliar with
abelian categories with this important concept. (The reader
who does not wish to learn about abelian categories can skip
this chapter, and can read Introduction, Chapter 2 with
"abelian groups" replacing "objects" throughout.)
We will assume in this chapter that the reader is familiar
with the concept of "category", and of the dual of a category,
as defined in the original, still excellent, reference [C.A.].
We recall a few definitions from [C.A.], those of mono­
morphism and epimorphism.
Defini tion. If C is a category and f: A -+ B is a map in C,
then f is a monomorphism iff whenever 0 is any object in C,
and g and h are any maps from 0 into A, such that
fog = f 0 h, then g = h.
A map f:A-+B in the category C is an epimorphism iff
f is a monomorphism considered as a map from B into A in
the dual category Co.
Equivalently a map, f:A -+ B is an epimorphism iff given
1
2 Section 1
any object D and maps g,h:B ->- D such that go f =h 0 f,
then g = h.
A map f:A ->- B in the category C is an isomorphism iff
.
A map that is an isomorphism is always both an epimorphism and
a monomorphism, but the converse is in general false.
Examples. 1. If C is the category of sets and functions (or
the category of groups and homomorphisms; or the category of
abelian groups and homomorphisms; or the category of rings and
homomorphisms; or the category of commutative rings and homo­
morphisms), then a map is a monomorphism iff it is one-to-one,
and is an epimorphism iff it is onto. In these cases, a map is
an isomorphism iff it is both an epimorphism and a monomorphism.
2. If C is the category of all topological spaces
and continuous functions, then a map is a monomorphism iff it
is one-to-one; an epimorphism iff it is onto; and an isomorphism
iff it is a homeomorphism. In this case, not every map that
is both an epimorphism and a monomorphism (i.e., that is one­
to-one and onto) is an isomorphism (i.e., is a homeomorphism).
3. If C is the category of all Hausdorff topo­
logical spaces and continuous functions, then let f: X ->- Y be
a map. Then f is a monomorphism iff f is one-to-one. f
is an epimorphism iff the set-theoretic image f(X) of f
is dense in Y (this is in general weaker than the condition,
IIf is onto"). f is an isomorphism iff f is a homeomorphism.
Therefore in this category also, not every map that is both a
monomorphism and an epimorphism is an isomorphism.
We will also assume, in this chapter, that the reader is
Categories 3
familiar with the notions of the direct product and direct sum
of an indexed family of objects in a category, as presented in
[C.A.], and also with the definitions and elementary properties
of the inverse limit (or direct limit) of an inverse (or direct)
system in a category indexed by a directed set, as covered in
[C.A.]. (This generalizes without change to the case in which
the indexing directed set is replaced by a directed class (i.e.,
a non-empty class (see [K.G.]), perhaps even a proper class, with
a preorder such that for every x,y there exists a z such that
x:::. z, y:::. z) .)
Subobjects and Quotient-Objects
Let A be a category and let A be an object in A.
Then consider the class UK.G.]) MA of all pairs (B, f) where
B is an object in A and f:B -+ A is a monomorphism. Then we
have a natural pre-order on the class MA , by defining, when-
ever (B',f') and (B",f") are elements of M A
, that
(B",f")::. (B' ,f') iff there exists a map g:B" -+B' in the
category A such that f' 0 g = f". If such a map g exists,
then (since f' is a monomorphism) it is unique; and (since
fll is a monomorphism) it is a monomorphism. Therefore,
Lemma 1. Let A be a category, let A be an object in A and
let (B', f'), (B" , fll) E M A
. Then the following two conditions
are equivalent.
There exists an isomorphism
.
the category A, such that f' 0 8 = f ".
When these equivalent conditions hold, then the isomorphism
of condition 2) is uniquely determined.
Proof: 2) =- 1) We have f' 08 = fll, respectively f' = f" 08- 1 ,
and therefore, (B",f") < (B',f'), respectively (B',f') < (B",f"),
1) =- 2): Let 8 :B" -+ B' and T:B' -+ B" be maps such
that f'o8=f" and f"oT=f'. Then f'o8 o T=f"oT=f'=f' oid B
4
, Similarly
follows since f' is a monomorphism. Q.E.D.
Definition 1. Let A be a category and let A be an object
in A. Then a subobject of A is an equivalence class of
pairs in MA , where two pairs (B' ,f') and (B",fll) are
equivalent iff the two equivalent conditions of Lemma 1 hold.
Given two sUbobjects S' and S" of A, we say S' < S" iff
there exist (equivalently: for all) (B' , f ') E S' and
(B",f")ES", we have (B',f') 2. (B",f")
We next make a convention. Given a category A (which
mayor may not be a set) and an object A in A, then by the
(*) strongest form of the Axiom of Choice (see [K.G.]) , there
exists a subclass Mi of MA that contains exactly one repre­
sentative element from each equivalence class. We will call
such a class a complete class of representatives for the
subobjects ~ It is a class containing exactly one repre-
sentative pair (B',f') for each subobject {(B' ,f')} of the
object A, and consisting only of such representative pairs.
Having chosen such a fixed complete class of representatives
for the subobjects of A, by a class (or set) of subobjects of
A we will mean a subclass (or subset) of the complete set of
(*)In [K.G.], Godel proves that this strong form of the Axiom of Choice is consistent with the other, standard, axioms of his set theory, assuming that those other axioms are them­ selves consistent. (The statement of this strongest form of the Axiom of Choice is, that "There exists a well-ordering on the class of all sets".)
6 5ection 2
representatives M~. Also, since the complete class of repre-
sentatives M~ is a subclass of MA , and since MA is a
pre-ordered class, we have that M' A
is a pre-ordered class.
In addition, given two subobjects 5' and 5" of A, if
(B', f') and (B", f") are their unique representatives in
we have by Definition 1 above that 5' < 5" as subobjects of
M' A
A iff (B',f') < (B",f") in the complete class of representa-
tives M~.
