Category Theory, Homology Theory and Their Applications II

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z~irich

92 Category Theory, Homology Theory and their Applications II

1969

Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, June 24 - July 19,1968 Volume Two

Springer-Verlag Berlin. Heidelberg. New York

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. 9 by Springer-Verlag Berlin- Heidelberg 1969 Library of Congress Catalog Card Number 75"75931 Printed in Germany. Title No. 3698

Preface

This is the second part of the Proceedings of the Conference on Category Theory, Homology Theory and their Applications,

held at the Seattle Research Center of the Battelle

Memorial Institute during the summer of 1968. The first part, comprising 12 papers, was published as Volume 86 in the Lecture Notes series. Following the Table of Contents,

there is appended a list of papers to be published in

subsequent volumes. It is again a pleasure to express to the administrative and clerical staff of the Seattle Research Center the appreciation of the contributors to this volume, and of the organizing committee of the conference,

for their invaluable assistance

in the prepa-

ration of the manuscripts.

Cornell University,

Ithaca, January,1969

Peter Hilton

T a b l e of C o n t e n t s

H. B. B r i n k m a n n

Relations

S. U. C h a s e

Galois objects

P. D e d e c k e r

T h r e e - d i m e n s i o n a l n o n - a b e l i a n c o h o m o l o g y for g r o u p s

R. R. D o u g l a s ,

for g r o u p s and for e x a c t c a t e g o r i e s and extensions of Hopf Algebras

H-spaces

C. E h r e s m a n n

C o n s t r u c t i o n de

K. W. G r u e n b e r g

C a t e g o r y of g r o u p e x t e n s i o n s

M. A. K n u s

Algebras graded by a group

F. W. L a w v e r e

Diagonal Arguments

S. M a c L a n e

Foundations

B. M i t c h e l l

On the d i m e n s i o n o f o b j e c t s a n d c a t e g o r i e s

F. U l m e r

.....

10 .

. . . . . . . . . . . . . . . . . . . . . . . . structures

libres

. . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

and Cartesian Closed Categories

for c a t e g o r i e s a n d sets

Hochschild dimension E. R o o s

I

32

P. J. H i l t o n

a n d F. S i g r i s t

J.

......

Kan extensions,

cotriples

74 105 117

. . .134

. . . . . . . . . .

146

III

. . . . . . . . . . . . . . . . .

Locally Noetherian categories

65

. . . . . . . . . . . . .

and A n d r 6 ~ o ~ o m o l o g y

....

165 197 278

Papers to appear in future volumes

J. F. Adams

Generalized cohomology

J. Beck

On H-spaces and infinite loop spaces

D. B. Epstein and M. Kneser

Functors between categories of vector spaces

D. B~ Epstein

Natural vector bundles

P. J. Freyd

New concepts in category theory

J. W~ Gray

Categorical fibrations and 2-categories

R. Hoobler

Non-abelian sheaf cohomology

M. Karoubi

Foncteurs d6riv~s et K-th~orie

F. E. J. Linton

Relative functorial semantics

E. G. Manes

Minimal subalgebras for dynamical triples

P. May

Categories of spectra and infinite loop spaces

P. Olum

Homology of squares and factorization of diagrams

-I-

RELATIONS

F O R GROUPS

A N D F O R EXACT C A T E G O R I E S

by Hans-Berndt

Relations

(Correspondences)

defining

connecting

spectral

sequences.

and L e i c h t

among

Puppe

answer

the q u e s t i o n

satisfying

Applied

as desired.

fractional

calculus.

A

category

note

is e x a c t

of a cokernel

unique

up to

(1)

The

former

categories

chasing

and

for

or d i f f e r e n t i a l s

MacLane,

a n y e x a c t (1) c a t e g o r y and thus b e i n g

p . 1 8 and A x i o m s

Puppe,

in

Hilton

for groups.

categories

(1)

category,

A construction

We give

was

out

mentioned in

we r e c o v e r

[i0!

a positive for any

we o b t a i n the

of r e l a t i o n s

[4] then d e s c r i b e d

for some m i n o r

the case

, if it has

and

changes

a calcustandard

for a b e l i a n

this case b y m e a n s

independently

functorial

an a b b r e v i a t e d

is i n c l u d e d

a zero o b j e c t

and

of a

also c o n s i d e r e d

after

choice.

[I0].

p.9-19]

version

readily

Standard or in

No addition

of

[1].

now.

if e v e r y m o r p h i s m

The d e c o m p o s i t i o n

[9~ 1.7-15,

is q u a s i e x a c t

of Buchsbatun.

of g r o u p s

b y a kernel.

in M i t c h e l l

terminology

shown

can be c a r r i e d

of g r o u p s

the p r o b l e m

a calculus

for the p u r p o s e s

It w a s also

exact

to the c a t e g o r y

admits

can exist.

below

small

(w

[3].

followed

found

KI-3]).

described

Hilton

suited

of r e l a t i o n s

and c o l o c a l l y

is, e x c e p t

isomorphism

m a y be

Lambek,

[I0].

are such that

sition

categories

b y Riguet,

Applied

For exact

[6] and C a l e n k o

The p r e s e n t

I.

studied

axioms

relations

w a s g i v e n b y Puppe

made

operations

such c a l c u l u s

to a l o c a l l y

categories

The c h a n g e s

for d i a g r a m

cohomology

The c o n s t r u c t i o n

of h o m o m o r p h i c

by Leicht

tool

order

whether

[10~4.18,

that at m o s t one

lus of r e l a t i o n s category

certain

(Puppe

to the problem.

category.

a useful

others.

in the b e g i n n i n g p.17]

higher

They have been

posed

of r e l a t i o n s

4.15,

morphisms,

provide

Brinkmann

is a compo-

turns

properties

out to be of e x a c t

[2].

is r e q u i r e d

as in the exact

_

Exact additive

categories

group

we o b t a i n we w o u l d

obtain

of r e l a t i o n s

compatible

: K

)K

which Z

consists

K

Beispiel

w i t h one object. a

of a c a t e g o r y

A, p.23].

Adjoining

(natural)

as its m u l t i p l i c a t i v e

satisfying

we w r i t e

[101w

does n o t e v e n admit

w i t h composition)

(conversion)

simplicity

[1Osl.3,

category

(Puppe

is a c a t e g o r y

a field h a v i n g

(i.e.

For

of i n t e g e r s

an exact

2. A c a t e g o r y K

Z

n e e d not be a b e l i a n

a zero o b j e c t

addition,

K, a

(natural)

order

relation

order preserving

A~ = A

and

for o b j e c t s of d a t a

since

else

group.

and a c o n t r a v a r i a n t

for the triple

The

f~

= f

(K, i

The a r r a n g e m e n t inition.

The m a p s

(Identities

Lemma

f < g

g

< ff~g

MK

hence

restrict

Examples: (abelian

of

are maps).

2.3

Proof:

of the p i c t u r e

to

and

~ fg#g

K

refers

form a s u b c a t e g o r y Obviously

f,g

( f

~ MK

MRS =

for

(sets),

groups).

(2) fg m e a n s first

f

then

imply

g

involved

in the def-

contains

all o b j e c t s

of

f = g.

I < f f@,

no further

f ~ MK_, f#

MRS,

MK_, w h i c h

of d u a l i t y

we h a v e

using

so far c a r r i e s

MK_, since

to the two types

= S,

f~

< g

structure

and

g g

< I.

than b e i n g

a category

(@

does not

n e e d not be a map).

(pointed

sets),

MRG = G

(groups),

M R A b = Ab

-3-

3.

Let

out

that

[10]!

K

be a c a t e g o r y MK_

4.8,

precise

(locally

p.15]).

We w i l l

form of these

especially MK

is a

nice

[10! 4.15,

K

of relations.

Examples:

The axioms

for

K3b,

(E.g.

however,

replace

expressed and

K3

by:

Also p.

15-16]

satisfied

for

that

K1-3

RAb.

K

such

u = m ee m

that

u

= u

(subquotient)

(Axioms

to b o t h e r

K,

KI-3

the reader

the e m b e d d i n g

of

MK

[10] w i l l be c a l l e d

for

in this case m a y be w e a k e n e d u

on

is up to i s o m o r p h i s m

of

Except

are i m p o s e d

category

details

and that

axioms

axioms

exact

it turns out

satisfying

Every

in the form

= u

K3b

2

(symmetric

such that

m,e

of

[IO]!

with

the

into

K

by

a pseudoexact

satisfied

recover

uniqueness

idampotent)

are m a p s

is

determined

t h e y are also

so as to still

it turns

may be

and

mm

= I

exists

a

e e = 1.

4. We can n o w state

Theorem

C

pseudoexact

sets.

the

following

result:

4.1 Let

be a l o c a l l y

category

and c o l o c a l l y

of relations

K

small

such that

The s m a l l n e s s c o n d i t i o n

of the t h e o r e m

The c o n s t r u c t i o n

is b e l i e v e d

pendent

below

exact

Then

category.

there

C = MK.

is o n l y u s e d

to insure

to g i v e an h o n e s t

that

mathematical

K

has hom

object

inde-

of this assumption.

Let

E_xx d e n o t e

categories

and

both with

their

w

are

small)

not go into e n o u g h

[10! 4.8-4.12, p.17].

If further

and c o l o c a l l y

axioms.

category

RG.

of relations.

Psex

4.1 r e s u l t s

K Ex~----~--~MPsex

the

(illegitimate)

(illegitimate)

appropriate

functors.

in that we d e f i n e d

is an e q u i v a l e n c e

To m o t i v a t e category

the

category

We c o n s i d e r

of l o c a l l y

a functor

Ex

categories

to the r e s u l t s K

~Psex

of

such that

(M is the e x t e n s i o n

to be given, a diagram

and c o l o c a l l y

of p s e u d o e x a c t

With regard

of c a t e g o r i e s

the c o n s t r u c t i o n

of relations.

category

w e assume

that

of

K

small

exact

of r e l a t i o n s

[10] m e n t i o n e d

in

the pair "maps"

to a functor).

is a p s e u d o e x a c t

-4-

f

-

g

(4.2)

in

MK

and analyze when

(4.3)

f@g

< g t f t@

(4.4)

f@g = g'f'@

and

We have Lemma 4.5

f~g < g'f'e,

iff the diagram commutes in

Lemma 4.6

f~g = g'f'#,

iff the decomposition in

MK (fg' = gf').

MK_ of the diagram to

(4.7)

yields in

(I) a pushout,

(2) a pullback and (3) and (4) bicartesian

(pullback and pushout)

MK . A proof is given in [2], the hypotheses may be weakened to the form mentioned

in w

Examples

(See [2; Anhang]).

A square as 4.2 in If

C

MK_

such that

fSg = g'f'~

will be called fully commutative.

is abelian, Hilton [4; Theorem 3.3, p.258] described fully commutative squares

by the existence of a completion to a commutative diagram

(4.8)

such that the inner square is bicartesian~

This is readily seen to be equivalent to

-5-

the c h a r a c t e r i z a t i o n

5_:.

We are

morphism cwordC

thus A

given

led to the

f )B

in

of c w o r d s

Every

on

object

Every map For

every

If

a

C

We c a n h e n c e

of

A---~fB map

~

nor

having

).

concatenation

of

from

A

This

Let

R

the

will

will

by

9 f#

a l s o be d e n o t e d

from if

C

~

.

A,

is a c w o r d

from

B

to

~ | ~

is a c w o r d

finite

diagrams

C

from

(A

(5.3)

(f | g,fg)

) A,A)

f

abbreviates

r extends

~ R

for

all

to an a n t i a u t o m o r p h i s m

fl

between

A.

cwords

defined

by

A, f

)

*

mI (5.4)

(m I ~ m 2,m I

(5.5)

(m ~ e ~ , e 'e ~ m')

(3) The

terminology

| m~)

is due

~ R

g "

,

*

;

m2

, if

is a p u l l b a c k

if

is b i c a r t e s i a n

~ R

to Freyd.

to

f#

We h a v e

~ .


category

is the i d e n t i t y

of

f ~ C

under

by

KC:

(a*,~*)

= cwordC/~

on objects.

this

~ R.

Also

functor w i l l

f.

5.6 Let

relation

C
.>-
>- ~

>->

> .~

squares

are

.,

.,

fully c o m m u t a t i v e ~.~

w

5.8 then

(2).

in

>.,

4.5 and 4.6 are true



9

diagram

of r e l a t i o n s

every relation

[5! P r o p o s i t i o n s

be c o n s t r u c t e d

(2) in 6.1

in s t a r t i n g

formed

as p u s h o u t

5.7.

for g r o u p s in

1,2,

for groups,

theorem

RG

(6)

to the

p. 47,48]).

however,

and e n o u g h form

None

since

of the

5.8 fails

n e e d not be b i c a r t e s i a n

for

(or fully

A pushout

o

for groups, commutative

where

m

(pushout

in 5.6 is

RG

is n o t n o r m a l ~ e = cokm,

pullback

be a p u l l b a c k ~ m = kere

and h e n c e

!). K G

cannot

with order

be

fully

relation

as

.

Certainly

we c o u l d h a v e

directly

using

position

is a s s o c i a t i v e .

.
%ox(~) k

the e x t e n s i o n

contains

that

V(S)

whose

n th p o w e r s

= x-l~(x). all n t h

is p r e c i s e l y

which

S

is

Galois

group

are

in

a normal, J,

V(S)

• J

Now,

roots

the p a i r i n g

'> U n

1

J in

defined has

for

field

introduced

and

exponent

elements

the s p e c i a l extension above

of

is de-

by

S, i t is e a s y

of n o n - z e r o

Thus,

separable

determines,

since

of

the s e t k~

~S

-

~ 0

the m a p

an e l e m e n t

a

that

in

and

by

Proposition

) H o m ~ ( J , U n)

the i n c l u s i o n

a one-

a coboundary

U(S)

homo-

to the G a l o i s

X],

Theorem

> V(S)

therefore

n,

an a r b i t r a r y

regard

in

that

is

left

Un

with

x

in

exponent

U(k) ~

U(S),

exists

denote

S

[19, C h a p i t r e in

U(k)

of

>

of H i l b e r t ' s

there

that

is in %o: J

for a l l

We m a y

fact

that

%OCt) = x-l~(x)

~S:

x

values

has

k, w h e n c e

for e x a m p l e ,

J

J

i n the s u b g r o u p

the s u b r i n g

is p r e c i s e l y

Suppose

cocycle

Since

n to see

of

case k

coincides

S in with with

22

that arising in the c l a s s i c a l

formulation

of K u m m e r

(see, e.g.,

The m a p p i n g

(9) assigns

cl(S)

[i, pp.

in

X(Jk)

Ex~(Ho~J,U

19-22]).

the e l e m e n t

n) , U(k)),

cl(~ s)

theory to

in

and the techniques

of

[i, pp.

19-22]

can then be used to show that this m a p p i n g is a w e l l - d e f i n e d isomorphism

of abelian

(3) becomes

more apparent

H o m ~ ( J , U n) ~ A(kJ,k)

groups.

The relation b e t w e e n

if one observes

= A((Jk)*,k),

A(

)

(9) and

that denoting k-algebra

h omomorphi s ms. In order to obtain in a quite tative

similar

fashion.

We define

such that

x-lu(x)

If

(7a) .

x

is in

formula

V(S)

V(S)

Routine

R

V(S),

be a commutative,

antipode,

for all

is a subgroup

and

we define

of

u in A*, U(S)

a mapping

S

be a Galois x

in

with

containing ~x: A*

cocommu-

> R

U(S) u(x) U(R). by the

-

computations,

just discussed,

algebras,

= x-lu(x) entirely show

that

(u in A*) similar ~x

to those of the special

is a h o m o m o r p h i s m

of R-

and the sequence b e l o w is e x a c t 0 ~>

where

A( )

arrows

denote

tively.

A

(i), we p r o c e e d

to be the set of all

is in

~x(U)

case

Let

finite H o p f R - a l g e b r a with

A-object.

as in

the i s o m o r p h i s m

denotes

U(R)

> V(S)

~ A(A*,R)

R-algebra homomorphisms,

the i n c l u s i o n

map and the map

(i0)

and the u n l a b e l e d x

> ~x'

respec-

In this case the latter map is not in g e n e r a l surjective.

- 2S-

Now,

the left A * - m o d u l e

in the u s u a l way,

structure

a right A*-module

on

structure

S

induces,

on

S* = H o m R ( S , R ) .

A

F o r the s p e c i a l

case

in w h i c h

the s u r j e c t i v i t y

of the m a p

f ollows.

generates

If

z

is an R - a l g e b r a

S* V(S)

S*

homomorphism,

as r i g h t A * - m o d u l e s ,

A*

> ~(A*,R)

m a y b e s e e n as

as an A * - m o d u l e we define

~: A* ---> R

and

f: S*

~ R

by the

equation f(zu) S

is a f i n i t e l y

exists

a unique

( ): S* @ S shown

that

generated x in S

> R x

is in

V(S)

T

and

in g e n e r a l

faithfully

can be c h o s e n

.

R-module,

f = (

and thus

.,x),

pairing.

there

where

It can t h e n be

9 = ~x" S*

n e e d n o t be i s o m o r p h i c

it is at l e a s t a p r o j e c t i v e

The techniques

be u s e d to s h o w t h a t for s o m e

projective

such that

as a r i g h t A * - m o d u l e ,

of r a n k one.

(u in A*)

is the d u a l i t y

Although A*

= ~(u)

of,

for e x a m p l e ,

S* | T ~ A* | T flat c o m m u t a t i v e

[5,

w

to

A*-module can then

as r i g h t A* | T - m o d u l e s R-algebra

T.

In fact,

to be of the f o r m r

T

=

H

(ll)

Rxi

i=l where

Xl,...,x r

the J a c o b s o n

are e l e m e n t s

radical,

is the l o c a l i z a t i o n subset g~nerated antee

easily

by

such that

of x i.

R

of

n o n e of w h i c h

R,

x I + ... + x r = i, a n d

at the m u l t i p l i c a t i v e l y The preceding

that a base-change

are in

from

R

paragraphs to

T

Ri

closed then guar-

renders

the

- 24

sequence

(i0)

a short

It is thus of s h e a v e s algebra

in t h e

S,

(ii)

easily those

and

that

form

x in S

this

collection

properties

thus

gives

rise

dual

to the

then

consider

category the

A

to a b e l i a n

groups

such

with

below

S | T

S

the

sheaves

for any

S

- F(d0),

(i = 0,i)

transformation

S

is a s h o r t

conditions

The

hold

exact

of

category,

0 . > F' - ~ in

[2]),

and

category We may in this F

d:

S

from --->

S

|

-

> S | T | T

is an a b e l i a n

to

a n d ele-

functor

covering

checks

dual

R-algebras.

(covariant)

F(S | T) F(d0)

a natural

to

on the

F(S

F

sequence

P>

| T 8 T)

defined

d O (s | t) = s | 1 | t, d 1 (s | t) = s | t | i. is s i m p l y

One

axioms

reader

of a b e l i a n

that,

is of the

S | T.

topology

is a

is e x a c t

> F(S) F ( d ) ) ~:

S

category of

0

refer

of c o m m u t a t i v e

object

the s e q u e n c e

we

T

satisfies

R-

an R - a l g e b r a

(for the d e f i n i t i o n

to a G r o t h e n d i e c k

An

topology.

of m a p s

sequences

a commutative

to be

in

groups. exact

where

x | 1

topology

of w h i c h

S

of

> S | T, to

short

Given

context.