It might be, given a category A and an object A in A,
that a complete classof representative;; M' A
for the subobjects
of A is a set. If this is so, then it is independent of the
choice of the class of representati ves M~, and we say that
the subobjects of A form a set. Also, in this case, the
cardinality of M~ is independent of the choice of the complete
class of representatives M' A
and we then call this cardinal
number the cardinality of the set of all subobjects of A.
(The proof of these observations is that given any two complete
classes of representatives M' A
and
for the subobjects of
such that S(B,f) is equivalent to (B,f), fora11 (B,F) in
Remarks 1. The reason for our introducing the notion of "a
complete class of representatives for the subobjects" is as
follows. In Godel's set theory, [K.Gl, one cannot speak of a
class some of the elements of which are proper classes. But,
by Definition 1 above, if A is a category that is not a set,
and A is an object in A, then it is possible for some, or
even all, of the subobjects of A to be proper classes. (E.g. ,
Subobjects and Quotient-Objects 7
this is the case if A is the ca~egory of abelian groups and
if A is any object in A; or if A is the category of sets
and if A is any non-empty set). Therefore, in Godel's set
theory, in such circumstances, one cannot speak literally of
"a class, or set, of subobjects of A" in the literal sense of
Definition 1; so instead we speak of "a class, or set, of the
representatives for the subobjects of A", as in the above
convention.
2. In the most common categories A, there is usually
a "natural choice" of a complete class of representatives
for every object A, making it unnecessary to invoke the
Axiom of Choice in that case.
3. v·lhere not inconvenient, we will state results in
M' A
such a way so as to not necessitate using the above convention
(i.e., so as not to speak of "classes or sets of subobjects").
This is usually not di fficul t to do--al though some theorems
become sloppy in their statement unless one uses the above con-
vention, and then we will use the above convention.
Examples. 1. If A is the category of sets, then for every
object A in A, a complete class of representatives for the
subobjects of A is the set of all pairs (B,l) where B is
a subset of A and l:B ... A is the inclusion.
Notice in this case that if A is a non-empty set then
every subobject of A in the sense of Definition 1, except
for the empty subobject, is a proper class. Notice also, in
this case, that the subobjects of A form a set, and are in
natural one-to-one correspondence with the set of all subsets
of A.
8 Section 2
2. If A is the category of all groups (or abelian
groups, or rings, or commutative rings, or fields), then for
every object A in A, as in Example 1, a complete set of
representatives M' A
all pairs (B,l) where B is a subgroup (or, respectively,
subgroup, or subring, or subring, or subfield) and l:B -+ A is
the inclusion. Again, the subobjects of A form a set, in
natural one-to-one correspondence with the set of all subgroups
(or, resp~ subgroups, subrings, subrings, or subfieldsl of A.
3. If A is the category of all topological spaces,
and A is an object in A, then a complete set of representa-
tives for the subobjects of A is given by {(B,l)} where B
is a topological space such that the underlying set of B is
a subset of the underlying set of A and such that the topology
of B is finer than the induced topology from A; and 1 is incl.
4. Every pre-ordered class 0 defines a category
AD' such that the objects in AD are the elements of 0,
and such that
o r,1 otherwise.
Let 0 be the pre-ordered class of all ordinal numbers (as
defined in [K.G.]l with the reverse of the usual pre-ordering.
Then for every object A in AD each Eubobject of A con-
sists of a single element and is therefore a set. The class of
all subobjects of A is a proper class, and is in natural one­
to-one correspondence with {ordinal numbers B such that
a < e for the usual ordering}.
Subobjects and Quotient-Objects 9
Let A be a category and let A be an object in A.
Suppose that we have chosen a complete set of representatives
for the subobjects of the object A. Then given a class S of
subobjects of A, then by a supremum,or infimum, of S we
mean a supremum, or infimum, in the complete class 0 f representa­
tives MA. We write L B, respectively n B, for this B{:S B ES
supremum, respectively infimum, if it exists.
For example, if A is the category of sets, then the sup-
remum of a set H of subobjects of a set A is the usual
union, and the infimum is the usual intersection. If A is
the category of abelian groups, then the supremum of a set H
of subobjects of an abelian group A is their sum, B= (B, l )EH
and the infimum is {finite sums of elements of U B}; (B, l) EH
the set-theoretic intersection. In the category of groups, the
supremum of a collection of subgroups of a group is the sub-
group generated by the set-theoretic union; the infimum is the
set-theoretic intersection.
Definition 10. Let A be a category and let A be an object
in A. Then a quotient-object of the object A in the category
A is a subobject of A in the dual category AO • Thus, ex-
plici tly, a quotient-object of A is an equivalence class of
pairs (B',p'), where B' is an object in A and p':A->-B
is an epimorphism; where (B' ,p') is equivalent to (B",p")
iff there exists an isomorphism 8:B' ->- B" such that
p"=8 o p'. A complete class of representatives E' for the A --
quotient-objects of A in the category A is a complete class
of representatives E' A
for the subobjects of A in the dual
category AO • This is in a natural way a pre-ordered class;
10 Section 2
we can speak of a class (or set) of guotient-objects of A,
and of the supremum or infimum of such a class when the sup-
remum or infimum exists.
1 10. Examp es. In Example 1 above, a complete set of representa-
tives for the quotient-objects of a given set A in the cate-
gory of sets A, is the set of all pairs (B,p) where B is
the quotient-set of A by some equivalence relation, and where
p:A -+ B is the natural mapping.
20 • In Example 2 above, a complete set of reps. for
the quot.-objs. of a given obj. A in A is the set of all pairs
(B,p) where B is a quotient-group (resp.: a quotient
group; a quotient ring; a quotient ring; A) and p is the
natural map into the quotient (resp.: ibid, ibid, ibid,
30 • In Example 3 above, a complete set of representa-
tives for the quotient-objects of a topological space X are
the pairs (Y,p) where the underlying set of Y is the
quotient-set of the underlying set of X for some equivalence
relation, and where the topology of Y is coarser than the
quotient topology; and where p: X -+ Y is the natural mapping.