S

goes

of a b e l i a n

to c o n s i d e r

a coverin~

of a G r o t h e n d i e c k

mentary

A

following

of the

sequence

natural

we define

homomorphism form

exact

-

by

A map

in

S

functors. and

F t'

a sequence

>

if a n d o n l y

-

0

if the

following

-

sequence

0

) F' (S)T(S);> F(S) p(S)-_ F'' (S)

(12a)

T,

-

is an e x a c t

Given there

sequence

of a b e l i a n

any o b j e c t

is a c o v e r i n g

s uch t h a t

S

of

d: S

p (S | T)(x)

II,

w

-

groups

A

and

for any o b j e c t

x"

> S | T

in F "

a n d an

S

of

(S) ,

A.

(12b)

x in F(S | T)

= F"(d)(x'~.

F or the p r o o f s Chapter

25

Note

of t h e s e

finally

facts we r e f e r

that,

since

S

to

[2,

is an a b e l i a n

*

category, Chapter then

we m a y d e f i n e

XII].

In p a r t i c u l a r ,

Ext~(F,G)

l ence c l a s s e s

E x t s(-,-)

m a y be v i e w e d

Now, A,

given

we n o t e

that

tative Hopf T-algebra A | T-object. cussion

It then

; G A

A | T with

is a s h o r t e x a c t the a b e l i a n

U(T)

> V( S | T)

V ( S | T) noting

T-algebra

and

of

G

in

> F

are a b e l i a n

as in

antipode,

sheaves,

g r o u p of e q u i v a of the f o r m -

(i0) a n d

and

easily

S

[13,

> 0

is a f i n i t e

> Vs

> U in

Vs(T)

T

an o b j e c t

commutative, S | T

cocommu-

is a G a l o i s

f r o m the p r e c e d i n g

dis-

S;

> A* where,

> 0 for

T

an o b j e c t

= V ( S | T)

and the m a p s

~ AT(HOmT(A

| T, T),T)

of

and

> A~*(T) = A(A*,T)

t hat the ro les

and

in

-

sequence group

S

follows

t h a t the s e q u e n c e

for e x a m p l e ,

as the a b e l i a n

; E

and

~S: 0

A,

F

of s h o r t e x a c t s e q u e n c e s 0

of

if

as,

homomorphisms) R, A,

S | T, r e s p e c t i v e l y .

S

are d e f i n e d

as in

(~ (i0),

are n o w b e i n g p l a y e d by The m a p p i n g

(i) s e n d s

deexcept

T, A | T,

cl(S)

in X(A)

-

to

cl(~s ) in Ext I(A*,U),

26

-

and c o m p u t a t i o n s

of an e s s e n t i a l l y

r o u t i n e n a t u r e e n s u r e that it is a h o m o m o r p h i s m of a b e l i a n groups. In order to o b t a i n the i n v e r s e map we t u r n to c o n s i d e r a t i o n s

Ext~(A*,U)

> X (A) ,

of a m o r e H o p f a l g e b r a i c nature.

G i v e n a short e x a c t s e q u e n c e ~: 0 ~ in

S,

U---~ E

> A*

we observe that the f u n c t o r

Hopf algebra

P~.

then guarantee R-algebra arises

H

The t e c h n i q u e s

that

E

U

is r e p r e s e n t e d by the

of f a i t h f u l l y

flat d e s c e n t

is l i k e w i s e r e p r e s e n t a b l e by some Hopf

(see, for example,

[14, p. III.

17-6]), w h e n c e

from a sequence A* ~

of a b e l i a n c o g r o u p objects

H in

> RZ A;

i.e.,

tative Hopf R - a l g e b r a s w i t h antipode. that

> 0

H

is an RZ-object,

with

commutative,

cocommu-

It is then easy to see

aH: H

> H | RZ

the c o m p o s i t e

map -

H AH-_ H @ H H @ ~> H @ I ~

Therefore,

with

Hn =

as r e m a r k e d earlier,

{z

in

H/all(z)

H

H =+~

|

= z |

tn}.

is a Z - g r a d e d R - a l g e b r a -

Hn

Since

it follows

of

is even a H o p f s u b a l g e b r a of

whence

the image of

H0 p

lies in

H0,

Hn

is a Hopf al-

gebra homomorphism, H,

that each

eH

thus p r o v i d i n g

is a s u b - c o a l g e b r a H.

Moreover,

a Hopf a l g e b r a

-

homomorphism morphism

p: A

2 7 -

We then obtain a c o a l g e b r a homo-

> H 0.

-

H I | A* H I |

P_~ H I | H 0

the u n l a b e l e d a r r o w d e n o t i n g the m u l t i p l i c a t i o n map of generated projective

the a p p r o p r i a t e r e s t r i c t i o n of

H.

Finally,

H1

is a f i n i t e l y

R-module, w h e n c e d u a l i z a t i o n of

yields an R-algebra homomorphism S = H I.

(13)

> HI

It turns out that

S =

aS: S (S,~ s)

> S | A,

(13) with

is a Galois A - o b j e c t , i

and the map cl(S)

(i) then sends

in X(A).

cl(~)

This c o m p l e t e s

of the i s o m o r p h i s m

in Ext~(~*,U)

to

our sketch of the c o n s t r u c t i o n

(I).

A final r e m a r k r e g a r d i n g

the G r o t h e n d i e c k

w h i c h w e use to d e f i n e our sheaves.

topology

N o t e that the c o v e r i n g s

in this t o p o l o g y are of a v e r y r e s t r i c t e d type;

in p a r t i c u l a r ,

the t o p o l o g y is m u c h c o a r s e r than,

the f a i t h f u l l y

flat t o p o l o g y on R

A.

is a local ring,

sheaf on groups)

A

Indeed,

for example,

for the s p e c i a l case in w h i c h

it is t r i v i a l l y v e r i f i e d that every pre-

(i.e., c o v a r i a n t

is a sheaf.

functor from

This is p r o b a b l y

in the d e d u c t i o n of the i s o m o r p h i s m

A

to a b e l i a n

the m o s t i m p o r t a n t step

(2) from the i s o m o r p h i s m

(i). The f o r e g o i n g m a t e r i a l

is a r e s u m e of a lecture pre-

s e n t e d at the B a t t e l l e c o n f e r e n c e on c a t e g o r i c a l cal algebra,

Seattle,

June-July

1968.

and h o m o l o g i -

D u r i n g the conference,

-

S. Shatz and D. Quillen isomorphism

28-

suggested

(i), the ingredients

Shatz formula, cohomological

a spectral

an alternate

of which include

sequence

classification

argument,

of principal

This method is similar to that of H. Epp using sheaves

in the faithfully

approach to the the Cartier-

and a well-known

homogeneous

spaces.

[8], who derived

flat topology,

for the special

case of group schemes whose duals are of multiplicative (i.e.,

A

has the property

fully flat commutative abelian group

J).

that

R-algebra

A | T ~ JT T

(i),

type

for some faith-

and finitely generated

29-

REF [1]

Artin, E., and Tate, J., Class Field Theory~ Princeton University mimeographed notes, Princeton,

New Jersey

(1960). [2]

Artin, M., Grothendieck Topologies~ Harvard University mimeographed notes, Cambridge,

[3]

Bourbaki,

(1962).

Chase, S. U., Harrison,

D. K., and Rosenberg,

"Galois Theory and Galois Cohomology

Chase, S. U., and Rosenberg,

Alex,

the Brauer Group, " Memoirs Amer 79, [6]

52; 15-33,

(1965).

"Amitsur Cohomology

and

Math. Soc., Vol. 52; 34-

(1965).

Chase, S. U., and Rosenberg, Kummer Theory,

Chase, S. U.,

Alex,

"A Theorem of Harrison,

and Galois Algebras,"

Vol. 2_~7; 663-685, [7]

Alex,

of Commutative

Rings," Memoirs Amer. Math. Soc. j Vol. [5]

(1962).

N., Alg~bre Commutativej Chapters I-II,

Hermann, Paris [4]

Massachusetts

Nagoya Math. J.,

(1966).

"Abelian Extensions

and a Cohomology

Theory

of Harrison, " 375-403 in Proceedings of the Conference on

Categorical Algebraj La Jolla (1965), Springer-Verlag, New York Inc., [8]

(1966).

Epp, H. P., Commutative Group Schemesj Harrison's Theorem,

and Galois Extensionsj Ph.D. Versity,

(1966).

thesis, Northwestern

Uni-

-

[9]

Harrison,

Columbia Uni-

(1965).

D. K., "Abelian Extensions of Arbitrary Fields,"

Trans. Amer.

Soc., Vol.

Math.

106; 230-235,

(1963).

, "Abelian Extensions of Commutative Rings,"

[11]

Memoirs Amer.

[12 ]

-

Giraud, J., Cohomologie Non-Abelienne, versity Notes

[lO]

30

Math.

Soc.

52; 1-14,

(1965).

Hasse, H., "Invariante Kennzeichnung Galoisscher K6rper mit Vorgegebener Galoisgruppe," Vol. 187; 14-43,

J. Reine Angew.

Math. ,

(1950).

[13]

MacClane, S., Homology, Academic Press, New York,

[14]

Oort, F., Commutative Mathematics,

[15]

Vol.

(1966).

Zeitsch., Vol.

105; 128-140,

(1968).

, "A Cohomological Description of Abelian Galois Extensions,"

to appear.

Shatz, S., "Cohomology of Artinian Group Schemes Over Local Fields," Ann.

of Math., Vol. 79; 411-449,

(1964).

Serre, J. P., Groupes Algebriques et Corps de Classes, Hermann, Paris,

(1959),

(Act. Sci. Ind. #1264).

, Corps Locaux, Hermann, Paris,

[19] Sci. Ind. [20]

Berlin,

Orzech, M., "A Cohomology Theory for Commutative Galois

[16 ]

[18]

(Lecture Notes in

15), Springer-Verlag,

Extensions," Math.

[17]

Group Schemes,

(1963).

(1962),

#1296).

Strasbourg University Department of Mathematics, Algebriques," 1965-66.

(Act.

"Groupes

Seminaire Heide lberg-Strasbourg Annee

-

[21]

Sweedler, Applied 276,

[22]

M. E.,

31

-

"The Hopf Algebra

to Field Theory, " J. of Algebra, Vol.

9 "Structure

Wolf,

as

8; 262-

(1968). of Inseparable

Annals of Math., Vol. 87; 401-410 9 [23]

of an Algebra

P. , "Algebraische

Theorie

Extensions,"

(1968).

der Galoisschen

Algebren,"

Deutscher Verlag der Wissenschaften, Math. Forschungsberichte III, Berlin,

(1956).

-

THREE

DIMENSIONAL

32

-

NON-ABELIAN

COHOMOLOGY

FOR

GROUPS

by Paul Dedecker

The

So far non-abelian cohomolog7 has been discussed in dimensions n < Z.

following is an effort to discuss dimension n = 3 and solves the m a i n difficulties sO H o w e v e r an n-dimensional theory for

that just polishing w o r k remains necessary. n > 3 remains

a remote

The present

report

I am grateful Wesleyan offered 1.

target. is a direct

continuation

for the hospitality

University,

of t h e C e n t e r

during the final phase

by the University

of [ 2 ] . for Advanced

of t h e r e d a c t i o n .

Studies,

Typing was

kindly

of K e n t u c k y .

Introduction. We shall assume

as developed few facts.

in e a r l i e r

to serve

dimensions

0 and

took so much

* Research belge

papers

as coefficient

1.

1958

group.

with the

[1],

category

time to take off.

[2], [ 3 ] .

theory,

Let us however

is

why the

1958 w e h a v e t o

22~ and

a

of G o n l y in

2 and that is the reason

g r a n t No.

n < 2,

recall

>7~1 o f G - g r o u p s

for a cohomolog 7 theory

As I have shown since

by NAT0 Research

et de T o p o l o g i e

n-dimensional

Then the category

I t i s n o t s o in d i m e n s i o n

supported

d'Alg~bre

is familiar

since

Let G be an arbitrary

satisfactory

theory

the reader

Centre

-

replace 3 1 7~2 -~ ~ i "

33

-

by the category 7~2 of crossed groups which has a forgetful functor This concept was introduced by 5. H. C. Whitehead under the n a m e of

crossed module.

V e r y roughly speaking a crossed group (or crossed module)is a

group endowed with an additional structure related to its group of automorphisms. In dimension 3 w e shall have to introduce a n e w category ~ 3

whose objects w e

shall call supercrossed groups and which has a forgetful functor ~ 3

-~ ~ 2 "

As is

well known, if the functor H0(G, -) has values in groups, this is not the case of the functor

HI(G,-)

whose values are often described

However

this can be made much more

category

of p o l y p i [1, I, II], [3].

a category functor

precise

Similarly

known as the category

as pointed sets

H2(G,-)

So f a r t h e s t r u c t u r e

to say enough to show that they are very curious

in t h e

t a k e s i t s v a l u e s in of v a l u e s

H 3 ( G , -) is n o t s u f f i c i e n t l y k n o w n t o g i v e t h e s e o b j e c t s a n a m e .

however

[6] S e r r e ) .

since this functor has values

the functor

of s p i d e r s .

(e.g.

and interesting

of t h e

I hope animals

with nice feathers. Of course and

the usefulness

(ii) it p r o d u c e s

sequence

q2 functorial

of t h i s c o h o r n o l o g y t h e o r y ~ i s t h a t

a cohornology exact sequence

in t h e c o e f f i c i e n t s

category,

associated

a notion respected

(i) it is f u n c t o r i a l

with a short exact

by the forgetful functors

Let us m e n t i o n h e r e that C i r a u d [5], [61 d o e s not p r o d u c e a t r u l y cohomology,

nor a true exact cohornology-sequence.

....

HI(A,,) _~ H2(A ,) -~o H2(A) ~ H2(A,,)

He has in fact a

s ection

associated

to a short exact sequence

of s h e a v e s

A' -~ A -~ A"

In t h e r e t h e

-

symbol

o

does not represent

theory

in 3 1 .

However

(showing that

our H 3 to the more

this defect in Giraud's

a nice paper

beautiful theory more

to generalize

This trouble

when one does not go into a good category

could be easily avoided and certainly

important

2.

occurs

but wants to remain

this and making Giraud's

-

a map but a relation.

t h i s k i n d of H 2 is n o t f u n c t o r i a l ) of c o e f f i c i e n t s

34

could be written

accessible.

general

clarifying

It w o u l d t h e n b e

situation considered

by Giraud.

Quick review of 3-coh.o_mology. We r e m i n d

a n d 13 a r e

that a crossed-group

(arbitrary)

groups,

of

13 o n t o H b y a u t o m o r p h i s m s ,

if

ae

13, h e H , (i) (ii)

H,

If a ~ 13,

morphisms commutative

A = (H, p,13, ~) i n w h i c h H

p is a homomorphism

p : H - ~ rl a n d 9 i s a n a c t i o n

subject to conditions

(i) a n d (ii) b e l o w .

t h e r e s u l t of a acting on (or t w i s t i n g )

If h , k (

A morphism

A is a system

then

h ( H,

h is d e n o t e d b y ah c H .

Pkh = k h k - 1 . then

p(ah) = a. p h . a

A -~ A' b e t w e e n t w o c r o s s e d

j : H -~ H' ,

In these,

y : 13-* 13'

compatible

-1

groups

is of c o u r s e

with the structures,

a p a i r of i.e.

making

the squares

ll

Y > If'

H

j >

13 X H

@>H

(2..1)

This defines the category

~2

H'

II'

X H' @'~l H'

and the forgetful functor ~2

-" ~ 1 "

(~7~I = the

- 35

c a t e g o r y of g r o u p s )

is d e f i n e d by

(H, p, rI,@) ~ T h e o r e m 2.1.

-

H .

T h i s f o r g e t f u l f u n c t o r h a s no s e c t i o n .

R o u g h l y speaking, this m e a n s g r o u p of a u t o m o r p h i s m s

that it is not possible to associate to a g r o u p a

in a functorial w a y .

If t h e t h e o r e m w e r e f a l s e , t h e r e w o u l d be no n e e d of c r o s s e d g r o u p s to d e f i n e non-abelian cohomology. exists.

To the b e s t of m y k n o w l e d g e no p r o o f of t h e t h e o r e m

H o w e v e r t h i s n o n - p r o v e d t h e o r e m is r e s p o n s i b l e f o r t h e s n a g g y

o in

Giraud's theory. For

G

an arbitrary group, w e denote C0(G,A)

= C0(G,H)

= H ,

0-cochains,

CI(G,A)

= CI(G,H)

= App(G,H)

C2(G,A)

= A p p ( G , II) X A p p ( G X G , H )

,

l-cochains, ,

2-cochains,

w h e r e App d e n o t e s t h e s e t of a l l m a p s in b e t w e e n t h e u n d e r l y i n g s e t s .

Moreover

we d e n o t e by Z 2 ( G , A ) t h e s e t of 2 - e o c h a i n s

c G):

(2.2.a)

~(S)h(t,u) 9 h ( s , t u )

(2.Z.b)

~(s).

~(t)

--

ph(s,t).

(~,h)

such that (s,t,u,...

= h(s,t) 9 h(st,u) , ~(st)

.

An a c t i o n (2.3) is d e f i n e d by

* : CI(G, A) • Z Z ( G , A ) --~ Z2(G, A)

- 86-

a*(~,h) ~'(s)

= (~',h')

,

a:G-~H,

= pa(s) 9 ~(s) ,

h'(s,t)

T h i s is a g r o u p a c t i o n , c o n s i d e r i n g

(~,h) e Z Z ( G , A ) ,

= a ( s ) . ~(S)a(t) 9 h ( s , t ) , a(st) -I .

CI(G, A) as a g r o u p in t h e o b v i o u s w a y .

The

orbits form a set (thick 2-cohomology) HZ(G,A)

= Z2(G,A)/CI(G,A) .

"

There are other actions

(z.3)

E] : 11 •

A) -.. CI(G, A)

(2.4)

: 11 •

-* Z 2 ( G , A )

w h i c h a r e d e f i n e d by t h e f o l l o w i n g f o r m u l a s in w h i c h ac

11,

aE

CI(G,A),

(~,h) c Z Z ( G , A ) : a

aWa

=

a',

~(s) = a . ~ ( s ) . a

a'(s)

-1 d e f = a~(s) ,

=

a(s);

h(s,t)

= ah(s,t) .

T h i s s a t i s f i e s t h e f o l l o w i n g " d i s t r i b u t i v i t 7 law!' av[a*~,h)]

= (~Oa) * ( a V ( ~ , h ) ]

w h i c h shows t h a t t h e a c t i o n V t r a n s f o r m s t h e c o r r e s p o n d i n g o r b i t of a V (~, h). WV2(G, A).

t h e w h o l e o r b i t of (4, h) u n d e r C I into

T h i s t h u s d e f i n e s a n a c t i o n of 11 onto

T h e s e t of o r b i t s in ]~2(G, A) u n d e r t h i s l a s t a c t i o n is c a l l e d t h e t h i n

2 - c o h o m o l o g y and d e n o t e d

-

3 7 -

H2(G, A)

= ]HIE(G,A)/n.

W e can also consider the crossed product (or semi-direct product) r

= CI(G,A)~I]

which has as underlying set the product CI(G, A) x 11 with multiplication law (b,~). (a,a) = (b. (~ •

a),13a) .

W e then define a group action

|

1- x ZZ(G, A) -~ ZZ(G, A)

by

(a, ~) | ($, h) = a * [ ~ V ( $ , h ) ] . T h i s o b v i o u s l y p r o d u c e s an i s o m o r p h i s m

H2(G,A)

~ ZZ(G,A)/F

.

A 2 - c o c y c l e (4, h) is said to be neutral if h : G X G w h i c h we s h a l l d e n o t e by 1, t a k i n g e v e r y p a i r ( s , t ) this case

-*H is the trivial ma p,

i n t o t h e u n i t e l e m e n t of H.

~ : G -~ TI is a good s t a n d i n g h o m o m o r p h i s m .

A 2-dimensional coho-

m o l o g y class in H 2 (G,A) or in H2(G, A) is said to be neutral if it contains a neutral Z-cocycle. L e t 8 : G -~ 11 be a h o m o m o r p h i s m .