Section 3
Abelian Categories
Definition 1. A pointed category is a category e, together
with the additional data, the giving, for every pair A,B of
objects of e, of
(Zl) A map, 0A,B from A into B in the category e,
such that
(Z2) For every object D in e and every map f:D+A,
respectively g:B+D, we have that 0A,B of=OD,B'
respectively g oOA,B =OA,D
Given a pointed category e, the maps 0A,B for all objects
A,B in e are called the points. We often write 0A for
0A, A E Home (A, A) , for all objects A.
If e is a category that admits the structure of pointed
category, then the points are uniquely determined by the cate-
gory structure of e.
Proof: In fact, if we had another set of points (0' ) A,B A,B objs.
in e obeying axioms (Zl) and (Z2) above, then using
(Z2) ,
11
12 Section 3
Examples 1. A pointed set is a pair (S,so) where S is a
set and So E S. Then we have the category of pointed sets,
such that for all pointed sets (S,so) and
(S,so) -.. (T,to ) are all functions f from
(T,to )' the maps:
category.
pointed category.
is a pointed category.
4. The category of sets and functions is not a pointed
category (since, e.g., Hom{sets} (S,<jl) = <jl).
5. The category of non-empty sets and functions is
not a pointed category. (Proof left as an exercise) .
6. The category of rings (or of commutative rings)
with identity and homomorphisms of rings (that preserve the
identity element) is not a pointed category. (Since e.g.,
Hom( {oJ ,if) = </». Similarly for the category of all non-zero
rings (or of all non-zero commutative rings) and such homomor-
phisms.
Definition 2. An additive category is a pair
(C (+ ) ) where C is a category, and , A,B A,B objects in C'
where, for every pair of objects A,B of C, +A,B is a bi-
nary composition on the set HomC(A,B), such that
(ADl) HomC(A,B) together with the binary composition
+A,B is an additive abelian group, and
(AD2) Whenever A,B,D are objects in C and
Abelian Categories 13
have that
k O(f + g) = k 0 f + k 0 g in Home (A, D) and
(f+g) oh=foh+goh in Home(D,B).
Given a category ,e, a doubly indexed system
(+A,B) A.B cbjects in e that obeys axioms (ADI) and (AD2) ,
will be called an additive structure on the category e.
As we shall see in the next Remark, every additive cate­
gory is pointed; but the converse is in general false (vide
infra) .
Remarks l. Let e be an additive category, and let A,B,D
objects of e. Let k EHome(B,D) (respectively: let
hE Home(D,A». Then the first part (respectively: second
part) of axiom (AD2) is equivalent to the assertion that the
be
function: f + k .Of (respectively: f + f oh) is a homomorphism
Home (A,D) of abelian groups from Home(A,B) into
tively: into Home(D,B». Therefore, if
zero element of the additive abelian group
(respec-
(1)
whenever
respectively: 0A,B Oh=OD,B'
resp.: hEHome(D,A), and
(2) k 0 (f-g) =k of-k og, and (f-g) oh=f oh-g oh,
whenever f, g E Home (A, B), k E Home (B, D) and hE Home (D, A) •
Equations (1) imply that every additive category has a
natural structure as pointed category, by taking the additive
identity element of HomA(A,B) for the point 0A,B' for all
objects A,B in A.
(ADl') HomC(A,B) together with the binary composition
+A,B is an additive group (that is not necessarily
abelian) .
Then we claim that axiom (ADI'), in the presence of axiom (AD2) ,
is equivalent to axiom (ADl).
Proof: Suppose we have axioms (ADl') and (AD2). Then given
any objects A,B in the category C and any two maps
f,g€HomC(A,B)' by axiom (AD2) and the associative law, we
have that
(Hg) o idA + (f+g) 0 idA =f+g+£+g.
On the other hand,
f+g+f+g f+f+g+g in P.omC(A,B).
Since Home (A,B) is a group, by the left and right cancella­
tion laws we have
f+g=g+f. Q.E.D.
~xample 7. Let e be the category consisting of one object,
call it P, and such that Home(P,p) = (the multiplicative
monoid of the ring 7. of integers). (This means that
Home(P,p) =7. as set, and the composition is given by multiplica-
Abelian Categories 15
tion of integers). Then there exists an additive structure on
the category C as in Definition 2 above, such that C be-
comes an additive category. Namely, take the usual addition lot
"+" of integers. However, there are actually 2 0 different
additive structures on the category C such that C becomes
an additive category.
Proof: In fact, if IT is any permutation of the set of
rational primes, then by unique factorization there exists a
unique automorphism a IT
of the multiplicative monoid ( lI, .)
for all positive primes p. Given
two integers n,m, define
Then the binary composition "t" on lI, together with the
usual multiplication, makes P into a ring. Therefore, the
datum, "+" IT '
additive structures "+" on the category C are distinct for IT
different permutations IT of the set of all rational primes.
Q.E.D.
Example 7 above shows that, unlike a pointed category, an
additive category cannot in general be thought of as being a
category that obeys a certain set of axioms; but the additive
structure must, in general, be thought of as a new set of data.
However, we will see in Corollary 1.1 below, that unlike the
situation in Example 7 above, for a wide class of categories,
there is at most one additive structure. First, we prove a
Proposition, that is quite important in its own right.
Proposition 1. Let A be an additive category, let n be a
16 Section 3
non-negative integer and let AI' .•• ,An be objects in A.