~

(G,A)

= Hom0(G,A)

the set of e-crossed h o m o m o r p h i s m s

W e then denote by = ZlelG, A) C

CIIG, A)

f:G-~ H, i.e. the set of m a p s

f:G-~ A

In

-

38

-

such that f(st)

=

f(s)

9 els)f(t)

.

An a c t i o n is d e f i n e d : *:H•

Z1o(G,A) -~ Z10(G,A)

,

b y putting h*f

= f',

f'(s)

= h- f(s). O(S)h'l 9

The orbits for this action form the set

Hle(G,A) - zle(o, A)/H 9 We a l s o d e n o t e by 0 H i ( G , A)

-- H 0

t h e s u b g r o u p of H c o n s i s t i n g of t h e 0 - i n v a r i a n t e l e m e n t s

h E H, i, e.

such that

h = 0(S)h f o r a n y s E G. F i n a l l y w e s h a l l d e n o t e by

t h e c o r r e s p o n d i n g s e t s e n d o w e d w i t h t h e s t r u c t u r e of n o t ov_ly h a v i n g a " c l o u d " of n e u t r a l e l e m e n t s but a l s o h a v i n g a m o n g t h e m a p r i v i l e g e d o n e , n a m e l y t h e cohomology class containing the 2-cocycle

(0,1).

T h e s e t ]E2(G, A) is a l s o d e n o t e d by E x t l ( G , A) .

-

39

-

To be able to define cohomology exact sequences w e need to define the notion of a short exact sequence in the category ~F~z of crossed groups.

This will be a

diagram

A

(z.s) giving

rise

to the commutative

11'

I

diagram

)

.'T

) H'

I) A J>A,,

. )

t

II

~ > 11"

.T H

."T

. Y H"

J

) i

satisfying the following conditions: (i)

the lower

row is a short

exact

sequence

of groups

;

(ii) 11' -~ 11 is an isomorphism and ~:II -~ 11" is epimorphic. In view of condition (ii) w e shall put 11' = 11 and replace this diagram by

(Z.6)

(Z) 1

n

----

p'I

I

~ J

.)

H

) H'

n

:

)

n,,

:p" .) H "

' )

I

J

Let e : G -~11, e " : G - - 11,, be h o m o m o r p h i s m s then possible

to define

it is

maps

A 1 1 A = AE:H~

such that e" = ~/. e.

0

= AE:H

,,(G,A")-- H (G,A') I i I HZ(G, Extl(G, A') , A = AE:He,,(G,A")-" A')

such that the following canonical square c o m m u t e s

-

1 H o m e , ,(~, A"I ~k-~

Hle ' ' ( G ' A ' ' ) 1 - ~ A

40

-

E x t l ( ~ , A') = ~I 2 ( G , A ' )

H2(G'A')

Moreover Theorem 2.2. s e q u e n c e (E) i n ~ 2 1

All w h a t we h a v e d e f i n e d is f u n c t o r i a l a n d e v e r y s h o r t e x a c t g i v e s r i s e to c o h o m o l o g y e x a c t s e~uenc e s:

0 A") A0 _~ H~(G, A') _~ H0(G, A) _~ He,,(G, -~

(2.7)

2 -. HI(G,A ') -~ H Ie (G,A)-~ He,,(G,A,, i AO ) Al-- H ~ (G, A' ) -- H ~ (G,A) --* HO,,(G; A"), 1

, -~ Home(G,A' ) -~ H o m e ( G , A ) -~ Home,,(G,A,, ) &-~

( z . 8)

&l Extle(G, A, ) In t h i s s t a t e m e n t , endowed with structures

Extle(G,A )

Extle,,(G,A")

e x a c t n e s s m e a n s t h a t t h e s e t s in t h e s e s e q u e n c e s a r e w h i c h m a k e it p o s s i b l e to a n s w e r t h e f o l l o w i n g t w o

que s t i o n s : Problem

(a).

W h e n is an e l e m e n t in one s e t ( e x c e p t t h e l a s t one) t h e image

of t h e p r e c e d i n g a r r o w ? Problem

(b).

W h e n do two e l e m e n t s in one s e t h a v e t h e s a m e i m a g e t h r o u g h

the next arrow? F o r e x a m p l e e a c h s e t is p o i n t e d o r h a s a c l o u d of n e u t r a l " d r o p l e t s " w i t h a m o r e distinguished one.

T h i s a l l o w s to s o l v e q u e s t i o n (a).

But t h e s t r u c t u r e n e c e s s a r y

to s o l v e q u e s t i o n (b) is m o r e c o m p l i c a t e d a n d m o r e i n t e r e s t i n g .

For example each

-

41

-

H o m f ( G , A) is a polypus and each Ex===~tI(G,A) = I~Z(G, A) is sitting under an object which I would like to call a lobster, na me ly a bigger animal which has pinces.

3. Approach of three-dimensional cohomology. This aims at solving question (a) for the last object in the sequences (Z.6) and (2.7). 1 To be m o r e geometric let us remind that any ~ e Ext0(G,A) can be repres e n t e d by an i s o m o r p h i s m c l a s s of d i a g r a m s 11 A~

(3. I)

I'~ I

1

> H i'-~X

>1

>G

a homomorphism

with the r o w an exact sequence of groups and

s u b j e c t to

c onditions (i) (ii)

p = ~.i, f o r x ~ X,

h c H,

i(~Xh) = x . i h . x

-1

T h e s e d i a g r a m s a r e f u n c t o r i a l in t h e s e n s e the a m o r p h i s m s

(Z.l) produces a sort of push-out diagram 11 1

)

H

i

)

H ' ---->

)X

)G

>

(3.2) O

>i

1

(j, •j

in '~z

as

in

-

u n i q u e up to i s o m o r p h i s m .

42

-

In p a r t i c u l a r t a k e H' = H,

II'

= II, ~/ = id,

and let

a

j =j

: H - ~ H be t h e a u t o m o r p h i s m i n d u c e d by s o m e a e I I .

T h e n (3.Z) c a n be

c o n s i d e r e d as an i s o m o r p h i s m b e t w e e n two d i a g r a m s of t y p e (3.1).

These larger

i s o m o r p h i s m c l a s s e s c a n be i d e n t i f i e d w i t h the e l e m e n t s of t h e t h i n HZ(G, A). P r o b l e m (a) f o r the l a s t o b j e c t in the s e q u e n c e (2.7) is e q u i v a l e n t to: s u p p o s i n g t h a t in (Z. 1) j a n d ~ a r e e p i m o r p h i s m s

and we h a v e in (3.2) t h e

bottom part, namely I] !

1

>H'

is it p o s s i b l e to f i n d s o m e d i a g r a m

>X'

>G

> 1

(3.1) f i t t i n g into ( 3 . 2 ) ?

And s i m i l a r l y f o r t h e

l a s t o b j e c t of s e q u e n c e ( 2 . 6 ) . L e t us n o w go b a c k t o t h e e x a c t s e q u e n c e (Z.4) in ~7~2 and s u p p o s e t h a t A C I] is the k e r n e l of N : I I -~ I I " ,

l

so t h a t we h a v e t h e e x a c t s e q u e n c e

> A

> n

> n,,-->l

N e x t , l e t us c o n s i d e r a Z - c o c y c l e ( ~ " , h " ) e z Z ( G , A " ) . p o s s i b l e to f i n d a 2 - c o c h a i n (~,h) w i t h v a l u e s in A, w h i c h l i f t s ( ~ " , h " ) in t h e s e n s e t h a t

k:GXG

) A

n a m e l y (~,h) c C 2 ( G , A )

5 (4, h) = ( ~ " , h " ) .

#

,

k:GXGXG

It is t h e n c e r t a i n l y

~

We t h e n h a v e f u n c t i o n s

H'

such that (3.3)

~(s)~(t)

(3.4)

r

= Ms,t) ph(s,t)~(st)

h(s,tu) = k(s,t,u), h(s,t), h(st,u)

-43

-

It thus s e e m s that the pair (k, k) is a candidate to the title of 3-cochain and ultiH o w e v e r another lifting ( ~ ' , h ' )

mately of 3-cocycle.

of ( ( ~ " , h " )

is o b t a i n e d by

applying to (6, h) a deviation (a, a) such that

(~',h')

= ( a , a ) . (~,h)

= (a. r

a.

h)

where

a:G-~A are arbitrary

functions.

and

a:GX G-~H'

T h e n e w lifting ( ~ ' , h ' )

then produces a pair

(k',k')

and

one w o u l d l i k e to c o m p u t e i t a s a f u n c t i o n of t h e i n i t i a l (k, k) and t h e d e v i a t i o n (a, a). U n f o r t u n a t e l y s u c h a f o r m u l a d o e s n o t e x i s t and e a s y c a l c u l a t i o n s y i e l d : k'(s,t)

= a(s).~(S)a(t).k(s,t).ph(s't)a(st)-I

k'(s t,u) = a(s)~(S)a(t,u).~(S)L '

a(s)

(t,

T h e r e a l t r o u b l e is t h e p r e s e n c e ~(s) L a(s)

(t,u)

. p a ( s , t ) -I ,

u). [~(s) ph(t, u) ]a(s ' tu). k ( s , t, u). ph(s, t ) a ( s t ' u)-I .a(s, t) -1

of t h e e x p r e s s i o n

=

a(s)~(S)h(t, u) ~(S)h(t, u) -1

and w e r e a c h h e r e the f i r s t c r u c i a l p r o b l e m in t h e c o n s t r u c t i o n of 3 - c o h o m o l o g y . Moreover,

in b o t h f o r m u l a s a p p e a r the f u n c t i o n ~ w h i c h m e a n s t h a t we h a v e to add

it to t h e p a i r

(k, k).

But we a l s o h a v e to i n c o r p o r a t e

L:A •

a function

x G x G -~ H'

the v a l u e of w h i c h at (a, 6, s , t ) we w a n t to d e n o t e

L(a,~,s,t) = !L(s,t) = ~L a

s,t

-

44

-

and which is given by

~a L ( s , t ) This is actually ~h(s,t)

c H,

a nice function

when twisted

element

of the smaller

inverse

crossed

morphism

by

group

s i n c e it m e a s u r e s

H'

Also, L:A

for

the modification

-~ H '

(namely

undergone

being by necessity

~, s , t f i x e d ,

an

this function is

its inverse

by

is a crossed

an homo-

or one has

a~ L = a(L).

We can however

-I 9

a ~ A , this modification

homomorphism

A -~ H'

= a.~h(s,t).~h(s,t)

consider

the more

simple

function

K:A•

(3.5)

a L)

-~ H'

defined by (3.6) which measures

K(a,s,t)

= aK(s,t)

the non invariantness

of

= ah(s,t).h(s,t)-i

h:G x G-~ H by a c A.

!L(s,tl = This can be clarified

as follows.

Consider

C I = CI(A,H')

and define a group action c 1] and K ~ C I,

~*K:A

then have

_lo K] the set = App(A,H')

* of 1] onto C I (composition product) -~ H'

We

is defined by

such that for

-

45

-

or equivalently

Similarly,

if K is a s in ( 3 . 5 ) ,

then we get

it c a n be i d e n t i f i e d w i t h a m a p

K:Gx

G-~ C1 and

~*K:G XG-. C 1. With this formalism it is clear that ~aL(s,t) : a ( ~ * K ) ( s , t )

(3.7)

To simplify the notations w e shall a l s o w r i t e ~ * K

.

= ~K a n d

~L(s,t) = !K(s,t) .

(3.8)

W e then want to consider

a

3-cochain as a system (k, k, 4; K, ~) of five

functions k:GxGXG k:GXG ~:G

-~ H', -- A ,

-~ 11 ,

K : A x G XG -~ H ' , ~:GxG

-~ I I .

NOW, s t a r t i n g w i t h a l i f t i n g (4, h) c C 2 ( G , A ) of ( ~ " , h " ) c Z 2 ( G , A " ) , obtained such a system

Thus

[ w i t h 11 ( s , t ) = p h ( s , t ) ] w h i c h we s h a l l d e n o t e b y

(k, k, };K;T]) = [ ] (~,h} .

(3.9) D

represents

we have

s o r t of a c o b o u n d a r y o p e r a t o r

[7 :C2(G,A) -* C3(G, ?) .

- 46

-

At t h i s p o i n t t h e l e c t o r c a n e a s i l y c h e c k t h a t the p a i r s

(a, a) f o r m a g r o u p

C2(G, B) w h e r e B is t h e c r o s s e d g r o u p ( H ' , p' , A , ~ )

in w h i c h ~ is t h e a c t i o n of

A to H '

M o r e o v e r t h e r e is a g r o u p

r e s t r i c t e d f r o m t h e one ~' of II onto H '

action

cZ(G' B) x CZ(G, A) - CZlG, A), ((a,a),(~,h))~-~(a,a).

(~,h)

= (~',h')

We a l s o h a v e a g r o u p a c t i o n

(3.10)

* : C2(G, B) x C3(G, ?) -- C3(G, ?) ((a, a), (k, k, J~;K;rl))n,--~ ( k ' , k', +' ;K' ;1"1' )

such that (3.11)

0[(a,a).(~,h)]

= (a,a)*[O (~,h)] .

This action is explicitly defined by putting (3 12) 9

k'(s,t,u) = a(s)~(S)a(t,u).J~(S)K'- " [~(s)~t'U)]a(s , tu) a(s)

~,uj.

9 k(~ t , u).

rl(s, t ) a ( s t ' u) -1. a(s, t)-1 ,

(3.13)

k'(s,t) = a(s). ~(S)a(t). k(s, t).Tl(s' t ) a ( s t ) - l , pa(s, t) -1

(3.14)

(3.15)

(3.16)

~'(s)

i K'(s,t) ,l'(s,t)

= a(s).~(s),

= t~a(s,t).l K(s,t).a(s,t)-I = pa(s,t).n(s,t)

.

,

-47

-

T h e e x i s t e n c e of t h e a c t i o n (3.10) giving r i s e to f o r m u l a 3-cochain (k', k', ~';K' ;~ ' ) derived f r o m

(k, k, ~;K;~]) and t h e d e v i a t i o n (u, a).

(3.11) s h o w s t h a t t h e

(~', h') can be c o m p u t e d in t e r m s of

Thus the 5-uples

(k, k, ~;K;~) a r e r e a l l y

s e n s i b l e c a n d i d a t e s to t h e t i t l e s of 3 - c o c h a i n s and 3 - c o c y c l e s . N e x t we h a v e to f i n d out i d e n t i t i e s s a t i s f i e d by t h e 3 - c o c h a i n s d e f i n e d by

(3.9).

These identities will define a subset

Z3(G,?) C

C3(G, ?)

of 3 - c o c y c l e s and we s h a l l h a v e to k n o w w h a t

? means,

namely what the system

of c o e f f i c i e n t s is. We r e a c h h e r e the s e c o n d c r u c i a l p r o b l e m , t h e c o m p o n e n t k : G x G • G -~ H '

n a m e l y d e r i v i n g a n i d e n t i t y on

This identity should generalize the well known

c onditi on (3.17) (6k)(s,t,u,v) = ~(S)k(t,u,v)-k(st,u,v)+k(s,tu, v)-k(s,t, uv)-k(s,t,u)

of the c l a s s i c a l t h e o r y .

T h i s we do as f o l l o w s .

(3.18)

= ~(S)h(t,u).h(s,tu).h(st, u)-l.h(s,t)-I

k(s,t,u)

= 0

We h a v e

t o g e t h e r with

(3.19)

~(S)h(t,u)

= k(s,t,u).h(s,t).h(st,u).h(s, tu)-i .

We t h e n u s e (3.18) to w r i t e $(S)k(t, u, v) -- $(s)$(t)hlu, v). $(S)hlt, u v ) . $(S)hltu, v) -1. $(S)hlt, u ) - i This is then transformed using (3.3) and four different f o r m s of (3.19) to yield

- 48 -

~(S)k(t,u,v).k(s,t,u). 9H(s't'u)k(s,tu,v ) =

(3.2o.1)

[~ (s, t) ),(s, t). n (s, t)k(st, u, v). N( s, t, u, v).

pH( st, u, v) ]k(s, t, u v ) .

In t h e r e one has = h(s, t) 9h(st, u) 9h(s, tu)

-1

(3.21)

H(s, t,u)

(3.22)

N(s, t, u, v) = X(s, t). n (s, t)H(st, u, v). n (s, t)H(st, u, v) , I

, .

M o r e o v e r ~ ( s , t ) r e p r e s e n t s ph(s,t) and can be c o m p u t e d in t e r m s of the cornponents X, ~ by the f o r m u l a (3. ~0.II)

6(s) ~(t)

= x(s, t ) . ~ (s, t). ~(st) .

L a s t but not l e a s t the function N can be c o m p u t e d out of the function K, mo that (3.20. I) is actually an identity on the 5-uple (k, k, ~;K;~).

The a s s i d u o u s l e c t o r

will indeed c h e c k that N(s,t,u,v)

-1 -1 -1 = kX. nlkK1). ~nI(XK2 )n~lTI2n 3 ( k K ; 1 ) . n n l n 2 n 3 n (xK.I)

with

R e m a r k 3.1.

~K - ~(s, t) K(s' t) ,

XKI -- ~(s, t) K(st' u)

AK2 = X(s,t)K(stu, v)

xK 3 = X(s,t)K(st, uv)

It is useful to o b s e r v e that f o r m u l a (3.20.1) obviously has the

shape of (3.17) and it also has the shape of the d i a g r a m

49-

S @ (T@(U@V))

/\

S ~ k T , U, V

S~

((T~ U)(~ u /~R:S, T(~U, V

ks, T, U ( ~ V ~

(S(~)(T(~U)) (~ V

(S@T) (~ ( U @ V )

f kSxT'U'V~

S, T, U x v

((S~)T)(~)U) ~ ) V

the cornmutativit7 of which is a relation among the canonical isomorphisms kS, T,U:(~S~T) ~ U -~ S ~

(T~U)

expressing the defect of associativity of the tensor product of modules.

The only

snag is that the first m e m b e r of (3.20.I) should be written as ~(S)k(t, u, v). pk( s, t, u). pH( s, t, u)k( s, tu, v). k(s, t, u) .

But precisely the function k is an obstruction to the associativity of a product defined on the set H X G

by

(m. s). (n, t) = (m.~(S)n.h(s,t),st) where

m , n e H,

s,t e G .

R e m a r k 3.2.

Let h be an e l e m e n t of H and a an e l e m e n t of A .

We then

define the function 3h :A -~ H', putting (8h)(a)

=

8 h a

=

ah.h-I

.

The s a m e f o r m a l i s m applies when h is a function f r o m s o m e set E to H; then %h i s a f u n c t i o n f r o m

II x E t o H '

(e.g.

the function

K of ( 3 . 6 )

is just

%h f o r

-

h:G

XG

-* H).

50

We h a v e a l s o the f u n c t i o n

-

6h:A

-~ H '

(6h)(a) = 6 h = h. ah -I a

W e call the operators

6 and a the coboundary and antiboundar)r operators.

a crossed h o m o m o r p h i s m w h i l e Considering that the group

6h is

Oh = (6h) -I is an inverse crossed h o m o m o r p h i s m .

I] operates onto H by automorphisms

and onto CI(A,H ')

by the composition product ':', it is clear that O and 6 are equivariant, namely 8(~)h) = q~':' (8h), We c a n a l s o c o n s i d e r then observe

that,

6(~h) = ~ * (6h) .

8h a s a f u n c t i o n in b e t w e e n t h e l a r g e r

groups

I1 a n d H a n d

in t h i s f r a m e , = ( a ~ h . h - 1 ) . ( ~ ) h . h -1) -i = (a ~h). (0~h) "I .

8 a ( $ h ) = a~h.4)h-1

M o r e generally, if K : A

-~ H' is an inverse crossed h o m o m o r p h i s m ,

formula (justifying the writing

a

one has the

K instead of K(a))

(3.Z3)

a(~*K)

= a~K. ~K -I .

This can be seen as follows:

K verifies

a~K = a(K). aK so t h a t (3.24)

(aK)-I = a(_IK)

or

K(a) -I = a(K(a-l)) .

a

Then one proves (3.23) by w r i t i n g

a(fl*K) : ~( _lafl K) : fl[~-l(a~3K). _ i K] : aflK.(~K)

-I

-

A c c o r d i n g to p r e v i o u s n o t a t i o n s , A to

H'

A onto H '

51

-

the s e t of c r o s s e d h o m o m o r p h i s m s

is d e n o t e d Z I ( A , H ') = Z I ( A , H ' ) .

from

( R e m i n d t h a t @ is t h e a c t i o n of

in t h e c r o s s e d g r o u p B of ( 3 . 1 0 ) . )

We s h a l l d e n o t e b y

SI(A,H ,) = SI(A,H ,) t h e s e t of i n v e r s e c r o s s e d h o m o m o r p h i s m s .

Remark homomorphism,

3.3.

W e also want to show that if K : A - * H'

is an inverse crossed

so is also ~* K for any ~ ~ 11. Indeed

(~*K)(a~)

=

*[K(~-la~)]

=

~[~-la~K(~-l~).K(r

= a~K (~-I~). ~K(~'la~)

=

a[(~. K)(~)]. (~* K)(a) .

We n o w d e r i v e t h r e e m o r e i d e n t i t i e s on t h e 5-uple (k,k, ~;K;,]) = D (~'h)" One is j u s t t h e f a c t t h a t K is an i n v e r s e c r o s s e d h o m o m o r p h i s m , (3.ZO.III)

a~K

= a( K ) . a K

or

a~K(s,t)

To o b t a i n t h e n e x t i d e n t i t y we t w i s t f o r m u l a

a~(s)h(t,u)~ah(s,tu)

= a[~K(s,t)].aK(S,t)

namely .

( 3 . 4 ) b y a to g e t

= ak(s,t,u).ah(s,t).ah(st,u)

@

Then inserting in the spot 9 the trival expression ~( S)h(t, u) -i. k( s, t,u). h( s, t).h( st, u)" h( s, tu) -1 yields

~(s)a L(t, u)" k( s, t, u). h( s, t). h( st, u)- h(s, tu) -I. ah( s, tu) = a k ( s , t , u ) . ah(s, t). a h ( s t , u )

-

52

-

and finally, putting @(s,t,u) = 9H(s,t,u): (3.20.IV}

~(S)K(t'u)'k(s't'u)'e(s't'U)[aK(S'tU)]a

ak(s't'U)'aK(S't)'Vl(s't)aK(st'u)"

:

The analogy between this formula and the fundamental identity (2.2. a) on 2-cocycles is c l e a r . We f i n a l l y h a v e t h e f o l l o w i n g i d e n t i t y , w h i c h p l a y s a -role s i m i l a r to t h e s e c o n d identity (2.2.b) (B.20.V)

for 2-cocycles:

~(s)k(t,u).[~(S)rl(t'u)]k(s,tu).pk(s,t,u)

We d e r i v e it f r o m

= k(s,t).n(s't)k(st,

( 3 . 3 ) , c o m p u t i n g in t w o d i f f e r e n t w a y s t h e p r o d u c t r

u). ~(t). r

When a and ~ a r e e l e m e n t s of 1], w e h a v e w r i t t e n

a~ = a~a-1

4.

D e f i n i t i o n off t h r e e - d i m e n s i o n a l

cohomolo~y

We h a v e n o w o b t a i n e d g o o d c a n d i d a t e s f o r 3 - c o c h a i n s , of f u n c t i o n s

(k, k, ~;K;~) a s a b o v e .

(B.20. I - I I - I I I - I V - V )

are candidate

s y s t e m is n o t y e t c l e a r .

Among them, those satisfying the conditions 3-cocycles.

However so far the coefficient

A candidate would be the s y s t e m = (H',9',11,

in w h i c h A f = ( H ' , p '

n a m e l y the 5-uples

,I], @~) is a c r o s s e d

g r o u p and in w h i c h A is a n o r m a l

s u b g r o u p of 1~ s u c h t h a t

p'H'C

AC

~;A)

n

-53

-

H o w e v e r w e s h a l l f a c e m o r e d i f f i c u l t y w h e n t r y i n g to d e f i n e a c o n n e c t i n g m a p and a longer exact sequence,

T h i s w i l l o b l i g e u s to i n c o r p o r a t e

more

structure

in

the coefficient system.

Going back to a Z-cocycle (4",h") e Z 2 (G,A"),

we have lifted it into

(4 h) e C z (G, A) and formed ,

m (4, h) : (k, • 4;K;n) ~ Z3(G, A') However

(4,h) is defined up to a pair (a,a) , CZ(G,B)

(where B = (H',p',A,@)

is the crossed group appearing in (3.10)). Replacing (4,h) by (4' ,h') = (a.4, a.h) yields a system (k',k',4';K' ;T]') related to the preceding formulas

(3.12 ) to (3.16)

which precisely represent the action (3.10). They also represent what we shall call the first variation of the 3-cocycle (k, k, 4;K;T]). The second variation shall be obtained by moving the Z-cocycle (4", h") along its thick cohomology class, namely applying to it the action (2.3). To that effect w e consider b" :G -~ H" and produce b"* (4",h") = (~",h--"). Then b can be lifted into s o m e function b :G -~ H and the lifting (4,h) of (4",h") is then transformed into b* (~,h) D (~,~)

= (~,h'-) w h i c h is a l i f t i n g of ( ~ " , h " ) .

= (k, k , ~ ; K;n) in t e r m s

The problem

is t h e n to c o m p u t e

of [] (6, h) = (k, k, 4 ; K ; n ) a n d s o m e o p e r a t i o n to be

described inside of the coefficient system.

H o w e v e r it will turn out that such an

operation cannot be described in terms of our candidate A' : this will lead to the third crucial step in the development of 3-cohomology. Easy computations yield the following k(s, t, u) = b(s). 4(S)b(t) 94(s)4(t)b(u). k(s, t, u). ~ (s, t). 4(st)b(u)-l. 4(S)b(t) ~i. b(s)-i

- 54 or, putting ~(s) = pb(t) , (4.1)

~(s,t,u) = ~(s)'~(s)~(t)'~(s)~(t)~(U)k(s,t,u).~(s)'~(s)~(t)[8~(s)~(t,)b(u)]-l; =J.

),(s,t)I

(4. z)

~(s,t) = ~(s)"~(s) (t)k(s,t) ;

(4.3)

"~(s)

= ~(s).J~(s)

;

aK(S,--t) = [aab(S)]. ~(s)[aa~(S)b(t)]. ~(s). ~(s)~(t)[aK (s, t)].

(4.4)

9 ~(s). $(s) ~(t).~(s, t). ~(st)-I

(4. s)

[Oab(st)]-1

~s, t) = ~(s). ~(s)~(t).n(s, t). ~(st) -I

In t h e s e f o r m u l a s

we have set

(4.6)

8a~h = aa(~h)

= a[a(~h)] = a[~*(ah)].

There is finally a third variation if we allow the Z-cocycle (~", h") to m o v e into its thin cohomology class, specifically if we m o v e it under the action (Z.6) of If"

This however produces the very sweet formulas (4. 7) to (4.11) . Suppose

(Jp",h")

is brought by a"

If I'

into

',hfT):

~,,(s) = a"J~"(s),

~"'(s,t)

= ~"h(s,t) .

Lifting a" ~ I f " b y ~ e 11 brings the lifting (~, h) into a lifting (~, h~) of (~", ~") such that N

=

s) = ~.

jp(

s).

Then D (~,~) = (~,~,~;K;~) is given b y

-i

his,t)

= ~h(s,t) .

-

55

-

(4.7)

k(s, t,u) =

(4.8)

k(s, t)

(4.9)

~(s)

~k(s,t,u)

= ~•

,

,

= "~(s) , e~

(4. I0)

K(s, t) = a ( ~ * K)(s,t) o r K = ~ * K , -i

(4. n)

=

TI

m1~

=

Let us n o w g o b a c k to the s e c o n d variation.

We would have liked to eliminate candidate possible

coefficient) because

out of formulas

the group

(4.1) t o ( 4 . 5 ) .

we have to know the expression

of pb = ~ i s of c o u r s e

no trouble

H (which does not appear Unfortunately

8b w i t h b : G -~ H .

since it lies into

in our

this is not The presence

A,

T o go a r o u n d this difficulty w e a s k o u r s e l v e s u n d e r w h a t condition t w o elements

b a n d b'

of H

p r o d u c e at the s a m e

pb = pb'

Suppose

b' = b.x,

x e H.

px = I or

and

time

8b = 8b'

Then one should have xe

ZH

= Ker

p, the center of A

and a

for e v e r 7 a e A,

x = x.

T h e s e c o n d condition m e a n s

that x belongs to the II-invariant s u b g r o u p of

A - i n v a r i a n t e l e m e n t s of H.

T h u s the intersection of this g r o u p a n d

ZH

is a

-

normal

and even

quotient

group

rl-invariant

E = H/P

subgroup

and the quotient

P

56

-

of H. map

We thus

want to consider

• : H -~ E f i t t i n g

the

into the diagram

H

(4.1Z)

]3

which

defines

There

exists

well

as

uniquely also

8 and

a canonical

A in the sense

(

CI(A,H')

'% a n d i n w h i c h action

K'

o f rf o n t o

E

=

K i ,

8'

such that

= ai,

i:H'

-~ H .

K: i s e q u i v a r i a n t

as

that

Suppose n o w w e have a m a p

,%(r

= r

b:G--

H producing m

can then transliterate the delicate formulas

,

m ~ E , ~, =

11.

~.b:G--

E.

We

(4.1) and (4.4) as follows (we put

~(s) = era(s) and use a convention similar to (4.6)): (4.1)'

k(s, t, u) = ~(s)"~(s) ~(t)" ~(s)~(t) ~(U)k(s, t,u). ~(s)-~(S)~lt)[A~ls)~lt) mlu)]-i ;

Ms, t)-l

(4.4')

aE(s't) = [,%am(S)]'~(s)[,%~(S)m(s)]'~(s)'~(s)~(t)[aK(s't)]'a

~(s).~(s)~(t).n(s,t).~(st)-l[,%am( st)]-i Definition 4.1.

B y a super-crossed group w e m e a n A

as follows:

= (H,p,n,@;A;E,@,

E, and t h e r e f o r e

Without Using

type,

with

to this

of m a n i f o l d s . E~

which

Sq > E > Sn is a f i b r a t i o n a n d E > q = n, it f o l l o w s t h a t t h e r e is a c r o s s -

Whitehead on

10-manifold

3 is j o i n t w o r k w i t h J. R o i t b e r g .

PROOFS

Suppose an H - s p a c e .

emerges

of S e c t i o n

of these;

of a s y s t e m a t i c

manifolds

products

were

A new H-space,

closed

in p r o d u c t s

homotopy

~

that these were the

type).

as a b y - p r o d u c t

of d i f f e r e n t

in

S 7, is a c o u n t e r e x a m p l e

two d i f f e r e n t i a b l e

constructed,

products

conjectured

(up to h o m o t o p y

for n o n - c a n c e l l a t i o n

Moreover,

S 7, RP 7, and p r o d u c t s

in this p a p e r as a s m o o t h

is a 3 - s p h e r e

section

the o n l y k n o w n H - s p a c e s

it h a d b e e n m i l d l y

only H-spaces

67-

q < n.

H*(E;Z2).

algebra

on one

H * ( E ; Z z) ~ H * ( s 2 n - I ; z 2 ) ,

-

then

it f o l l o w s

thus,

(q,n)

exterior

=

from

[i]

(1,2)

or

algebra

or

15,

then

P2E

three. with

8 and

~2(e)

lead

by

or

be

polynomial

~2~3(e)

=

_ ~2

[2],

algebra

(mod 2)

~2~3(8)

and

i;

E P-~

from

show

Lemma

(7,1),

If

S3

then

X

or

that

=

- B2 ~ 0 see

filtered, B

(of

at h e i g h t together

~3~2(S),

of p r o o f Sn

e,

(mod 2), [4]).

is a f i b r a t i o n

Theorem (q,n)

E;

2 will

and

E

follow

c a n n o t be

(3,4),

(7,7).

> X has

> S4

is a f i b r a t i o n

the h o m o t o p y

type

of

and

X

is an

S 7.

2

If or

can

n = ii

1

H-space,

Lemma

(7,3),

(q,n)

of the H - s p a c e

truncated

~2(B)

(for d e t a i l s

Sq

with

on t w o g e n e r a t o r s

and

~3~2(e)

if w e

l) ;

an

that

to s h o w

(torsion-free)

n + I, r e s p e c t i v e l y ) ,

suppose

i,

n >

H*(E;Z2)

plane

of as a

H-space.

(1,7),

then

is a f i b r a t i o n ,

is a h o m o t o p y - a s s o c i a t i v e Theorem

is n o t

H*(E;Z2)

it s u f f i c e s

> Sn

thought

~ 0

(recall

7

(7,15).

to a c o n t r a d i c t i o n

Now

If

be t h e p r o j e c t i v e

may

KU(P2E)

filtrations

(3,4).

S 7 ---> E

let

truncated

and,

(7,11)

If

2n - 1 = 3 or

on o n e g e n e r a t o r ,

H*(S q • sn;z2), c a n n o t be

that

68-

(7,1),

S q ---~ X

then

> Sn

the u n i v e r s a l

is a f i b r a t i o n , coverin@

X

of

with X

has

(~,n) the

=

(1,7),

-

homotopy

type of

S 7 .

Remark: associative

Lemma

The u n i v e r s a l

H-space

H*(X;Z)

X

of a h o m o t o p y -

a homotopy-associative

is a h o m o t o p y - a s s o c i a t i v e

~ H*(S 3 • sT;z)r

: H3(X;Z3 ) ~

Now exact

is a g a i n

covering

H-space.

3

If

pl

69

H7(X;Z3)

let

sequence

then

(q,n)

=

the S t e e n r o d

(7,3)

: HT(E;Z3)

and

implies ~

which

contradicts

Finally,

(q,n)

~

that

S7

(i.e.,

P~ = P~

9 P~

The S e r r e

that

the c o n c l u s i o n

for e s s e n t i a l l y

space

3 Steenrod

S3-bundles

by e l e m e n t s and

~n-I (S3) ~

of L e m m a

the same H-space

algebra).

S3-BUNDLES

principal

are c l a s s i f i e d

isomorphism

(7,7)

in the m o d

We consider

the c l a s s i f y i n g

X = E.

can not be a h o m o t o p y - a s s o c i a t i v e

3.

bundles

operation

H7(S7;Z3)

is an i s o m o r p h i s m ,

reason

and

is n o n - t r i v i a l .

for c o h o m o l o g y i*

H-space

let

~n(B) .

e
E 8

Eu

P~ --=>

and

Ee8

nSO S --~ B

2

Eu8 Theorem

U e n+3

3

E~ ~ E 8 < ~ > This

classes.

a CW-structure Eu =

Theorem

and homotopy

(James,Whitehead)

has

E

maps

B,

= Esu;

= E~

x S3

if

8 0 9 P~ = 0.

4

Let

8 = Zu.

Then

8 0 9 P~ = 0

if

ptp%~,~_1,.

--

where To p r o v e

~ 6 ~6(S 3) this we

measures consider

the non-commutativity

the Puppe

sequence

.

~3~

=

2

of

$3.

of t h e i n c l u s i o n

of

0,

-

the

fibre

in

E

readily

i> E

identify

q:

Y E ~ 7 ( S 4)

S n + S n+3

where

is

> Sn

s0

=

(Z~,

le

Whitehead s

if

/(Z-l) T

9 u,

=

Corollar[ I prime

to

k~

or to

9 u =

(leo,s

, and

Let

I ~ 1 mod

ko,

k = p,

where

s

is p r i m e

to

p

2.

E~ ~ E T ~

= Sp(2),

Then

9 74 ~>

is a d j o i n t

be

s O

of

Thusr

to

9 Pe

= 0,

,

ET~

s ~ •

is

. ~ 2 ,3 ,

and

mod

then

an

p,

if

E 8.

8 = 7~.

x S 3 = ET~

3 and

k 0 = gcd(k,24),

Then

a prime

~ = ~,

E~

k,

order

8 = s

E~ ~

Let

Theorems so

(s0,0),

[e,e]

hence

and

• S 3 = E 8 • S 3, a l t h o u g h

E~

= qj ,

" ~!4~)

• S3 = E8 • S3

if

Now

9 Y

e

example,

from

Pe

1 E ~ 3 ( S 3) 9

[e,e]

for

follows

Now

Z(s 2

Thus,

This

Z4~),

O.

i.

Corollary

as

'0>

(s

9 u

Ee

Ee

S4

components,

s0 9 q =

(s

product

73e~

9

9

its

map.

" q =

" q = s ~

of

u)

and

= the

y

Hopf

is a d j o i n t

e E ~4(B) s

Now

sn + sn~3

in terms

the

9 q = e

and

)

u, u

where

-

, S3

We

71

4 and H-space 9

Then

x S3 the

fact On

that

~r9 ( S 3 )

dimensionality

= Z39

-

grounds

it follows

it is the total

that

72

-

it cannot be any of the known H-manifolds;

space of a p r i n c i p a l

S3-bundle

over S 7.

73-

REFERENCES [1]

Adams,

J. F., "On the non-existence

of elements of Hopf

invariant one", Ann. of Math. 72; 20-104. [2]

Adams, J. F., "H-spaces with few cells", 67-729

[3]

(1960).

Topology ~;

(1962).

Dougla% R. R. and F. Sigrist, which are H-spaces"

"Sphere bundles over spheres

Rendic. Acc. Naz

Lincei 44; No.4

(1968). [4]

Douglas,

R. R. and F. Sigrist,

spheres and H-spaces",

E5]

Douglas,

"Sphere bundles over

(To appear) 9

R. R. and F. Sigrist,

"Homotopy-associative

H-spaces which are sphere bundles over spheres", appear).

(To

-

74-

CONSTRUCTION DE STRUCTURES LIBRES Charles Ehresmann

INTRODUCTION. Le but de cet article est de donner un crit~re d'existence de p-structures libres et des applications de ce critSre, qui fait intervenir un foncteur auxiliaire dont

p

est une restriction;

par exemple si

foncteur d'oubli relatif ~ l'univers M

p

P

est le

de la cat~gorie des O

homomorphismes

entre structures d'un certain type, P

ra @tre le foncteur analogue relatif ~ un univers ~

pourauquel

O

appartient M . Contrairement au th~or~me d'existence d'adO

joint de Freyd, ce crit~re impose des propri~t~s

(telles

que l'existence d'un "assez grand nombre" de produits) sur P, non sur le foncteur donn~ 8tre g~n~ralis~es

p. Les hypothSses peuvent en

(volt [i]), mais ici nous avons cherch~

indiquer des conditions simples, r~alis~es dans la plupart des exemples. Comme application,

nous obtenons des th~or~mes d'exis-

tence de limites inductives ou de structures quasi-quotients (voir aussi

[i]), et un th~or~me de compl~tion "maximale"

d'une cat~gorie en une cat~gorie ~ limites projectives et inductives d'une certaine espSce, avec conservation de limites donn~es. Nous avons montr~ ailleurs r~sultat a des consequences

[2] que ce dernier

int~ressantes pour l'~tude d'une

75

-

notion g~n~rale de "structure alg~brique" d~finie comme r~alisation,

dens une cat~gorie quelconque, d'une esquisse

(i.e. d'un graphe multiplicatif muni d'une famille de transformations naturelles).