Then
(A) If there exists an object S in A, together with
maps 71 i :S'" Ai and 1. :A. + S, such that
(1)
, .•. ,7I n
in the category A (in the category-theoretic sense, see
[C.A.]), and the data (S,ll' •.• 'In' is a direct sum of
(B) Conversely, if (S,7I l
, ••• ,7In ) is a direct product of
Al, ... ,An in the category A, then there exist unique maps
1.:A. +S, l,::,i,::,n, such that equation:; (1) and (21 above hold. l l
(cl Similarly, if (S,ll, •.. ,ln) is a direct sum of
Al, ••• ,An in the category A, then there exist unique mappings
7Ii:S+Ai, l~is.n, such that equations (II and (2) above
hold.
n
L 6 .. f j
f. :B + A. l l
where 0 .. = J 1 lOA. ,A.
J1
Abelian Categories 17
satisfies the existence part for
a direct product. Conversely, if e is any map such that
'IIi 06=f i
then
6 = ids 0 6 = (1 I'll 1 + ... + 1 n 'II n) 0 e = 1 1 ('II 1 e) + ... + 1 n ('II n e )
11fl + •.• + 1n f n = f,
(4) e=f.
(B) =!> (A): For each integer i, 1 ~ i ~ n, since
(S,'TT 1
, ..• ,'TT n
, ... ,A n
, there exists
A. ,A. '11.01 = 1 J
{
j = i,
But then, 'TT j (1 1 '11 1 + ••• + 1n'TTn) = 'TT j ,
11 '11 1 + •.. + 1n'TTn = identity of S.
1 ~ j ~ n, and therefore
The proofs that (A)~ (C) and (C)~ (A) are similar--
or can be deduced from "(A) ~ (B)" and "(B) =!> (A)" re-
spectively by passing to the dual category. Q.E.D.
18 Section 3
additive category A, if the direct product of finitely many
.objects exists, then so does the direct sum, and they are
canonically isomorphic. Also, conditions (1) and (2) of part
(A) give a way of defining the direct product (equivalently,
the direct sum) of finitely many objects in an additive cate­
gory, in a way not mentioning universal mapping properties.
Both of these observations are special to additive cate-
gories.
Corollary 1.1. Let A be an additive category such that, for
every object A in A, there exists a direct product A x A
of A with itself. Then there exists at most one additive
structure on that category A.
Proof: Suppose that there exists an additive structure on the
category A. If A is an object in A, then let (A x A.1T l
,1T 2
)
be a direct product of A with itself. Then by Proposition 1,
part (B), we have unique maps 1 1
, 1 2
: A + A x A such that condi­
tions (1) and (2) of part (A) of Proposition 1 hold. But, since
is a direct product of A with itself, by the
universal mapping property, condition (1) alone determines the
mappings 1 1 ,1 2 , Since the point 0A=OA,A depends only on the
category structure, it follows that the mappings and
depend only on the category structure. Then, by the Proposi-
tion, part (A), we have that (A x A, 1 1
,1 2
is a direct sum of
Abelian Categories 19
Then depends only on the category structure of A. But
direct sum of A with itself, it follows th.at
If now B is any (possibly different) object irt the category
A, and if f,g:B->-A are maps, then let
(f,g):B ->-A x A
be the unique map such that 7Tl ° (f,g) =f,7T 2
0 (f,g) =g. Then
(f,g) depends only on the category structure of A. And by
equation (3) we have that,
(4) f+g=PA o (f,g).
Since the right side of equation (4) depends only on the cate­
gory structure, it follows from equation (4) that the additive
structure of A is uniquely determined by the category
structure. Q.E.D.
Remark: Of course, applying Corollary 1.1 to the dual cate­
gory, it follows, likewise, that: If A is any category such
that, for every object A in A, the direct sum: A e A of
A with itself exists, then there is at most one additive
structure on the category A; i.e., that there is at most one
20 Section 3
way of making A into an additive category.
Let C be an arbitrary pointed category. Let I be a
set, and suppose that (Ai)iEI is a family of objects in the
category A indexed by the set I. Suppose that the direct
sum,
IT A. iU ].
of the objects Ai' i E I, exist. Then we define the canonical
8: 6l A. -+ IT A. iO]. iO].
all i E L where 71.: IT A. -+A. , 1. j EI J 1.
if i-lj,
the canonical projections, resp.: injections, all i E I.
Then
Corollary 1.2. If A is an additive category, if I is a
are
finite set and if Ai' i E I, are objects in A such that the
direct sum 6l A. or the direct product iEI 1.
they both exist, and the natural mapping
8: 6l A. -+ IT A. iEI 1. i EI 1.
IT A. i EI l.
exists, then
Proof: By proposition 1, the indicated direct sum and direct
product both exist. Let (S 'TIl' ... ,TIn,ll' ... ,In) be as in
part (A) of Proposition 1. Then by the defining property (3)
of e, we have e = ids' which is indeed an isomorphism.
Q.E.D.
Remarks 1. If A is an additive category, I is an infinite
set, and A. 1
i E: I, are objects in A, such that the direct
sum Gl A. and the direct product IT A. both exist, then the iEI 1
iEI 1
natural map e is in general neither a monomorphism nor an
epimorphism. (In the category of abelian groups it is of
course always a monomorphism.)
tive categories such that, for every object A, the direct
product: A x A exists. More precisely,
Corollary 1.3. Let A be a pointed category such that, for
every obj ect A inA, the di rect product A x A exi s ts .
Then A admits an additive structure iff
(1) For every object A in A the direct sum AGlA
exists, and
(2 ) For every object A in A, the natural map
e:AGlA-+AxA
is an isomorphism.
Proof: (We only sketch the proof, since we will make no use
of this elsewhere.) Necessity follows from Corollary 1.2.