O. QUELQUESP PPELS. Nous nous plaqons dens le cadre de la Th~orie des ensembles avec existence d'au moins deux univers M M

tels que

E M . N a i s en f a i r

0

seraient

0

aussi

valables

la

plupart

et

O

O

des raisonnements

dans une T h f i o r i e a v e c e n s e m b l e s

classes, en prenant pour M

la class, de t o u s l e s

et

ensembles

O

et pour ~l~ments de ~

des classes. Dens le dernier w

l'axi-

O

ome du choix est librement utilis~ dens M

O

La terminologie et les notations sont ceux de [~ , dens l'index

duquel

se trouvent

les

mots non e x p l i c i t e m e n t

d~fi-

nis ici. Les autres notions que nous allons rappeler figurent dens le cours

Univers. 1 ~ Si

[4].

Ensemble M

E

d'ensembles tel que: o appartient ~ M l'ensemble P(E) de ses parties O'

estun

~l~ment e t u n e

pattie de M O"

2 ~ Si (Ei)ie I e s t par un ~l~ment

I

une famille d'~l~ments de Mo index~e

de M , sa r~union appartient ~ M . O

O

3 ~ Ii exist, un ensemble infini appartenant ~ M . O

(Contrairement

~ la d~finition de Grothendieck,

geons pas que M

nous n'exi-

soit un ensemble transitif). O

Application9

Une application

d~sign~e par le triplet x § f(x)

f

de

(M',~,M), o~ de

M

M

dens

M'

est

est la surjection sur

f(M) C M'.

76

Si

f

-

est l'injection canonique de

~crit

M C M'

dans

M', on

f = (M' l,M) M d~signe toujours la cat~gorie pleine des applications

associ~e ~ un univers M . O

Cat~gor~e. La cat~gorie sition

(y,x) + y.x

de

M C C•

parfois, C ), et l'on pose de

C"

est

(C, Horn (R,A) F

--~ E.

R

) A

theorem

) H2(G,A)

Then

and this

that

. ) 0

is exact.)

Remark. spondence exactly

with

co(AIE)

(~,o) :

so that objects

(AIE)

introduce

A morphism

x~* = x I" of

elements

in the sense

(A,x), w h e r e

x E H2(G,A).

is free on a set in o n e - o n e

--~

of

G, t h e n

8R

corresis

of S c h r e i e r .

(A 11EI ) , t h e n

clearly

i ) c o ( A 1 IEI).

We now are p a i r s

F

the n o n - u n i t

a factor-set

If e*:

If

Mod G

(~G

a second

A 6 Mod G

category,

(= r i g h t

is a m o d u l e

F : (AIE)

G-modules)

homomorphism

is the c a t e g o r y

in M a c L a n e ' s

~G' w h o s e

of

terminology:

---~ (A, co(AIE))

H2(G,

objects

and

~: A

) A1

)-pointed

[4] , p.53.)

Then

-

is a functor from (2).

,r

[~I

107

-

%.

to

is surjective

(= full and representative).

This is a completely elementary result. Nevertheless,

[i], p.179.)

it leads to very rapid proofs of some important

results in group theory: the centre-commutator tation theorem for

e.g., the splitting theorem of Schur,

theorem of Schur, the Magnus represen-

F/[R,R].

The surjectivity of what follows~/

(Cf.,e.g.,

In particular,

F

is also constantly needed in

for the characterization

of

epimorphisms and monomorphisms: (3). if r a

(~,~)

is an epimorphism in

is an epimorphism in

epimorphism in

Mod G.

OG

~!

if~ and only

if, and only if, a

is an

Ditto for monomorphi_sms.

We may now define projectives

and injectives in the

usual manner.

(4). r

tive in

(A] E)

is injective in

is in~ective in

[G!

if, and only if,

ifLand A

only if,

is injec-

Mod G. Thus both our categories have enough injectives.

But the injectives do not seem nearly as interesting as the projectives

and we shall say nothing more about injectives here.

It is easy to manufacture examples of projectives.

108 -

Take any free presentation, = R/[R,R], and

F = F/[R,R].

(R,x), where

these objects sense.

as in Then

(i) above, (RIF)

X = co(R[F),

is projective

is projective

in

in

~.

6GI

In fact,

can be regarded as "free" in a certain natural

In any case, both categories have enough projectives. It is difficult

the relation between

to say anything n o n - t r i v i a l

different

is a simple consequence

of Schanuel's

lemma: are free objects,

R--I~___ V2 ~'------R2 | Vi

then

'

IS G-free of rank = rank F i. A more profitable

comparison

about

The following

free objects.

[5).

whe re

and write

project turns out to be the

of mi'nimal pro~ectives:

called minimal if every e p i m o r p h i s m

of

is n e c e s s a r i l y

A ~ 0 ).

an i s o m o r p h i s m

(and

The following statements

(i)

(AIE)

(ii)

r CA[E) =

(ii!)

there exists (R,x) =

Moreover,

s arily G-projective.

in

~;

is ~ro~ective

a free pair

(A,x) 9

the module

of projectives-

are equivalent:

is projective (A,X)

(P,O). P

is

to a projective

(ALE)

we first state a characterization

C6) 9

(AIE)

The projective

in

in~;

(R,x) (%

so that

has productsl)

(iii) is neces~

-

It follows

109

-

that a projective

(A,x)

is minimal

if,

and only if, there does not exist a splitting of the form (A,x) = (B,y) @ If

G

is finite,

H2(G,

(P,0)

.

projective)

= 0

and so we

have: (7). minimal

For finite

G, the projective

if, and only if, A

scalars.

Let

of all pairs

K

give

x

QG

to restrict the discussion

corresponding

be a commutative

A 6 MOdKG

let

Then

(A,x)

QKG

the category

and morphisms for

A(K ) = A | K Z

to

to a change of

ring and

A similar definition

A 6 MOdG,

' / ~ X(K ). ,

is a functor:

%

(A,x), with

KG-homomorphisms.

If

of

is

has no projective direct summand.

It is often necessary certain subcategories

(A,x)

to be

IKGI.--

and

A

> A,K , ~J

) (A(K) ,X(K )) = (A,x) (K)

) ~KG"

Everything we have done so far works, with suitable modifications, free objects"

in in

~KG ~KG

and

IK-~GI. In particular,

are of the form

(R'•

(K)"

w 3

H e n c e f o r t h we shall assume

G

is finite.

the "K-

-

110

-

Our concern will be with rings:

fields,

Z, ~(p)

following

= local ring at

coefficient

p, Zp = p-adic

integers.

IGI

and

tives. phisms

Suppose

K

(A,x),

(B,y)

If

(R'•

to

is a field of c h a r a c t e r i s t i c are finitely

(K) ' (S,T) (K)

(A,x) , (B,y),

generated

dividing

minimal

projec-

are K - f r e e pairs with epimor-

respectively,

then by

(5) and

(6)

(iii), A 9 P |

where

P, P'

projective

(KG) m

B 9 P' | (KG) n

are KG-projective.

summands

(8).

A

~

(by

(7)),

Since

is minimal

have no KG-

the K r u l l - S c h m i d t

t h e o r e m implies

B.

(If the c h a r a c t e r i s t i c (A,x)

A,B

projective

if,

of

K

is prime

and only if,

to

x = 0

IGI, then and

A

is inducible. )

(9). IGI,

If

K

is a field of characteristic

then ~ny two finitely

generated

dividing

minimal p r o j e c t i v e s

are

isomorphic. To complete gical

information

about

In view of same as that in

the proof of

R(K)"

A

and

(6) (iii),

(9), we need more

cohomolo-

B.

the cohomology

Now we have

in

the following

A

is the

result.

-

(i0).

Let

K

111

-

be any commutative dZ ~-~ Hq(G,K),

(i)

H q+2 (G,R(K))

(ii)

H 2 (G,RcK)) - ~

K/n~j,

ring.

for all

where

Then

q > 0;

n = [G[,

and

d (iii) H!(G,R(K)) ideal of

where

~

is I the augmentation

KGo If we had used Tate cohomology,

Remark, and

gG r

~

(iii) would be special cases of (i).

due to Tate

(cf. Kawada

0 ---~ r ---) KF ----) KG --~ 0 mentation ideals of

MOdKG. where

{/~r

We know (xi)

~: F ----) G

K = ~

is

and so, if

{,g

yields are the aug-

KF, KG, respectively,

0 ---> rl~r ----) {l{r ---) g --~ 0

(11).

(ii)

[3]).

The homomorphism

Proof,

The case

then

is KG-free on all

freely generate

F

and that

(i-

is exact in

x i) + fr,

~/{r ~ ~(K).

we

also have the exact sequence (12). and (12) yield

0 --~ g --~ K G - - ~ (i);

(ii) since

0.

Obviously

(Ii)

Finally,

(ii)

also

CKG)G ---~ K - - ~ gives

K--~

Hl(g) - ~

(KG) G = KT, where

H 2(RCK )- ) T = Z x. xs G

112

yields

(iii)

(~/{r)G __) gG

since

We may now complete characteristic basis

of

dividing

H 2 (G,A)

the i s o m o r p h i s m

-

and

(8).

(9) .

JG J . y

in the

zero map.

So

is a f i e l d of

nK = 0

Thus

a basis

K

a n d so

H2(G,B).

of

x

Let

is a #

be

Then x~* = ky

a nd h e n c e

~

gives

Remark 9

Zp, f o r any

is

(A,x)

an i s o m o r p h i s m :

The proof

of

N\

J

(9) o b v i o u s l y

(B,y)

9

also works

if

K

P.

w 4

Result characterization

(I0) c a n be u s e d of p r o j e c t i v e s

We first determines

e

in

E,

way

(AIE) , let

of r e p r e s e n t a t i v e s

in

QKG' w h e r e

show that every

in a n a t u r a l

Given

to g i v e a p u r e l y

element

an e l e m e n t

T =

is a field 9

(A,x)

in

in

QKG

HI(G,A).

be a t r a n s v e r s a l

(t i)

of the cosets)

K

cohomological

A

of

in

E

a n d for e a c h

let tie = ai, e ti(e)

Put ed T = n t -I i i(e) Then

dT

is a d e r i v a t i o n

of

ai

E

,e

ti

in

(e) A.

If

S

( = set

is s e c o n d

113 -

transversal,

then

an element, that when transfer char K

ds

call A

is cohomologous

it the transfer

is central

in

homomorphism.) dividing

class,

E, then

dT

of as an element

Ker d T

conjugate:

another (13).

proof

If

HI(E,A).

have this

of Schur's

~ = Z x 6 G,

(Note

is the ordinary K

a field with

in

class

HI(G,A).

When

is a subgroup

and since all such subgroups

So we obtain

and so the transfer

Remark due to B. Wehrfritz: an automorphism,

d T.

in

In our case of

JGJ, Ad T = 1

can really be thought

to

A ~

A

is

complementary form,

to

A

they are all

theorem!

then the isomorphism

x6G

(i0) (iii) maps

~

to the transfer

class

of

(R,x)

-

(14). IGJ, then

If the field

(A,x)

is projective

(i)

HI(G,A)

K

has c h a r a c t e r i s t i c in

QKG

H2(G,A)

dividing

if, and only

has d i m e n s i o n

if,

one over

K

and

has d i m e n s i o n

one over

K

and

is a trivial

consequence

the transfer (ii)

9 (K)

class ~ 0;

x~O. Proof. (10) and

(6).

epimorphism Then

#

One d i r e c t i o n

Conversely, r

induces

(R'•

assume

(K)

isomorphisms

(i) and

of

(ii) and pick an

) (A,x). Hi(~,(K))

N)

Hi(A),

i = 1,2,

114

and hence

H2(p)

projective

(cf.

P

(K

= 0

where

[5], chaper

-

P = Ker~.

Thus

9) and hence

P

is KGsplits over

R(K)

is a field!). Somewhat (15).

if, for all

similar arguments yield

(A,x)

is projective

p_ and any Sylow p-subgroup (i)

H I (Gp,A) = 0,

(ii)

H2(Gp,A) order

(iii) A

is

(16). is projective

If

in

~pG'

(A,x)

~

for all

if, and only

Gp ,

is cyclic O n

xRes,

of

IGpl ; and T-free.

There is also a localization

tive in

~_

in

principle:

is finitely generated,

if, and only if,

(A,x) (Tp)

then

(A,x)

is projec-

p.

w 5

/-%

ZG (B,y)

~I

P/IGI

We return now to minimal projectives. Z(p)

and write

are in the same genus (17).

M G = M | Z G. if

(A,x) G ~

Let

We shall say

(A,x) ,

(B,y) G.

Any two finitely generated m i n i m a l projectives

are in the same genus.

-

We omit the proof having

had crucial

115

here,

-

but I wish

help from Irving

to a c k n o w l e d g e

Reiner with one part of

this proof.

Finally, of this

theory with m u c h

Let

K

We shall call Frattini

(AIE)

extensions

This was

[2].

It follows

the following

is m i n i m a l

of some

extension

The Frattini Frattini

to

if

(AIE)

IGI.

A & Fr(E),

is m a x i m a l

(AIE)

It is not at all obvious

dividing

the

if any

is n e c e s s a r i l y

that m a x i m a l

Frattini

In fact they do and any two are isomor-

established

by Gasch~tz

from our theory

(for

K = ~p)

as a consequence

of

in 1954 (9) and

theorem:

(18). then

E.

even exist.

phic.

connexion

earlier work of Gasch~tz.

a Frattini

from another

an isomorphism.

a surprising

be a field of c h a r a c t e r i s t i c

group of

epimorphism

IGI,

we m e n t i o n

(AIE)

If

K

is a field of c h a r a c t e r i s t i c

is m a x i m a l

projective.

Frattini

if, and only

dividing if,

(AIE)

-

116

-

REFERENCES

[1]

Artin,

E. and J. Tare,

Class Field Theory, Benjamin,

New York, 19679 [2]

Gasch~tz, Gruppen,

W.,

"Uber modulare Darstellungen

die von freien Gruppen induziert werden",

Math. Z. 60; 274-286. [3]

Kawada,

9; 417-431.

J. Fac

Bull

Amer

Math9 S o c ,

(1965).

Serre, J.-P.,

SQ

(1963).

S , "Categorical Algebra"

71; 40-106. [5]

(1954)9

Y., "Cohomology of group extensions"

Univ. Tokyo, [4] MacLane,

endlicher

Corps locauz, Hermann,

Paris,

1962.

-

ALGEBRAS

117

-

GRADED BY A GROUP

by Max A. Knus*

INTRODUCTION

A Brauer

theory

for ~ / 2 X - g r a d e d

d e v e l o p e d by C. T. W a l l in on a l g e b r a i c we

K-theory,

try to define

an arbitrary simple

algebras Before

Bombay,

such a theory

abelian

algebras

group G.

(See also H. Bass, 1967,

[3]).

a structure

algebras.

a class of graded algebras w h i c h

Clifford

algebras.

many s u g g e s t i o n s

class of

graded tensor product.

we study

thanks

This

group, we prove

t h e o r e m for graded central simple

My sincere

graded by

We first define graded central

a suitable

the Brauer

Lectures

In this paper,

for algebras

and give some examples.

is closed under

defining

[7],

algebras was

Finally,

generalize

are due to Michel Andre

and discussions.

*This w o r k was c a r r i e d out under a grant from the N a t i o n a l Science F o u n d a t i o n and a fellowship from the S c h w e i z e r i s c h e N a t i o n a l f o n d s .

for

- 118

io

-

GRADED CENTRAL SIMPLE ALGEBRAS

By algebra, w e s h a l l m e a n a finite d i m e n s i o n a l associative field

algebra

A

w i t h u n i t over a c o m m u t a t i v e

K.

Let

G

not necessarily

be a group, w r i t t e n a d d i t i v e l y , b u t abelian.

A G-@raded al@ebra

A

is an

a l g e b r a w h i c h is given t o g e t h e r w i t h a d i r e c t s u m d e c o m p o s i t i o n as a m o d u l e A =@ A g6G g w h e r e the A ' s g

are s u b s p a c e s

of A, in s u c h a w a y

that

AA cA g h g+h The e l e m e n t s

of

K

are h o m o g e n e o u s

of d e g r e e

zero.

A

h o m o m o r p h i s m of g r a d e d a l g e b r a s is a h o m o m o r p h i s m of algebras A

subspace

~: A-4~ B I

of

such t h a t A

of the i n t e r s e c t i o n s

~(Ag) c Bg,

is g r a d e d if it is the d i r e c t su/n INA . g

We call the g r a d e d a l g e b r a are n o p r o p e r g r a d e d

g 6 G.

(two-sided)

a l g e b r a is n o t n e c e s s a r i l y

A

ideals.

s i m p l e if there A graded simple

a s i m p l e algebra, b u t the

f o l l o w i n g e a s y g e n e r a l i z a t i o n of the T h e o r e m of M a s c h k e is true.

-

Theorem

characteristic over

K,

A

be

of

then

x

0

A.

Since

= i.

Hence

does is

Let

A

graded

not

algebra.

divide

a semisimple

x = r. x , gEG g

is s i m p l e Trace(x)

the

dimension

= Dim A

of

algebra.

x ~ 0, b e

graded,

If the

we

in

may

is n o t

the

radical

assume

zero

that

and

x

is

K

not nilpotent.

Let

Therefore

~:

~(gl

group

In p a r t i c u l a r

Let

to s i m p l i f y

~(0,h)

K.

= ~(g,0)

A = 9 A gEG g

the n o t a t i o n ,

a, d e g r e e

graded

algebra

x

A

or

of

be

such

A

This

must be

of

G

zero.

in

means and for

g i ' h.1

in

G.

= I.

a G-graded

we

A

algebra.

shall write

~(a,b)

In o r d e r

for

b).

~

be

is

central

that,

ax = ~(a,x)xa

be

of

a pairing

= ~ ( g , h I) ~ ( g , h 2)

Let n o w

of

K*

+ g2 'h) = ~ ( g l 'h) ~ ( g ~ , h )

~ ( g , h I + h 2)

~(degree

the r a d i c a l

G x G ~

the m u l t i p l i c a t i v e

A

a simple

K A

Proof.

of

-

1.1

Let

A

119

a fixed pairing. if

the

xa = ~(x,a) ax

We

say

that

only homogeneous

the elements

for all h o m o g e n e o u s

for a l l h o m o g e n e o u s

a

in

A

a

in

are in

K.

-

Examples of ~raded 1.2

central

Let

G

trivially

on

class of

H z (G,K*).

Take

al~ebras

be the second

coefficients

K*.

-

simple

H 2 (G,K*)

group of

on the space

with

1 2 0

in

K*, where

a normalized

We define

of all formal

cohomology

cocycle

an associative

linear

G

operates

f

in any

multiplication

combinations

g6G g g with

coefficients

~

6 K

g

by setting

x We shall denote KfG.

If

f

group algebra cohomologous resulting K.

See

g~hf (g,h) Xg+h

=

g6G g g

h6G

g,h6

the G-graded

is the trivial KG

of

G

cocycles

algebra

thus

cocycle,

KfG

over

define

class is called

by

is simply

the

We remark

isomorphic

that two

graded

an algebra e x t e n s i o n

algebras. of

G

over

[8] for more details. An algebra e x t e n s i o n

Suppose

K.

constructed

now that

cocycle,

G

is abelian

and that

graded

f

simple.

is an abelian

i.e. f(g,h)

The algebra e x t e n s i o n satisfies

is obviously

= f(h,g) is central

the following Non-de@eneracy

g ~ 0, there exists

g,h 6 K

.

if the given p a i r i n g

condition: condition:

h 6 G

such that

for every ~(g,h)

g 6 G, ~ i.

The

121

-

In p a r t i c u l a r ,

the

group

algebra

then

is g r a d e d

central

simple.

It is e a s y if the means

group

G

that,

is

if

a primitive

G

n-th

to c o n s t r u c t finite

has root

direct

Pi p r i m e s Choose the

of

i).

1

a primitive

p~i-th

where

gi

s ~'

component 9:

of

g

)

K*

G • G

satisfying version ing.

of this

ug =

~u gi i i

space

root

if

~i

and

by

we

V K,

(this

K contains

be G

P l ~ P2 ~

all

i

=

of the

appearing

(*). = u gh

condition.

in

of

of the i - t h

The m a p is

a pairing

In the

only

the u s e

this

an a b s t r a c t

group.

therefore to see

first

special

dimensional

abelian

difficult

class

The

pair-

pairing.

G-graded algebra

graded

simple.

that

EndK(V)

"'''

"'" = Pq"

u gh = ~ugi hi i 9

a finite

any

that

de fine

considered

graded,

it is n o t

then

P m = Pm+l

for

~(g,h)

suggested

Let

G-cyclic

n,

such

the d e c o m p o s i t i o n

paper,

over

G

g,h 6 G

defined

is s i m p l e ,

Furthermore

of

a representative

Andr~

1.3.

EndK(V)

For

in

is

of o r d e r

" > rq

the n o n - d e g e n e r a c y

Michel

vector

(*).

is

K

G = 9 z/prlz

rm

sum

and

pairing

Let

sum decompositioa

and

direct

abelian

an e l e m e n t

(*) be t h e

a nondegenerate

is

-

graded

central

(for any p a i r i n g

Remark. algebraically a graded

A

is

which

9).

a finite

T h e map

V.

n-th

to

9 A

graded

J

is t r i v i a l ) .

automorphism

The

and

K

show that

(non-graded)

EndK(V)

is

is

for s o m e

as f o l l o w s 9 n

Let

is the o r d e r

defined by

---> ~ g a g

The r e s u l t

group

can e a s i l y

simple

One p r o c e e d s

,

g is a n o n - t r i v i a l

t h e n one

r o o t of I, w h e r e

~: A : a

cyclic

is c e n t r a l

algebra

space

be a primitive G.

G

as a g r a d e d

G-graded vector

of

If

-

c l o s e d and G - c y c l i c ,

algebra

isomorphic

1 2 2

of

ag 6 A g

A

is c l a s s i c a l

(if

A

is n o n - t r i v i a l l y

if the g r a d i n g

a l g e b r a is c e n t r a l

simple,

of

A

therefore

is i n n e r , Sag = u a ~ u -I for some

u 6 Ao

The e l e m e n t

scalar

multiple.

of

is c o m p l e t e l y

A

By d e f i n i t i o n

Ag = In p a r t i c u l a r , ~n

is t r i v i a l ,

to an e l e m e n t

w e see

determined

with

K

by

u,

A

of

u belongs

K.

to

is c e n t r a l Since

u

w e may a s s u m e

is a l g e b r a i c a l l y

the e n d o m o r p h i s m

4, the g r a d e d s t r u c t u r e

that

un

closed,

algebra

up to a

of

9

therefore ~ ~ 0

is d e t e r m i n e d

{a s A I ~ g a u = ua}

up to a s c a l a r m u l t i p l e , field

u

End

(V)

The a u t o m o r p h i s m

a n d thus is e q u a l

is o n l y d e t e r m i n e d that

hence we K

o

u n = i.

The

can i d e n t i f y

of a v e c t o r

space

A

-

V

over

K.

endomorphism Qu(t)

are

1 2 3

-

The m i n i m a l p o l y m o n i a l u ~g,

is

clearly equal

g 6 G.

Q

(t)

of the

t n - i.

to

The r o o t s

of

Set

Vg = {x 6 V I u(x) Since

Q u (t)

is a p r o d u c t

= ~gx}

of distin~

9

linear

factors

over

U

K, V=|

. g6G g

Finally

one v e r i f i e s

Ag = {

that

End K

l ~: V h ~

V g + h , V h 6 G}

as it s h o u l d be. 2% G R A D E D T E N S O R

A graded a igeb ras

A

and

tensor product B

is the u s u a l

PRODUCT

A | B

of two G - g r a d e d

graded vector

space

A @ B = ~ (A @ B) g6G g whe re (A @ B) g = h+h'| = g A h with

|

KBh S

the m u l t i p l i c a t i o n (a | b)

(a' | b')

a,a I 6 A The m a p defined

~

is a p a i r i n g

= ~(b,

and of

G

b,b' in

a j

)aa'

6 B K*.

| bb' .

The p r o d u c t

thus

is a s s o c i a t i v e .

Example.

Let

G = ~/2Z.

The u s u a l

graded

tensor

-

p r o d u c t corresponds

Let

~

to

respect

and

over

9.

@ C --~ A @ (B | C)

b u t in general not commutative,

V

to

All t e n s o r Let

A, B and

The tensor p r o d u c t is associative,

(A @ B)

v e c t o r spaces

(-1) gh.

now be a fixed pairing.

be graded algebras.

Let

-

~ (g,h) =

p r o d u c t s w i l l be taken w i t h C

124

Va

K.

be

,

A @ B ~ B | A.

finite d i m e n s i o n a l

The algebra

EndK(V)

G-graded

is g r a d e d and

~ n ~ (v~ ~ ~ n ~ (v') -~ ~ n ~ (v ~ v' ) if we de fine (f | f')

Proposition

A

A | B

and

x) fx | f'x'

B

are graded central simple

algebras,

is also graded central simple.

Proof. for example

~(f'

2.1

If then

(x @ x')

in

The proof

for the n o n - g r a d e d

cases

given

[4], can be used w i t h some trivial

modifications.

Let A*

of

A

A

be a graded

is i d e n t i c

to

A

algebra.

The opposite

as a vector space

algebra

and has the

multiplication a'b* = ~ ( a , b ) b a The algebra

A*

is obviously

.

central simple if

A

is so.

-

1 2 5

-

There is a n a t u r a l h o m o m o r p h i s m ~: A | A* ~ d e f i n e d by

of graded

algebras

EndK(A)

(a | b*)x = ~(b*,x) axb

if

G

is abelian.

P r o p o s i t i o n 2.2 is an i s o m o r p h i s m

The n a t u r a l map central

A

is

simple.

Proof. example

The same

as in the n o n - g r a d e d

See for

case.

[4].

3,

A STRUCTURE

A pairing ~(g,h)

if

= ~(h,g)

T H E O R E M FOR G - G R A D E D ALGEBRAS

is called s y m m e t r i c

q0: G • G --~ K*

for all

g,h

in

Suppose

G.

that

G

if

is abelian.

T h e o r e m 3.1

Let

~

be a s y m m e t r i c p a i r i n g

the n o n - d e g e n e r a c y of

G.

Then a graded

char K ~ d i m K A

H

and

H'

f

to a graded

Kf(H)

)

| of

graded by

G

is an abelian

G

A

such that

algebra of the type

.

such that

class

in

extension

H 2 (H,K*) of

H

is a m a t r i x a l g e b r a over a d i v i s i o n H'.

satisfying

subgroup

algebra

is i s o m o r p h i c

is the c o r r e s p o n d i n g

~n(D,H')

for any finite

central simple

are subgroups

G ~ H x H'. Kf(H)

condition

of

over ring

and K. D,

-

Proof. Z = The s p a c e

H =

Z

of

Therefore

Z

0

be

Z is

contains

is the

contained

have

Zh =

h

G,

defines

an a b e l i a n

Another

choice

of

cohomology

9

is

graded

subgroup

the

graded

Zh, h

is d i f f e r e n t

hence

simple. H

dimensional.

in

~,

homogeneous

in e a c h

zero

n a n d is

We

~ ~ 0.

Z h.

of

If

Zh, h 6 H,

all

space

center

6 H. from

~ 6 K,

The

are

We know

= f ( h , h ' ) X h + h,

x's h

class.

A,

the e q u a l i t y

2-cocycle the

be

xh ~ 0

therefore

a 6 A}

The

in

~n =

Z0x h = Kxh,

A

finite

xn h

of

Z = @ Zh, w h e r e h6H

G.

x h ~ 0~

in

all

center

of two n o n - z e r o

contained

X h X h,

same

of is

Choose

# 0,

for

since

Z 0' t h e r e f o r e

ZhX h c

Xh~,

A

Let

dimensional.

that

zero,

a n d is

of

in

the u n g r a d e d

The p r o d u c t

because K

-

graded,

a subgroup

Z 0 = K.

order

be

certainly

is n o t

finite

A, h e n c e

one

is

H

Z

{x 6 A I xa = ax

{h 6 G I Zh ~ 0}.

elements

must

Let

126

f

of

would

We have

H

with

define

thus

value

in

a cocycle

constructed

K*.

in the

the p a r t

Kf (H). Let n o w

B

B = zx

As

zx = xz

for

=

be

the

graded

centralizer

{x 6 A I xz = ~ ( x , z ) z x ~(z,x)xz,

all

z

in

Vz

Z,

E

z}

9

Vz 6 Z

of

Z or

in

A,

127

-

B = (9

{x 6 A I ~(z,x)

is s y m m e t r i c

element

! ).

f 6 HOmZ|

= 1

We s h o w

(Z ,A)

f((z I @ z*)x) 2

-

for all

that

is a m a p

z 6 Z}

B--~ H O m Z | f: Z

An

> A

such

= ~ ( f , z I @ z*)z | z*f(x) 2 I 2 = ~0(f,z I | z ~ ) q 0 ( z 2 , f x ) z l f ( x ) z 2

L et

P: H O m Z |

It f o l l o w s zf(1) B.

(Z ,A)

from

) A

= ~ ( z , f ( 1 ) ) f(1) z

for all the map

= zb, b 6 B, is a

non-degeneracy

condition

graded

central

simple

Hence

B ~--

n ~ (z)

z fb:

in

~

f(1).

that

Z, h e n c e

Z ~

f

A

f(1)

is i n

d e f i n e d by

Z | Z*-homomorphism.

By h y p o t h e s i s ,

Hom E

the m a p d e f i n e d b y

f((z | i) i) = f ( ( 1 | z) l)

On the o t h e r side,

f(z)

be

the p a i r i n g on

H.

~

satisfies

Therefore

and, by P r o p o s i t i o n (Z,A).

the

Z = Kf (H) 2.2,

By P r o ~ o s i t •

is

Z | Z* = E n d K ( Z ) .

A6 of

[2],

the map (Z,A)

: Z | K g i v e n by

0 (x | f) = f(x)

over

The

K.

corresponding

x | f(1)

=

which

an i s o m o r p h i s m

Z | B,

is ZB

(x | l) f(1)

and

A.

~ A

E n d K (Z) is an i s o m o r p h i s m isomorphism

= xf(1)

(see

of a l g e b r a s .

of v e c t o r

spaces

Z | B ---~ A is d e f i n e d b y

[2'] T h e o r e m Hence we

3.1.)

can i d e n t i f y

-

Let now or

x

be

in the g r a d e d

bx = ~(b,x)xb

xz = zx or

for

for

of

central, be in

because and

not difficult

therefore

is g r a d e d

and

Z

to see t h a t

B, xb = ~ ( x , b ) b x M(x,z)

= 1

and

xzb = ~ ( x , z b ) z b x x

is in

central.

any e l e m e n t B

of

Since

z 6 Z, w e can w r i t e

A, B

Z,

-

center

b 6 B.

zbx = ~(zb,x) xzb,

center

128

B

K,

the g r a d e d

is a l s o u n g r a d e d

of the c e n t e r

of

B

would

K.

It is

are d i s j o i n t

over

B

simple.

is g r a d e d

Since

char B

K ~ dimKA , char K ~ dim B a n d b y P r o p o s i t i o n 1. i., K is s e m i s i m p l e . A s e m i s i m p l e c e n t r a l a l g e b r a is s i m p l e ,

hence

B

is

(ungraded)

Let H' = of

H'

be

{g s G I ~(g,h) G.

Suppose

there exists

of

B,

Zh

know

Z

and

B

H N H' = 0. G=

Remark

would

B,

is a s u b g r o u p

h 6 H N H', h ~ 0. be

contained

are d i s j o i n t

that,

H'

of

if

over

Ag ~ 0

By

in

~,

K.

Hence

for a l l

but we

g 6 G, t h e n

H • Kso

Suppose The c o n s t r u c t i o n the n o n d e g e n e r a c y if

the s e t of d e g r e e s

= i, V h 6 H}.

definition that

simple.

K

showed

that

of E x a m p l e condition

is a l g e b r a i c a l l y in E x a m p l e

K

is a l g e b r a i c a l l y 1.1 g i v e s

closed

a pairing

for

Ho

Furthermore

closed.

If

G

1.2 t h a t

M

n

(K,H')

and G-cyclic.

satisfying H z (H,K*)

is c y c l i c , w e is the e n d o m o r p h i s m

= 1

129

-

algebra

of a vector

Corollary

order.

simple

to an algebra

be a l g e b r a i c a l l y

Suppose A

that

K

such that

End K(V)

2)

KG | End K(v)

where

V

Remark.

The hypothesis

Let

3.2.

~: G • G---~

G-graded

central

if there exist

simple

central

G

is ungraded.

that

above.

char K ~ dimKA

A direct proof

K*

4 in

is

can be

[7].

GROUP

be a fixed p a i r i n g

field

algebras

two G-graded

of

is isomorphic

immediately

of Lemma

in a commutative

A G-graded

End K(V)

from the remarks

the proof

G cyclic

is graded by

where

in C orollar~

and

types:

follows

following

G

closed

char K ~ dimKA

4. THE B R A U E R

group

H'.

is G-cyclic.

of the following

i)

not necessary given,

K

algebra

The proof

graded by

3.2 Let

prim~

space

-

vector

K. A

of abelian

We say that two and

B

spaces

are similar V

1

and

V

2

such that

A

En

Cv I)

as graded

algebras.

Let

classes.

By P r o p o s i t i o n

induces

a group structure

B

B(K,G)

En Cv2> be the set of e q u i v a l e n c e

2.1 and 2.2, on

B(K,G).

the tensor p r o d u c t The classical

-

(nongraded)

Brauer

group

1 3 0

-

is certainly

B (K)

c o n t a i n e d in

B (K,G). Exan~le and

G

that

4.1,

Let

cyclic of prime B(K,G)

of

of primes

order.

n

copies

of

9

Let

G

field.

Let

K G

contains

ALGEBRAS

abelian

group

and ~

K

a

constructed

be the n - f o l d graded

tensor

n

If

G

to

~)

of the group algebra

is cyclic

el, i = i, 2, m

of order

...n

is a p r i m i t i v e

subject

KG is d e s c r i b e d

to the relations

n

i < j,

m-th root of i.

cyclic, was

m, KnG

i = i, 2,...,

eie 4 = ge.e. ]l J

K G, G

B(K,G)

G.

OF C L I F F O R D

be a finite

ei = 1

of

is any finite

m

(with r e s p e c t

by generators

The

(non-graded)

d e s c r i b e d by Morris

structure

[6], w h e n

m

n

odd or

m

is even

and

shall give a complete abelian

groups.

3.2

is the n u m b e r

We shall use the p a i r i n g

in E x a m p l e 1.2.

w i t h itself.

G

n

closed

from C o r o l l a r y

If

Z/2Z, w h e r e

a p p e a r i n g in the order of

i

product

It follows

char K = 0, one can see ~hat

5. A G E N E R A L I Z A T I O N

G-cyclic

be a l g e b r a i c a l l y

is cyclic of order two.

cyclic group and product

K

~

has

a square r o o t in

description

for a r b i t r a r y

K.

We

finite

is

a

-

Let that

KG*

KG*

131

-

be the opposite

is t_h~ algebra

algebra

extension

of

of

G

to the cocycle

9)

n-fold

tensor p r o d u c t

KG* | . .. | KG*.

K G n

and

K G* q

are connected KnG* nG means

The proof

[i].

Clifford

algebras

K n G*

by the following

l| K2G*

ungraded

The algebras relations:

Kn+2G*

to the proof

given

and

and

!) tensor products.

(without

The algebras C ni

K

[| I K2G ~- Kn+2 G

[5] is similar

algebras

.

(remark

for the

corresponding graded

Write

over

KG

if

Cq

for Clifford

~G*

G

are the

is the cyclic

group

of order two.

KqG*, for

TO describe

completely

the algebras

KnG

and

it is therefore

sufficient

to know

and

KiG*

i = 1,2.

Using

the results

KIG = K • ...

KiG

of Morris

x K

[6], one obtains

n copies

K2G = M n (I{) n

G.

is the order of

that cyclic

KiG = KiG*.

If

n

is odd,

A more i n t e r e s t i n g

of order a power

contains

a primitive

not have

a square

of 2,

2r-th

root in

KIG* = K(~)

x...

K.

case is w h e n

n = 2 r.

root of

it is easy to see

We k n o w

1, ~.

that

Suppose

Then

• K(~)

2 r-I

copies

is

G K

does

-

where ~2

K (~ )

132

is the q u a d r a t i c

-

extension

of

K s u c h that

~.

=

K2Ge = ~42r-- I ( ~ ) where

is the q u a t e r n i o n i c

generators

and

n

algebra

subject

If

,,, h a s

results

a square

and p r o o f s

root in are g i v e n

given by

and

9

K, t h e n in

K

to the r e l a t i o n s

~2 = n2 = ~ ~n = -n~

over

[5].

K i G = KiG*.

Complete

-

1 3 3

-

REF~

[i]

Atiyah, M. , Bott, R. , and Shapiro, A. , "Clifford modules",

Topology, 3, [2]

Auslander,

(Supplement i) ; 3-38.

Mo, and Goldman,

(1965).

O., "Maximal orders",

Trans~ Amer. Math. Soc., 97; 1-24.

(1960).

[2~' ] Auslander, M. , and Goldman, 0. , "The Brauer group of a com/~utative ring", Trans. Amer. Math. Soc., 97; 367-409.

(1960).

[3]

Bass, H., Lectures on algebraic K-theory, Bombay,

[4]

Herstein,

Z., Noncommutative rings, Carus Publ. #15, MAA,

Providence,

[5]

(1968) .

Knus, M. A., A generalization of Clifford algebras, i0 p.

[6]

(unpublished) .

Morris, A. M., "On a generalized Clifford algebra",

Quart. ~. Math., [7]

(1967).

(Oxford Ser.), 18; 7-12. (1967).

Wall, C. T. C., "Graded Brauer groups",

J. Reine Angew.

Math., 213; 187-199. (1964).

[8]

Yamazaki, K,, "On projective representations extensions of finite groups", J. Fac. Sci.

Sect. I, i0; 147-195. (1964).

and ring

Univ. Tokoyo,

134

-

Diagonal Arguments

-

and Cartesian Closed Categories

by F. William Lawvere

The similarity between the famous arguments of Cantor, is well-known,

and suggests that these arguments

Russell,

G~del and Tarski.

should all be special cases of a sin-

gle theorem about a suitable kind of abstract structure.

We offer here a fixed-point

theorem in cartesian closed categories which seems to play this role. Cartesian closed categories

seem also to serve as a common abstraction of type theory and propositional

logic, but the authorts discussion at the Seattle conference observation will be in part described pear in Dialectica, Adjoint Functor",

elsewhere

["Adjointness

and "Equality in Hyperdoctrines

of the development in Foundations",

and the Comprehension

of that to ap-

Schema as an

to appear in the Proceedings of the AMS Symposium on Applications

of

Category theory]. 1.

By a cartesian closed category is meant a category C equipped with the follow-

ing three kinds of right adjointsz

a right adjoint 1 to the unique C

a right adjoint • to the diagonal

+ I

,

functor C ~ CxC,

and for each object A in C, a right adjoint C Ax(~ The adjunction transformations be denoted by

6,x

( )A to the functor C.

for these adjoint situations,

in the case of products and by

by A. Thus for each X one has XAA X

A >(AxX)

and for each Y one has Ax~ Given

f-AxX ~ Y, the composite morphism

YEA~Y.