Conversely, suppose that conditions (1) and (2) hold. Then
22 Section 3
for any object A in A, since 8 is an isomorphism, it
follows that if jl,j2:A+A61A are the canonical injections,
and if we define 11 =8 ojl,12=8 oj2' then (AXA,11,12) is
a direct sum of A with itself in the category A. Therefore
there exists a unique map in the category A,
PA:AXA+A,
such that
If now B is any other object in A, and f and g are maps
from B into A, then since (A x A, 'TT l' 'TT 2) is a direct
product of A with itself, there exists a unique mapping
(f,g):B+AXA
Define
f+g=PAo(£,g).
Then we leave it as an exercise for the reader to verify that,
for this definition of for all objects A,B in A,
that indeed A becomes an additive category. Q.E.D.
Examples: Example 1 above, the category of pointed sets, is
not an additive category: Since if (s,s) and (T,t) o 0
are
pointed sets, then the direct sum is "the disjoint union of S
and T with So glued to t " o (--Le., is (SX{O})u
(T x {U)I "', where (s,O) '" (t,l) iff s = So and t = to'
Abelian Categories 23
all s E S, t E T); and since the direct product is the usual
Cartesian product (SxT,(s ,t », o 0
and the natural mapping
into the product" (i.e., is the function such that 8(s) =
8(t) = (s ,t), all sES, tET); and although this o
function is a monomorphism, it is not an epimorphism, unless
either card(S) or card(T) is one.
Example 2 above, the category of groups, is likewise not
additive, since if G and H are groups, the direct sum is
the free product G * H and the direct product is the usual
Cartesian product: G x H. In this case, the natural mapping 8
is an epimorphism, but is not an isomorphism unless either G
or H is the trivial group (i.e., is of cardinality one).
Example 3 above, the category of abelian groups, is an
additive category, if for f,gEHomC(A,B), A,B abelian
groups, one defines f+gEHomC(A,B), by requiring that
(f + g) (x) = f(x) + g(x), all x (;A.
Examples 4,5 and 6 above are not additive categories,
since they are not even pointed categories.
Remark: If A is a pointed (respectively: additive) cate-
gory, then it is obvious that the dual category is also pointed
(respectively: additive); where the point from A in-
to B in AO is defined to be 0B,A in the category A; and
if the category A is additive, then the binary composition
~A,B on Horn (A,B) ('" HomA(B,A» AO
in is defined to be
24 Section 3
additive category A.
in the
Defini tion 3. Let C be a pointed category, and let f:A -+ B
be a map in C. Then by a kernel of f we mean a pair (K,l)
where K is an object in C and l:K-+A is a map in C
such that
(2) If (H, j) is any such pair(*),
then there exists a unique map j :H-+K o
such that the diagram:
is commutative--that is, such that j = 1 0 jo'
If f:A -+ B is a map in the pointed category C, then a
cokernel of f is a kernel of f considered as a map in the
dual category CO from B iLto A. Equivalently, a cokernel
of f is a pair (C,IT) where C is an object in C and
IT:B-+C is a map in C, such that
(1) IT of = ° , A,C and such that
(2) If (H,j) is any other such pair, then there exists
a unique map j :C-+H o
such that the diagram:
(*)that is, if H is any object in C and if j:H-+A is a map such that f 0 j = 0H,B '
Abelian Categories 25
is commutative--that is, such that jo 0 IT = j.
Proposi tion 2. Let C be a pointed category and let f:A -+ B
be a map in C. Then
(A.) Let K be an object and l:K -+ A be a map in the
category C. Then
(a) (K, l) is a kernel of f, if and only if the fol-
lowing three conditions all hold:
(A.O)
(A.I)
is a monomorphism, and
(A.2) Given any pair (H,j), where H is an object
in C and j:H -+ A is a map in C such that
f 0 j = 0H, B '
such that
th(re exists a map j :H -+ K o
(B.) Let C be an object and let IT:B -+ C be a map in the
category C. Then
(S) (C,IT) is a cokernel of f, if and only if the fol-
lowing three conditions hold:
(B.I) IT is an epimorphism, and
(B.2) Given any pair (H,j) where H is an object
in C and j:B -+ H is a map in C such that
j of=OA,H ' there exists a map
tha t j 0 0 IT = j .
j :C'" H o such
Proof: Assume condition (a). Then conditions (A.O) and (A.2)
26 Section 3
follow by Definition 3. On the other hand, let 0 be an ob-
ject and h,k:D~K be maps such that 1 0 h=l ok. Let
j=1 0 h=1 0 k. Then j:D->A is a map, and foj=f 01 oh=
OK, B 0 h = 00, B. By the universal mapping property, condition
(2) of Definition 3, we therefore have that there exists a
unique map j :0 ~ K o
such that But both h and k
obey this condition. Therefore h = k. This proves (A.2).
Conversely, suppose that conditions (A. 0), (A.!) and (A.2)
all hold. Then by (A.O), we have that f 01 = 0K,B. And, if
H is any object and j:H ~ A any map such that f oj = °H, B ,
then by (A.2) there exists a map jo such that 1 ojo=j·
Suppose that j~ is another such map, i.e. I that 1 oj~=j.
Then 1 0 jo = 1 j~, and since by (A.l) is a monomorphism,
this implies j 0 = j ~. Therefore there is a unique map j 0 as
in condition (2) of Definition 3. This proves (4).
The equivalence of (S) with (B.O), (B.l), and (B.2), fol-
lows from the equivalence of (a) with (A.O), (A.l) and (A.2),
by passing to the dual categoLY. Q.E.D.
Corollary 2.1. Let C be a pOinted category and let f:A ~ B
be a map. Then
(A.) If a kernel (K,l) of f exists, then the class of
all kernels of f is a subobject of A in the sense
of section 2, Definition 1.
(B.) If a cokernel (C,n) of f exists, then the class
of all cokernels of f is a quotient-object of B
in the sense of section 2, Definition 1°.
Proof: Suppose that a kernel of f exists. Then if (K,l)
and (K',l') are both kernels of f, then by Proposition 2,
Abelian Categories 27
(A.l) , we have that (K, 1 ) and (K', 1 ') E; M A
. By Proposition
2, (A.O), applied to (K',l'), we have that f ol' =OK',B
And since (K,l) is a kernel of f, by condition (A.2) we
have that there exists a map l' such that o There-
fore, by definition of the pre-ordering on MA (see section 2),
we have that (K', 1 ,) .s. (K, 1 ) in MA. Applying the same argu-
ment using that (K',l ') is a kernel of f, we obtain simi-
larly that (K, l) .s. (K' ,l ,) in MA. Therefore (K,l) is
equivalent to (K',l ') in MA , i.e., lie in the same sub-
object.
Conversely, if (K,l) is a kernel of f, and if (K',l ')
and (K,l) lie in the same subobject of A, i.e. I if
(K' , 1 ') 'V (K, l), then by section 2, Lemma 1, we have that there
exists a unique isomorphism e:K' ~ K such that lo e = 1 '. And
therefore clearly by Definition 3 (K',l ,) is also a kernel of
f. Therefore the set of kernels of f are an entire subobject
of A, as asserted.
The latter part of Corollary 2.1 follows from the former
part by passing to the dual category. Q.E.D.
If now C is a pointed category, and f:A .... B is a map
in C, then we define the kernel of f, denoted Ker f , to
be the class of all kernels (K, l) of f, if a kernel of f
exists. And we define the cokernel of f, denoted Cok f, to
be the class of all cokernels (C,n) of f, if a cokernel of
f exists. Then, by Corollary 2.1. if a kernel of f exists,
then the kernel, Ker f, is a subobject of A; and if a co-
kernel of f exists, then the cokernel, Cok f, is a quotient:-
object of B.
28 Section 3
Proposition 3. Let C be a pointed category and let Z be
an object in C. Then the following eight conditions are equiva-
lent.
C (Z. Z) .
( 3) Home(Z.A) has cardinality one. for all objects A
in A.
in A.
(5) Z is the direct sum of the empty set of objects.
(6) Z is the direct product of the empty set of objects.
(7) There exists (equivalently: For all) objects A in
A. there exists a map l: Z "* A. such that (Z. l) is
a kernel of idA:A -> A.
(8) There exists (equivalently: For all) objects A in
A. there exists a map 71:A-+Z. such that (Z.71) is
a cokernel of idA:A-+A.
Note: In condition (7) (respectively: (8)) one can replace the
map idA:A-+A with any monomorphism (respectively: epimorphism)
with domain (respectively: range) A.
Proof: Obviously (3)~(2)~(1). Suppose (1). Then if A is
any obj ect in the category A and f. g E Horne (Z. A), 'then
f = f 0 id = f 00 '" 0 = g 0 OZ. Z '" g 0 idz = g. Therefore Z Z.Z Z.A
HomC(Z.A) has cardinality 5-1. Since 0Z,AEHome(Z,A), we
have that Horne (Z, A) f 0. Therefore card Horne (Z, A) = 1, proving
(3). Passing to the dual category, we see likewise that
(4)~(2)~(I)~{4). We leave it as an easy exercise to the
reader to prove the equivalence of the remaining conditions and
Abelian Categories 29
(1), (2), (3) and (4). Q.E.D.
An object Z in a pointed category C that obeys the
eight equivalent conditions of Proposition 3 will be called
a zero object. It is immediate that, if a zero object exists,
then it is unique up to a unique isomorphism. We shall use the
symbol 0 to stand for the zero object.
Remarks: 1. If a zero object exists in the pointed category
C, then every object A has a smallest subobject--namely, the
equivalence class of the element where Z
is any zero object in C. We shall denote this smallest sub-
object of A by O.
2. In a pointed category C, if A and Bare
objects in C, then when there is no danger of confusion, we
shall denote the point 0A,B~ HomC(A,B) simply as O.
3. If C is a pointed category that does not have
a zero object, then we can "adjoin a zero object to C" if we
wish. Namely, let Z be such that Z is not an object of the
category C. Then define C' to be the category, having for
objects all the objects of C together with the one additional
object Z. Then define
(a set of cardinality one),
are objects of C,
if A or B = Z.
Then there exists a unique way of determining a composition such
that C' becomes a category and such that C is a sub-cate-
gory. And C' is a pointed category with a zero object.
We do not, however, insist on making this construction.
4. Let C be a category (not necessarily pointed),
30 Section 3
and let Z be an object in C. Then condition (3) of Proposi­
tion 3 is equivalent to condition (5). If Z obeys these
equivalent conditions, then Z is called a left zero object in
the category C. Passing to the dual category, we have like­
wise that conditions (4) and (6) of Proposition 3 are equiva­
lent. Z is called a right zero object if it obeys these two
equivalent conditions.
5. Let C be an arbitrary category. Then the fol­
lowing three conditions are equivalent.
(1) There exists an object Z in C that is
both a left zero object and a right zero object.
(2) There exists a left zero object
and a right zero object ZR in C, and HomC(ZR'ZL)
empty.
(3) There exists a right zero object and a left
zero object in C, and for all objects A,B in C, we have
that HomC(A,B) t 0.
a zero object.
6. By Remark 5 above, if Z is an object in a cate­
gory C, then Z obeys condition (1) of Remark 5 iff the cate­
gory C is pointed and Z is a zero object.
Example: The category of sets has a left zero object (the empty
set) and a right zero object (any set of cardinality one) .
Therefore, by Remark 5 above, the category of sets does not
have a zero object. More strongly, by Proposition 3, it fol­
lows that the category of sets is not pointed (a fact that's
obvious anyway). Similarly, the category of non-empty sets
Abelian Categories 31
has a right zero object (namely, any set of cardinality one),
but does not have a left zero object. Again, by Proposition 3,
it follows that the category of non-empty sets is not pointed
(a fact that is also obvious) .