AA,~ A

also assumed given, will

in the case of exponentiation

-

XAA~

(A•

x

will be c a l l e d transform

the

"A-transform"

135

-

A

of the m o r p h i s m

f. A m o r p h i s m

h:X ~ yA

is the A-

of f iff the d i a g r a m AxX

YeA is c o m m u t a t i v e , transform.

showing

Taking

cf~:1 ~ ~

in p a r t i c u l a r

the case

(d~pping

1 -- ~

is of t h a t

the indices

A , Y on

< a,~ one calls sume

~ the

in g e n e r a l Although

categories

"evaluation" that

natural

f is d e t e r m i n e d

we do n o t m a k e

as a l g e b r a i c

f can be u n i q u e l y

X = 1, one has t h a t e v e r y

and that e v e r y

a:l ~ A one has

that

form

~ when

f:A ~ Y

gives

for a u n i q u e

f.

from its ~rise

to a u n i q u e

Since

for e v e r y

t h e y are clear)

> ~ = a.f,

transformation!

note h o w e v e r

b y the k n o w l e d g e

use of it in this paper,

versions

recovered

of type t h e o r y

c a n be

of all

its

t h a t we do not as"values"

the u s e f u l n e s s further

a.f.

of c a r t e s i a n

illustrated

closed

by assuming

t h a t the c o p r o d u c t 2 = I+1 also exists

in C. It then

follows

(using

the c l o s e d

structure),

that

for e v e r y o b j e c t

A A• and so in p a r t i c u l a r

that

= A+A

2 is B o o l e a n - a l g e b r a - o b j e c t

in C, i.e.

that among

the m o r -

phisms 2x2x...x2 in C there truth

are w e l l

tables,

of B o o l e a n

determined

and that these

algebra.

morphisms

satisfy

Equivalently,

"C-attributes

X along

for e a c h X the

of type X" b e c o m e s

any morphism

of C induces

corresponding

to all the

all the c o m m u t a t i v e

Pc(X) of

~ 2

diagrams

finitary

(two-valued)

expressing

the axioms

set

= C(X,2)

canonically

an actual

contravariantly

Boolean

a Boolean

algebra,

homomorphism

and v a r y i n g

of a t t r i b u t e

136

-

algebras. among

The m o r p h i s m s

these

For a n y

I ~ 2

form

are the two c o p r o d u c t

"constant

truth-value.

Now noting

PC(1)

the B o o l e a n

injections

x:l ~ X

of type X"

-

algebra

of " t r u t h - v a l u e s " ;

w h i c h p l a y the r o l e s o f

and a n y a t t r i b u t e

"true"

and

"false",

~ of type X, x . ~ is t h e n a

that X• x

> 2 (2) CX

is a " b i n a r y o p e r a t i o n "

we c o u l d w r i t e

it b e t w e e n

X E ~'I

an e q u a l i t y of t r u t h v a l u e s ; the s u b s e t presses

of X c o r r e s p o n d i n g

the u s u a l

Returning surjective

iff for e v e r y

domain

an e v e n w e a k e r

for our

fixed p o i n t

there

exists

of the

x:l - X

of Z",

since

section)

inverse

n o t i o n of s u r j e c t i v i t y theorem.

a morphism

g is p o i n t - s u r j e c t i v e

from an e l e m e n t

as the c o n s t a n t

naming

@, o n e sees that the a b o v e e q u a t i o n

we d e f i n e

(as in the n e x t

transformation

limit of Z comes

concern,

"onto the w h o l e if

c~1 :i ~ 2 X

ex-

axiom.

z:i ~ Z

I; for e x a m p l e

then a natural

~,

to the a t t r i b u t e

to our i m m e d i a t e

so t h a t w e h a v e

= X.q),

if w e t h i n k o f

"comprehension"

i m p l y that g is n e c e s s a r i l y with

thus

its a r g u m e n t s ,

g:X ~ Z

with

xg = z. T h i s

t h e r e m a y be

X and Z are

does n o t

few m o r p h i s m s

set-valued

if e v e r y e l e m e n t

limit o f X.

to be p o i n t -

functors,

o f the i n v e r s e

In c a s e Z is o f the

c a n be c o n s i d e r e d ,

which

form

in fact s u f f i c e s

Namely g x

w i l l be c a l l e d w e a k l y p o i n t - s u r j e c t i v e for e v e r y

iff for e v e r y

there

is x s u c h that

a:l ~ A ( a,xg

>e = a . f

F i n a l l y we say t h a t an o b j e c t Y has the morphism Theorem

f:A -- Y

t:Y ~ Y

there

In a n y c a r t e s i a n

point-surjective

is

y:l ~ Y

closed

with

category,

fixed-p.oint p r o p e r t y

iff for e v e r y e n d o -

y.t = y. if t h e r e

exists

an o b j e c t A and a w e a k l y

morphism g

t h e n Y has Proof:

the f i x e d p o i n t p r o p e r t y . Let ~ be

the m o r p h i s m

whose

A-transform

is g. T h e n

for a n y

f:A ~ Y

there

-

is

xtl ~ A

such that

for all

any endomorphism

t of Y a n d

>g = a.f. let f b e the c o m p o s i t i o n

A6 A thus t h e r e

is x s u c h t h a t

a(Ab) The

=

< a,a

famed

Cantor's

theorem

Corollary

then

If there

A does

>. But t h e n

"diagonal

there

type

theory!

argument.

argument"

< a,a

exists

< x,x

>gt is c l e a r l y

a fixed p o i n t

just the c o n t r a p o s i t i v e

t:Y ~ Y

such that

yt 9 y

for all

could have been

relation,

( no e x p o n e n t i a t i o n )

v i t y as a p r o p e r t y

y:l ~ Y

then

for no

morphism).

that

set t h e o r y be f o r m u l a t e d

universe,

w e do n o t n e e d

In fact w e n e e d o n l y a p p l y the p r o o f o f our theorem,

ly, our t h e o r e m

o f our theorem.

morphism

for A the s e t - t h e o r e t i c a l

membership

for t.

Y = 2.

P a r a d o x d o e s not p r e s u p p o s e

t h a t is,

>Y!

>g

is o f c o u r s e

follows with

set-theoretical

products

>g =

exist a point-surjective

Russells

t >Y

y =

( or e v e n a w e a k l y p o i n t - s u r j e c t i v e

2.

g >AxA

for all a < a,x

since

-

a:l ~ A < a,x

Now consider

1 3 7

dispensing

with

with g entirely.

stated and proved

by simply phrasing

That

2 A for the

g:AxA ~ 2

as the

is, m o r e

general-

in a n y c a t e g o r y w i t h o n l y the n o t i o n

of

(weak)

as a h i g h e r

finite

point-surjecti-

of a m o r p h i s m A x X ~ Y!

however

discovering

require

thinking

elements

of X,

the l a t t e r

form

(or at l e a s t c a l l i n g

of such a m o r p h i s m

suggesting

it s u r j e c t i v i t y ! )

as a f a m i l y of m o r p h i s m s

that a closed category

is the

A ~ Y

"natural"

seems

to

i n d e x e d b y the

setting

for the the-

orem. In fact the m o r e products)

follows

the f o l l o w i n g full c l o s u r e

form of the t h e o r e m

from the c a r t e s i a n

remark. under

general

Notice

that

finite p r o d u c t s

just a l l u d e d

to

(for c a t e g o r i e s

closed version which we have proved,

it w o u l d

suffice

to a s s u m e

of the two o b j e c t s

A,Y)

C small

with

by virtue

(just take the

of

-

Remark

138

-

Any small category C can be fully and faithfully embedded in a cartesian closed

category in a manner which preserves any products or exponentials which m a y exist in C. Proof:

We consider the usual embedding C~__ ~ Cop

which identifies an object Y with the contravariant set-valued functor x ~

(x,Y).

By "Yoneda's Lemma" one has for any functor Y and any object A that the value at A of Y AY ~ ~C~ where the right hand side denotes the set of all natural transformations

from

(the

functor corresponding to) A into Y, so that in particular the embedding is full and faithful.

It is then also clear that the embedding preserves products

(in particular

if 1 exists in C it corresponds to the functor which is constantly the one-element set, which is the 1 of ~C~

For any two functors A,Y the functor C~C~

plays the role of ~ .

In particular if B A exists in C for a pair of objects A,B in C

then (C)BA - C(C,B A) J C(AxC,B)

-~COP(AxC,B)

showing that the embedding preserves exponentiation. Theorem

Let A,Y be any objects in any category with finite products

(including the

empty product I)~ then the following two statements cannot both be true a) there exists such that for all

~:AxA ~ Y

such that for all

f:A ~ Y

there exists

x:l ~ A

a=l ~ A < a,x >~ = a.f

b)

there exists

t:Y ~ Y

such that for all

y:l ~ Y

y.t&y. Proof:

Apply above remark and the proof in the previous section.

Of course the "transcendental" proof just given is somewhat ridiculous,

since the in-

compatibility of a) and b) can be proved directly just as simply as it was proved in the previous section under the more restrictive hypothesis on C. However we wish to

139

-

take the o p p o r t u n i t y o f an a r b i t r a r y denote

to m a k e

(small)

the s m a l l e s t

"definable"

braic ries

with definable alternative are

it u s u a l l y

to be at l e a s t p a r t l y definition.

a natural

rator have

to the)

call a natural

transformation given objects

operator

for all of $ C o p u n l e s s a partial

for C itself, Recall

result

is d e t e r m i n e d

c a l l the e l e m e n t s

(let the

latter

which

contains

C). One

structure

is d i f f i c u l t

theories

all p o s s i b l e

semantics

of alge-

if the e l e m e n t a r y

transformations for d e f i n a b l e

are o b j e c t s

is to con-

to o v e r s e e

enumerating

functorial

in the p r e s e n t

case.

theo-

are i d e n t i c a l ones.

Thus

The

latter

for e x a m p l e w e

in a c a t e g o r y

"values"

of the set

between

the e x p o n e n t i a l

in $ C~

(hence in C).

functional.

C = I~ h o w e v e r

in m o r e

"1 is a g e n e r a t o r

b y its

C

C with

fi-

of the

(func-

>D C

a natural

in t h a t d i r e c t i o n .

we c a n d e s c r i b e

that

(e.g.

embedding

operator

shall be s i m p l y a n a t u r a l

we would

this

substitute

If A , B , C , D

BA

tors c o r r e s p o n d i n g

of ~ C o p

that n a t u r a l

true

category

requires

of elementary

ones or at l e a s t a r e a s o n a b l e

led to the f o l l o w i n g

closed

canonical

into a h i g h e r - o r d e r

situations

semantics

the a b o v e

etc.! h o w e v e r

one h a s c o m e to e x p e c t

seems

nite products,

in m a n y

about

subcategory

a structure

p o i n t of v i e w since

or f u n c t o r i a l

are complete)

closed

operators,

On the o t h e r h a n d

theories

remarks

into a c a r t e s i a n

of e m b e d d i n g

functionals,

from a s i m p l e - m i n d e d definitions.

category

further

full c a r t e s i a n

of the s t a n d a r d w a y s sider

some

-

it m i g h t

In fact,

familiar

Note

C(I,X)

for

of p o i n t s

in p a r t i c u l a r

that

1 will

conceivably

be

if

C = I

not be a g e n e -

so for C, and we

in the c a s e t h a t 1 is a g e n e r a t o r

terms w h a t

for C" s i m p l y m e a n s

x.f:1 ~ Y

functors

a natural

operator

that a morphism

x : l ~ X.

In that c a s e

is.

f:X ~ Y

in C

it is s e n s i b l e

of X a l s o the e l e m e n t s

of X. T h e n a

function C(1,X) is i n d u c e d b y at m o s t o n e C - m o r p h i s m language

t h a t the

Proposition rator,

operator

are o b j e c t s

it is, w e

say by abuse of

o f C.

t h a t C is a c a t e g o r y w i t h

and t h a t A , B , C , D

I) a n a t u r a l

X ~ Y, a n d in c a s e

f u n c t i o n i_~s a m o r p h i s m

Suppose

~ C(1,Y)

finite p r o d u c t s

of C. T h e n

in w h i c h

I is a g e n e -

to

-

140

-

r

BA

>D C

is entirely determined by a single function Ir C (A,B)

> C (C,D)

9and 2) such a function determines a natural operator iff for every object X of C and for every C-morphism

fsAxX ~ B, the function (f) (Xr

C(1,CxX) is a C-morphism, where

(f) (Xr

~ C(1,D)

is defined by

,o,x

for any

=

co,

c:l ~ C, xzl = X, f denoting the composition x A

Proofz

f

~AxX ~AxI ~ Axx

> B.

We are abusing notations to the extent of identifying a morphism with its

A-transform via the bijections of the form

C(AxX,B) ~ C(AxX,B) ~ C(x,BA). Actually the given operator

r is a family of functions Xr C(X,B A)

> C(X,D c)

one for each object of C! the "naturalness" condition which this family must satisfy, is, via the abuse, that for every morphism

xsX ~ ~ X

of C,

the d i a g r a m

Xr

C(A~X,B)--- ~ C x C(A•

should commute. Now let tion Xr at a given

(CxX,D)

, B ) - - - ~ C ( C x X ' ,D) X'r

X' = I. Since I is a generator for C, the value of the func-

fzAxX ~ B

is determined by the knowledge,

for each element x of X

and element c of C, the result reached in the lower right hand corner by going across then down in the commutative diagram

-

141

-

Xr

c

C (AxX,B)

> C (CxX,D)

C(A,B)

> C (X,D)

>C(C,D)

But since

the

same r e s u l t s

X ~ are d e t e r m i n e d tion

is t h e n clear,

proposition

since

is just

To m a k e exponential

object

perfectly

c a n be d i s c u s s e d

thing

to do.

sional m a n i f o l d s categories

3.

Experts may wish

in the above

fixed-point

theorem

In order concerning

itself,

we

ducts. ples

of)

Thus

iff their

the m o r p h i s m s

are

(classes

of)

of)

formulas

with

ses of)

sentences

responding falsez

1 ~ A

(classes

of the theory.

rise

two

of)

constant

provable

to the class

There

is an

is h o w e v e r

and m o s t

to c o n s i d e r

then

and c o n s i d e r -

between

"natu-

finite-dimenparticular

whether

to o b t a i n

the

Tarski's

for a t h e o r y w i t h i n to a c a t e g o r y

C with

be e q u i v a l e n c e

formulas

terms,

in the t h e o r y whose

pro-

of

(tu-

in the theory.

An ~ 2

is a m o r p h i s m

finite

are c o n s i d e r e d

the m o r p h i s m s

morphisms

the t h e o r y

classes

(or terms)

is p r o v a b l e

of s e n t e n c e s

exist.

does not exist,

C to be these

while morphisms

there

of the

codomain

smoothest

section

in p a r t i c u l a r

In p a r t i c u l a r

asser-

in those cases.

truth

(or equality)

so that

of s e n t e n c e s

corresponding

where

two free v a r i a b l e s ,

n free v a r i a b l e s

to the class

I ~ 2

equivalence

are

terms w i t h

of taking

and let the C - m o r p h i s m s

of the theory,

whose

object

functions

of the p r e v i o u s

h o w a t h e o r y gives

A,2

logical

or C ~

the

second

statement

is an e x p o n e n t i a l ,

They may also wish

of d e f i n i n g

The

its v a l u e s

is a product.

domain

the r e s u l t

the t h e o r e m

in the

that m o r p h i s m s

one has a n y a p p l i c a t i o n s

the i m p o s s i b i l i t y

or terms

whose

given

the e x p o n e n t i a l

domain

all the f u n c t i o n s

first assertion.

of X r p r o v i d e d

notice

functions

considerations.

two o b j e c t s

formulas

equivalent

to c o n s i d e r

of s e c t i o n

the

is in m a n y c o n t e x t s

on r e c u r s i v e

first note b r i e f l y

Consider

whose

then across,

(f) (Xr

even t h o u g h

operators

to a p p l y

theorem

clear,

the m o r p h i s m s

ing t h e m to be the n a t u r a l ral"

of

naturality

instead morphisms

of d e t e r m i n i n g

down

1~, p r o v i n g

such as to assure

just b y c o n s i d e r i n g the p r o b l e m

function

b y going

the d e f i n i t i o n

the s i t u a t i o n

).

c

are o b t a i n e d

b y the one

!o

>C(1

1r

1 ~ 2 true:

and s i m i l a r l y negation

AxA ~ A are

(classes

are

(clas-

1 ~ 2

cor-

a morphism

is p r o v a b l e

in

-

the t h e o r ~ M o r p h i s m s make

2n ~ 2

no use of that e x c e p t If the t h e o r y

not n e e d

the n a t u r e

of those

correspond

to s u b s t i t u t i o n

~:A ~ 2

category

C with

theory.

Models

hom-sets

finite

introduction.

t h e o r y was

to those

with

a higher-order

arguments morphisms

We t h e n formula stant

arise

czl ~ A

~ 2

spelled

b u t we w i l l

such that

Anx2

~ not %

one, with

trivial

more

from the v a r i a b l e s

satisfaction

C ~.

We m a k e

cited

that the

of C are i s o m o r -

we could have

started

of a n y s i g n i f i c a n c e

note

no

b u t the cate-

assumed

all o b j e c t s

modifications

explicit

o f the

to the

that the p r o j e c t i o n

of the theory.

is d e f i n a b l e

in C such that

case

we get a

o f the two p a p e r s

tacitly

to

a unary

category

in the theory,

t h e o r y w i t h no c h a n g e

somewhat

with

functors

of C w e h a v e

composition

1 ~ 2, etc.)

the L i n d e n b a u m

to r e a d e r s

in w h i c h

that d e t e r m i n e s

Defining

composed

a~ not:

as c e r t a i n

w i l l be c l e a r

TM, but

one p o i n t

such that

a:l ~ A

by quantification

above

although

out above.

the s e n t e n c e

can t h e n be v i e w e d

or s e v e r a l - s o r t e d

say that

sat:AxA

2 ~ 2

2 = I+I,

w h i c h m i g h t be c a l l e d

in C i n d u c e d

form

fact that

w i t h not gives

single-sorted

To m a k e

An ~ A

not:

a constant

In our c o n s t r u c t i o n

of the

below.

is a m o r p h i s m

the

of this o p e r a t i o ~

a first-order

phic

to use

products

use h e r e of the o p e r a t i o n

in the

operations,

case:

not e x p l i c i t l y

of the t h e o r y

description

there

(for e x a m p l e

composed

gorical

all p r o p o s i t i o n a l

~:1 ~ 2

we w i l l

formula

include

for the f o l l o w i n g

is c o n s i s t e n t

for all m o r p h i s m s In p a r t i c u l a r

would

142

in the t h e o r y

for e v e r y u n a r y

for e v e r y c o n s t a n t

a the

formula

following

iff there ~:A ~ 2 diagram

is a b i n a r y there

is a con-

commutes

in C

a

i. < a,c

>A

>~

~

AxA

>2 sat

Here we i m a g i n e

taking

condition

traditionally

would

for c a G6del

number

be e x p r e s s e d

for

(one of the r e p r e s e n t a t i v e s

by requiring

that the

of)

~. The

sentence

a sat c w-~ a~ be p r o v a b l e

in the theory,

but

tegory

amounts

same thing.

this

to the

if C a r i s e s

from our c o n s t r u c t i o n

o f the L i n d e n b a u m

ca-

-

Combining the t h e o r e m Corollary

mean,

If s a t i s f a c t i o n

which

-

the a b o v e n o t i o n w i t h our r e m a r k a b o u t

of the p r e v i o u s

In o r d e r

143

section we have is d e f i n a b l e

immediately

often realizable.

to r e q u i r e Namely we

some

further

suppose

of c o n s i s t e n c y

assumptions

that there

and

the

in the t h e o r y t h e n the t h e o r y

to s h o w t h a t T r u t h c a n n o t be d e f i n e d w e

seems

the m e a n i n g

is n o t c o n s i s t e n t .

first n e e d to say w h a t T r u t h w o u l d on the theory,

is a b i n a r y

which

are h o w e v e r

term

subst AxA in C a n d a

("metamathematical")

>

A

binary relation

r ~ c(1,A)xc(1,2) between

constants

I)

For all

and s e n t e n c e s ~:A ~ 2

for w h i c h

there

is

the

c:l ~ A

following

holds.

such t h a t

for all

a:l ~ A

(a subst ic) F(a~) For e x a m p l e w e c o u l d o f the

sentences

applied unary

imagine

which

to a c o n s t a n t

formula

that

represent

dFu

means

G, a n d t h a t

a a n d to a c o n s t a n t

~, y i e l d s

the G 6 d e l

t h a t d is the G S d e l

subst

is a b i n a r y

c which happens

n u m b e r of the s e n t e n c e

number

operation

to be the G ~ d e l

of some one which,

when

number of a

obtained by substituting

a

into ~. Given a binary relation definable

in the t h e o r y

F ~- C(1,A) xC(1,2)

(relative

to

we

F ) provided

say that Truth

there

is a u n a r y

(of sentences)

is

formula Truth:A ~ 2

such t h a t 2) Again

F o r all

~:I ~ 2

the t r a d i t i o n a l

and

formulation would ~ a~

be p r o v a b l e ,

but

= ~.