Corollary 3.1. Let C be a pointed category. Let f: A -+ B be
a map in the category C. Then
(A.) If f is a monomorphism then Ker f exists iff a
zero object exists, in which case Ker f = 0.
(B.) Conversely, if the category C is additive, and if
either a zero object or Ker f
phism iff Ker f = 0.
exists, then f is a monomor-
Proof: (A.) If f is a monomorphism and if a zero object Z
exists then one verifies using Definition 3 that (Z,OZ,A) is
a kernel of f. On the other hand if a kernel (z,d of f
exists then Z obeys condition (7) of Proposition 3, as modi­
fied in the Note to Proposition 3, and is therefore a zero
object.
(B.) Suppose that the category C is additive and let
f be a map in C such that a kernel (z,d of f exists
with Z a zero object. To show that f is a monomorphism.
In fact, let D be an object in C and h,k:D-+A be
maps such that f ° h = f ok. Then f ° (h - k) = f ° h - f ° k = 0,
so that by the universal mapping property of Definition 3 there
exists a map t:D-+Z such that h-k=l °t. But since Z is
a zero object, both and t are zero maps, and therefore
h - k = lot is a zero map, and therefore h = k. Q.E.D.
Remark. Part (B.) of Corollary 3.1 does hold for some pointed
32 Section 3
categories that are not additive (e.g., the category of groups),
but fails to hold for some pointed categories that are not ad-
ditive.
Example. If C is the category of pointed sets, then let
(8,so) be any pointed set with
T={O,l} and let to={O}. Let
8 of cardinality.:: 3, let
f: S ~ T be the function such
that f (s ) = t , o 0
f (x) = 1, all xE;S-{s}. Then f is a map o
of pointed sets, Ker f = 0, and yet f is not a monomorphism.
Definition 4. Let C be a pointed category and let f:A~B
be a map in C. Suppose that a cokernel (C,n) of f exists.
Then an image of f is a kernel (I,k) of n. Then by Corol-
lary 2.1, part (A.), and by section 2, the dual of Lemma I, if
(C',n') is another cokernel of f. then there exists a unique
isomorphism 6 such that n' = en. It follows readily that the
definition of "an image of f" is independent of the choice of
a cokernel (C.n) of f.
Thus. for an image of f to exist, it is necessary and
sufficient that a cokernel (C,n) of f exist. and that for
some (equivalently: all) cokernels (C,n) of f. that a
kernel of n exist.
A co-image of f is an image of f considered as a map
in the dual category from B into A. Using methods similar
to that of the proof of Corollary 2.1, we see that
Proposition 4. If f:A .... B is a map in a pointed category, and
if an image (respectively: a co-image) of f exists. then
the class of all images (respectively: co-images) of f forms
a subobject of B (respectively: quotien~object of A) in
the sense of section 2. Definition 1 (respectively: section 2,
Abelian Categories 33
Definition 10 ) •
The proof is elementary and is left as an exercise for the
reader.
If C is a pointed category and f:A ~ B is a map in the
category C, and if an image (respectively: co-image) of f
exists, then by the image (respectively: the co-image) of f,
denoted 1m (f) (respectively: Coim(f», we mean the subob-
ject of B (respectively: the quotient-object of A) consis­
ting of the class of all images (respectively: all coimages)
of f in the sense of Definition 4 above.
We now make the conventions made in section 2, just after
Definitions 1 and 10. I.e., for every obj. A in C, fix a com­
plete class of reps. for the subobjects of A, and also fix a
complete class of reps. for the quotient-objects of A.
Therefore, in particular, iff: A ~ B is any map in C,
such that Ker f (respectively: Cok f. 1m f. Coim f) exists,
then we have a distinguished representative element (K,l)
(respectively: (C, n), ( I , k), (J , p) ) in Ke r f (respectively:
Cok f. Im(f), Coim f). We often call K (respectively: c, I,J)
the kernel (respectively: the cokernel, the image, the coimage)
of f, and use the notation Ker f (respectively: Cok f, 1m f,
Coim f) for this specific object K (respectively: C,I,J).
The map l:K ~ A (respectively: n:B ~ C, k:I -+ B, p:A -+ J) is
then called the canonical injection (respectively: the canoni­
cal projection; the canonical injection; the canonical projec­
tion) .
Theorem 5. Let C be a pOinted category, and let f:A ~ B
be a map in C such that Ker f, Cok f, 1m f and Coim f all
34 Section 3
exist. Let 1: Ker f ->- A, k: 1m f ->- B be the canonical inj ections
and let 71: B ->- Cok f and p:A ->-Coimf be the canonical projec-
tions. Then there exists a unique map
a:Coim f ->- 1m f
such that the diagram
(Ker f)~} f 'r~ICOk f) (Coim f)_a_:> (1m f)
is commutative.
Proof: We have f 01 ~O. Therefore, by definition of the co-
image, since (Coim f, p) is a cokernel of 1, there exists a
map e: (Coim f) ->- B such that e 0 p ~ f. Then 71 08 0 P = 71 0 f = O.
Since p is an epimorphism this implies that
(1) 7108=0.
Since (1mf, k) is an image of f, we have by Definition 4
that (1m f, k) is a kernel of 71. Therefore by equation (1)
we have that there exists a map a: Coim f -> 1m f such that
8 = ka. But then
kap=8p=f,
proving existence of u. Now suppose that 8 is another map
from Coim f into 1m f such that
kBp=f.
kap=f=kBp.
Since k is a monomorphism and p is an epimorphism, this
implies that a = B. Q.E.D.
If the hypotheses of Theorem 5 hold, then the map
)
is a map, then Cok f = T/f (S) (more precisely,
Cok f is (T/~, {to})' where "~' is the equivalence rela-
tion on the set T, t ~ t' iff either t = t' or both
t,t' ~f(S); and where to is the image of to under the natural
map: T~T/~). Imf =f(S),
-1 Coimf =5/f (to) (i .e. ,
the set-theoretic image, and
glued to a point") .
Therefore, in this pointed category, the factorization map
a: Coim f ~ 1m f is always an epimorphism-but is not, in gen-
eral, a monomorphism.
In the category of groups, iff: G ~ His a homomorphi sm
of groups, then -1
generated by f(Gj~ 1m f = (the normal subgroup generated by
f (G», and Coim f = G/Ker f '" f (G) , the set-theoretic image.
Therefore, in this pointed category, the factorization map
a:Coim f ~ 1m f is always a monomorphism, but is not always
an epimorphism.
In the category of abelian groups, if f:A ~ B is a homo-
morphism of abelian groups, then Ker f = f- l (0) , Cok f =
B/f (A), Im(f) = f (A) and Coim f = A/Ker f. Therefore in
this additive category, the factorization map a :Coim f -+ 1m f
36 Section 3
is always an isomorphism.
Example 8. Let C be the category of all topological abelian
groups and continuous homomorphisms. Then C is an additive
category, and the description of Ker, Cok, 1m, and Coim is
similar to case of the category of (abstract) abelian groups,
above. However, if f:A-+B is a continuous homomorphism of
topological abelian groups, and if C!: Coim f .... 1m f is the
factorization map, then C! is always both an epimorphism and
a monomorphism, but is not in general an isomorphism (since
Coim f has the quotient topology from A while 1m f has the
induced topology from B).
Example 9. Let C be the category of all Hausdorff to~ abelian
groups and continuous homomorphisms. Then if f:A -+ B is a map
in C, we have that -1 Ker f = f (0) with the induced topology
from A. Cok f = B/f (A), the quotient-group of B by the
closure of the set-theoretic imag~ with the quotient topology
from B. 1m f = f (A) , the closure of the set-theoretic image
with the induced topology from B, and -1 Coimf =A/f (0), the
quotient-group of A with the quotient topology from A. In
this additive category, the factorization map of a map f is
always a monomorphism,but is not in general an epimorphism.
The next definition is very important.
Definition 5. An abelian category is an additive category,
such that
(ABl) Finite direct sums of objects exist.
(AB2) Kernels and cokernels of maps exist.
(AB3) If f:A-+B is a map in A, then the factorization
map C!: (Coim f) -+ (1m f) is an isomorphism.
Abelian Categories 37
(ABl.l) There exists a zero object,
and (ABl. 2) If A and B are objec~ in A, then there
exists a direct sum A 6l B.
2. Since Imf =Ker(Cokf) , Coimf = Cok(Ker f)
axiom (AB2) implies that 1m f and Coim f exist.
3. In view of axiom (ABl) and Corollary 1.1, if a
category A admits an additive structure such that it is an
abelian category, then that additive structure is unique.
Therefore, an abelian category can be thought of, equivalently,
as being a category A, such that there exists an additive
structure on A such that axioms (ABl) , (AB2) and (AB3) all
hold.
4. Suppose that we have an additive category A such
that axioms (ABl) and (AB2) both hold. Suppose also that
If f:A -+ B is a map in A, then the factorization map
a:(Coimf) -+Im(f) is both a monomorphism and an epimorphism.
Then is A an abelian category? To the best of my knowledge,
this question has not yet been settled. (See, however, the
last paragraph of section 4 below.)
Example. Of Examples 1-9 above, the only one that is an abelian
category is Example 3, the category of abelian groups.
Example 10. The category of all left modules over a fixed ring
with identity R is an abelian category.
Example 11. The category of all sheaves of abelian groups S(X)
on a fixed topological space X is an abelian category.
Example 12. If A is a category, and if C is a category
that isa set, then we let AC, the exponent category, denote
38 Section 3
the category having for objects all covariant functors from
C into A, and for maps all natural transformations of func-
tors. (This useful notation is original to Joseph D'Atri of
Rutgers University, Newark.)
Then if A is an abelian (respectively: additive;
pointed) category, and if C is any category that is a set,
then AC ~s an abelian (respectively: additive; pointed) cate-
gory.
Example 13. Let A be an abelian category. Then the dual
category AO is an abelian category.
Example 14. Let A be an abelian (respectively: additive;
pointed) category. Then the category Co(A) of all cochain
complexes (en,dn ) indexed by all the integers is an abelian nE;?"
(respectively: additive; pointed) category.
Definition 6. Let A be an abelian category, and let
(1) f g
A -->B-->C
be a sequence of maps in the category A. Then we say that
the sequence (1) is exact at spot B iff Ker g = 1m f as
subobjects of B.
Example. Let A be an abelian category and let S be a cate-
gory that is a set. Then a sequence
F~G __ H
in AB is exact at spot G iff for every object B of S,
the sequence
is exact at spot G(B) in the abelian category A.
Similarly, if A is an abelian category, then a sequence:
A*-+B*-+C*
for every integer n, the sequence
n n n A -+B -+C
is exact at spot Bn.
A is exact at spot B* iff
In Example 11 above, in the category Six) of sheaves
of abelian groups on a fixed topological space X, a sequence
of sheaves of abelian groups in SiX) is exact at spot G
iff for every x E X, we have that the sequence of stalks:
F -+G -+H x x x
is exact in the category of abelian groups at spot G . x
For the rest of this section, we state several theorems,
for arbitrary abelian categories A, all of which are easy to
prove, or well-known, in the case that A is the category of
abelian groups. By the Exact Imbedding Theorem, see sect.4, we
therefore have these results in all abelian categories.
Proposition 7. Let A be an abelian category, and let
f g (1) A-->E~C
be a sequence of maps in A. Then Ker f = 1m g as subobjects
of B iff Coker f = Coim g as quotient-objects of B.
40 Section 3
Note: By Definition 6 above, an equivalent statement is:
"Then the sequence (1) is exact at spot B in the abelian
cat

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