Theorem

If the t h e o r y

binary relation

F

to the same b i n a r y

Proof-

If b o t h

I) a n d

dF~

require

category

is c o n s i s t e n t between

if

T r u t h ~-* ~; for

in the L i n d e n b a u m

~Truth

relative

d:l ~ A,

this

then t h a t the ~

and

just a m o u n t s

t h e n the d i a g r a m

to the e q u a t i o n

is d e f i n a b l e

sentences,

relation 2) h o l d

sentence

F~

and substitution

constants

dTruth =

then Truth

relative

to a g i v e n

is not d e f i n a b l e

144-

-

a

i,

>A

Ax

A

> 2

subst shows

Truth

that subst AxA

> A Truth

is a d e f i n i t i o n

We w i l l predicate. vability

of s a t i s f a c t i o n ,

also prove

Given

contradicting

an " i n c o m p l e t e n e s s

a binary

relation

is r e p r e s e n t a b l e

theorem",

F between

in the t h e o r y

the p r e v i o u s

using

constants

iff there

result.

the n o t i o n

and sentences,

is a u n a r y

formula

of a P r o v a b i l i t y we

s a y that Pro-

Pr:A ~ 2

such

that 3)

Whenever

dFa

then dPr = true

Theorem

Suppose

that

C, s u b s t i t u t i o n not c o m p l e t e Proof: ency implies

for a g i v e n b i n a r y is d e f i n a b l e

iff

u = true

relation

and P r o v a b i l i t y

F between

constants

is r e p r e s e n t a b l e .

and s e n t e n c e s

Then

the t h e o r y

of is

if it is c o n s i s t e n t . Suppose that

By completeness

But a) a n d b')

b)

on the c o n t r a r y

false

9 true.

that

C(i,2)

Condition

= {false,true}.

3) states

that

for

a)

u = true

implies

dPr = true

b)

~ 9 true

implies

dPr 9 true

b')

~ = false

implies

dPr = false

dFu

implies

together

with

completeness

mean

that w h e n e v e r

1

A p----~-~r 2

dF~

Our n o t i o n

of c o n s i s t -

-

is commutative,

145

-

i.e. that Pr satisfies condition

2) for a Truth-definition,

which by

our previous theorem yields a contradiction.

Notem

Our proposition

in section two can be interpreted as a fragment of a general

theory developed by Eilenberg and Kelly from an idea of Spanier.

-

146

-

F O U N D A T I O N S F O R C A T E G O R I E S A N D SETS

by S a u n d e r s Mac Lane*

I

INTRODUCTION

A p r e s s i n g p r o b l e m c o n f r o n t i n g c a t e g o r y t h e o r y is t h a t of p r o v i d i n g an adequate, approaches

precise,

and f l e x i b l e foundation.

Two

are c u r r e n t l y in use; n e i t h e r is really s a t i s f a c t o r y . One c u r r e n t a p p r o a c h uses the c l a s s - s e t d i s t i n c t i o n

p r o v i d e d by G ~ d e l - B e r n a y s

a x i o m a t i c s e t theory.

Here a small

c a t e g o r y is d e s c r i b e d as a s e t of m o r p h i s m s e q u i p p e d w i t h operation

of c o m p o s i t i o n

satisfying

an

the u s u a l p r o p e r t i e s , w h i l e

a large c a t e g o r y is a class of m o r p h i s m s w i t h c o m p o s i t i o n , h a v i n g the same p r o p e r t i e s . familiar

large c a t e g o r i e s

gory of all

(small)

This a p p r o a c h does p r o v i d e (the c a t e g o r y of all sets,

groups,

and for the f u n c t o r c a t e g o r y does n o t p r o v i d e

and

for the

the cate-

the c a t e g o r y of all s m a l l categories) AB

for

B

for the f u n c t o r c a t e g o r y

small. AB

However,

for

B

it

a large

category. A n o t h e r a p p r o a c h uses G r o t h e n d i e c k ' s n o t i o n verse

U - - a set s u c h t h a t the e l e m e n t s

membership

relation

of Z e r m e l o - F r a e n k e l t h e n a set

x

x 6 y

x 6 U

between such elements

a x i o m a t i c set theory.

A

w i t h the g i v e n form a model

U - c a t e g o r y is

of m o r p h i s m s w i t h c o m p o s i t i o n s u c h t h a t

* U n i v e r s i t y of Chicago. The i n v e s t i g a t i o n s s u p p o r t e d by an ONR grant.

of a uni-

x 6 U.

reported here were

-

147

-

This n o t i o n is e s s e n t i a l l y t h a t of a s m a l l category;

(better,

a U-small)

the c a t e g o r y of all U - c a t e g o r i e s is then a U ' - c a t e -

gory for some

larger u n i v e r s e

f u n c t o r categories,

U'.

In this a p p r o a c h w e may

p r o v i d e d one adds to set t h e o r y

t h a t e v e r y set is a m e m b e r of a universe. a x i o m of infinity.

Also,

form

the a x i o m

This is a s t r o n g

on this approach, n o one to my k n o w -

ledge has a d e q u a t e l y e x a m i n e d the r e l a t i o n c a t e g o r y of all rings in one u n i v e r s e

U

(say) b e t w e e n

the

and that of all rings

in some l a r g e r universe. C o m m o n to b o t h a p p r o a c h e s is the i d e a of u s i n g categories of d i f f e r e n t sizes In effect,

(small and large,

this is a use of c a t e g o r i e s w h i c h

models of set theory

is o n l y one set theory,

concern repeatedly all sets,

set theory).

it is a c o m m o n d i c t u m that there

t h o u g h p e r h a p s one n o t y e t c o m p l e t e l y

d e s c r i b e d by the u s u a l axioms Kowever,

U, in U').

lie in d i f f e r e n t

(that is, in a "variable"

A m o n g a x i o m a t i c set t h e o r i s t s ,

extent).

or in

(Zermelo-Fraenkel

the s t a n d a r d d e v e l o p m e n t s

plus

axioms of

of c a t e g o r y

theory

c o n s t r u c t s s u c h as the c a t e g o r y of all groups,

or all c a t e g o r i e s .

The only v i s i b l e a p p r o a c h to such

c o n s t r u c t s is some use of "variable"

m o d e l s of set theory.

Once the i d e a of a v a r i a b l e set theory is accepted, it b e c o m e s

clear that the set theory n e e d n o t be as s t r o n g as

Zermelo-Fraenkel axiomatics purposes

(ordinal n u m b e r s

To d e f i n e a category

require.

aside)

For m o s t m a t h e m a t i c a l

Z e r m e l o s e t theory is adequate.

a very m u c h w e a k e r s e t theory s u f f i c e s

a theory w i t h e l e m e n t s

and sets, b u t n o sets of sets.

-

This weak

axiomatic

called will

paper

be

which

because

an a b s t r a c t

the

general

is d e v o t e d

set theory.

a "school"

usual

form

A model

B,

C.

items

Everything

and p o s s i b l y to b e

read

there

is

and

x, y,

y

both.

"the

items

x

and

y.

lowing

axioms :

Extensionality

If

Ordered

will

is

a collection

z,

.--, w h i l e

x pair

item

adjoint

axioms

chief

functor be

will

development theorem,

replaced

is is

These

S

given

others

data

are

are

of the

which

called

a more

of

which

classes

class

to b e s u b j e c t e d

A",

x 6 A, and

to i t e m s

pair

B

if a n d o n l y

if

are

classes,

x 6 B,

then

and if

x

of the to the

given fol-

f o r all i t e m s

A = B.

Pair

If

<x,y)

= <x',y'>,

then

x = x'

and

A,

or a class,

relation

assigns

the o r d e r e d

some

an i t e m

a membership

an e l e m e n t

called

of t h i n g s ,

is e i t h e r

operation

<x,y)

and

in

by

for C l a s s e s

A

be

SCHOOLS

the s c h o o l

item

The

of one s u c h

distinction.

There

an o r d e r e d a new

x, x s A

in

classes.

distinction

II

are c a l l e d

of t h e s e w e a k

of F r e y d ' s

subschool-school

S

to the i n v e s t i g a t i o n

it has

set-class

A school

148

y = y'

.

-

Empty

149

-

Class

There

is

a class

g

with

no items

as m e m b e r s .

Unit Classes

To e a c h

item

x

there

is

{x}

class

with

x

as its

only element.

Cartesian

Product

To c l a s s e s whose

elements

are

A

all

and

B

there

the o r d e r e d

exists

pairs

A

a class

(x,y)

x B

xs

with

A

and

y s B.

Comprehension

If every

bound

low),

then

with

x 6 A

A

a class

variable there

The class

is

usually

is

and

is

that

property use

C

is

a formula

item with

consisting

in which

a "limit"

(see b e -

of all t h o s e

items

x

~(x).

class

C

constructed

i n this

axiom

schema

is the

written

{x]x ~ A and

it is a s u b c l a s s ~

~ (x)

a variable

a class

c = Note

and

used

to d e s c r i b e

only quantifiers

each quantifier

of

~(x)}.

the g i v e n

class

the e l e m e n t s

which

are

"limited".

is t o h a v e

one

of the

of

A

and that

the

the

subclass

is

By this we

forms.

mean

to

that

-

(Sy) y 6 B --for some given class

or

(u

ables

~ (Xl,... ,xn)

for items

of items may be defined

and c a r t e s i a n p r o d u c t n

with

t_here exists

C = {<Xl,.--,x n} Indeed,

limit of the quantifier.

recursion.

for each n a t u r a l n u m b e r

a formula

implies ----

( X l , . - . , x n}

using c o m p r e h e n s i o n show

z & B

B, called the

Ordered n-tuples by the usual

150

that to classes

one can

A1,..- , A n

all bouxld variables

and

limited vari-

a class

Ix! 6 A I , . . . , x n 6 A n and ~(Xl,--.,Xn)}

the c a r t e s i a n

product

r e p l a c e d by this s t r o n g e r

and c o m p r e h e n s i o n

comprehension

Here

are some e x a m p l e s

(i)

Any r e a l i z a t i o n

axioms

may be

a x i o m for n-tuples.

of schools.

("model")

of the Zermelo axioms

for set d%eory w i t h i t e m = class = set

(of the Zermelo model),

and w i t h the usual m e m b e r s h i p

and the usual d e f i n i t i o n

relation

of ordered p a i r in terms of membership. (2) the i t e r a t e d (U • U)

If

U

is any set

(in some set theory),

cartesian products

x U,-.-

of

U

w i t h itself,

s u b s e t of one of these p r o d u c t s one of these products.

U,U

• U,

and take

and "item"

The axioms

U

(U • U),

x

"class"

for a s c h o o l in the "atoms"

u' 6 U)

There

pairs

(u,u')

of atoms,

which simultaneously triples.

sense:

to be any

to be any e l e m e n t

that this school is "homogeneous" in the f o l l o w i n g

form all

then hold. (elements

are classes

of Note

u,

of o r d e r e d

and so on, b u t there n e e d be no classes

contain b o t h

Such h o m o g e n e o u s

o r d e r e d pairs

set theories

and o r d e r e d

are i n d e e d

appropriate

to

-

many

mathematical (3)

U, V,

and

theory)

form all iterated

factors

is o n e

of one

of t h e s e

of

This

"itsm" N;

Let

f: N

k, m E N

F: A

thus

if

is

By the

of two

Two

other S,

B

TWO

functions phism.

One

function phisms.

M

with

called

hand,

the

MXoM

Take

pair

ordered to b e

much

axiom,

of

any

of

pairs,

take

any s u b c l a s s

of

of the we

S

c a n be d e f i n e d a function

familiar may

all

the

define

the u s u a l

and

what

of

the

com-

collection as m o r p h i s m s ,

the

school.

is m e a n t

procedures.

by

a

Such

a

data O

domain

giving

The

functions

the c a t e g o r y

properties

construct

it a f u n c t i o n .

following

giving

M

function

with

one m a y

(of m o r p h i s m s ) > O

A

• B

and prove

following

M

these

are c l a s s e s

F c A

of

of one

numbers.

ordered

"class"

school.

comprehension

consists

classes

and

any s u b s e t

hold.

a category,

within C

N

of the s c h o o l ,

On the

category

and

functions

constitute

category

of

to b e

of the

school.

of n a t u r a l the

set

each

any e l e m e n t

a homogeneous

With

(in s o m e

in w h i c h

"class"

to b e

and define

a fixed

A

take

"item" is

g i v e n sets, products

and

= f(k,m).

a subset

of all c l a s s e s then

> N

be

are

the s e t

for a s c h o o l S

of a graph. posite

and

be



known

a function

G: A - - - >

axiom

then be

of

choice

of a c h o i c e

say

that

S

that

(a,b> that

If

F: B

B

with

the previous

(a,b>

with

Fb = a

then

for e a c h

This

apply

classes

to e a c h

E H,

form > A

or w i t h

c 6 A of

the

has

and

and

If

B: there

a function

exists (i.e.,

is e x a c t l y

of c h o i c e B ~ g,

To

this

G, w e

of a l l

those

find H a

not in

one

implies

domain

to the s e t

b s B

of school

there axiom

The

a choice

a 6 A is

in the

for a s e t

classes

A

item

there

F G F = F.

axiom

is

such

a school.

function

Hence we

any t w o

theory,

transformations

for

are n o

for

a cate-

described

there

simply

F.

may

how

set

and natural

for

6 G.

form:

weak

do,

such

G c H, s u c h

with

the

not

holds

a class

b 6 B

a graph) b 6 B

sets w i l l

following

S

the e x i s t e n c e

in a s c h o o l .

the

within

exactly

in a very

Functors

consider of

indicates

adequately

of a s c h o o l .

Next we usual

thus

there

the

is

pairs

image

of

-

Ill N O R M A L

A subschool

T

153

-

SUBSCHOOLS

of a s c h o o l

of a c o l l e c t i o n ,

closed

under

of the i t e m s

S

a collection

such

that

these

(under

the

if

is

T

though of

of

and

a subschool

this

class

elements unit

in

S

classes

school

T

pecularities normal

when

differ

restrict

do n o t

arise.

for e v e r y

B c A

in

that

the empty

unit

class

N any

is

their

class

{x}

in

that

the

of

of

of N

constructed

A N

of and

S

of

S).

Thus

class

empty

gT

'

class

of

T

none

T.

Similarly, in

to s u b s c h o o l s N all

of

N.

of w h o s e the

the sub-

A

an a p p l i c a t i o n

items

N • B

of

of the

x

B

N,

is its

of

of

remark

S

S

imply

that unit

such

to b e

the class

of two c l a s s e s

A similar

S.

is s a i d

conditions

class

of

in w h i c h

classes

These

x

S

those

those

product in

S

a school

of c l a s s e s

is t h e e m p t y

product

of

S.

all

of

of s o m e

classes

an e m p t y

and

N

of an i t e m

by

T

A subschool

cartesian

cartesian

S

in

form

original

products

are c l a s s e s

class

the

attention

class

with

S

~T

is i n

consists

operation,

relation

of the s u b s c h o o l

will

are i t e m s

and

n o t be

from those

x ~ A

S,

need

cartesian

with

in

there

are i t e m s

may

themselves

S

a class

and the

We

of

pair

of s o m e of the

of t h e m e m b e r s h i p

gT

S; it is s i m p l y

by d e f i n i t i o n

the o r d e r e d

two s u b c o l l e c t i o n s

restriction

S

holds

of for

comprehension

axiom. A normal

subschool

N

of

S

is

completely

determined

-

1 5 4

-

by the s p e c i f i c a t i o n of w h i c h c l a s s e s of

S

belongs

to

N,

in the f o l l o w i n g sense.

Proposition 1

If

S

is a school,

than a s u b c o l l e c t i o n

c o l l e c t i o n of all c l a s s e s of

S

classes of a n o r m a l s u b s c h o o l c o n t a i n s the e m p t y s u b c l a s s e s in

S

class of

subschool detail, x s A if

N

of

S

if a n d only if

any class of

and w i t h any two classes

A

and

S

x

of

for some class is in

S A

B

of

S

W h e n these c o n d i t i o n s h o l d the

is an i t e m of of

L

all its

is u n i q u e l y d e t e r m i n e d b y the c o l l e c t i o n

an i t e m

{x}

N

S.

of the

is the c o l l e c t i o n of all the

S, w i t h

t h e i r c a r t e s i a n p r o d u c t in

L

L

N

L.

In

if and only if

or, e q u i v a l e n t l y ,

if and only

L.

The p r o o f is left to the reader. This n o t i o n of a n o r m a l s u b s c h o o l i n c l u d e s many b a s i c examples,

m o s t n o t a b l y the n o r m a l s u b s c h o o l

sets w i t h i n let

G

a

G6del-Bernays

be any G ~ d e l - B e r n a y s

classes,

s o m e of w h i c h

axioms.

Then

the sets of

G

u n i v e r s e of sets and classes. universe;

a school

SG

in w h i c h

the s t a n d a r d c o n s t r u c t i o n

the

G-B

the items

are the c l a s s e s of

w i t h the g i v e n m e m b e r -

ship relation.

S i n c e w e h a v e c h o s e n to r e g a r d e a c h s e t as a and since the e m p t y

are

of o r d e r e d pairs,

and the classes

(special s o r t of) class,

G

Indeed,

that is a c o l l e c t i o n of

are c a l l e d sets, w h i c h s a t i s f y

determines

G, w i t h

c o m p o s e d of all the

class is a set,

155

-

since

the

every

subclass

in any

cartesian

G-B

of a s e t is

school

school where G-B

both

universe

which

V

Among

is

be

the

items

that

(x s y E V t s V).

and

satisfy

axioms

axioms

for

is

of

U, w h i l e

mal

subschool

the

and

finally

collection

L

subschool

classes

are

since

of all s e t s N

the

the

include

U the

set

x 6 V)

determines set

of

V

universe;

the u s u a l axiom

axiom schema

Zermelo

implies

Now

the

a set

a normal

the

for r e p l a c e m e n t .

a model

is

a Zermelo-Fraenkel

the u s u a l

Zermelo

but not

the

U

implies

weaker

a set,

determines

of sets w h i c h

theory.

of sets

the s u b sets

of

the

G.

Let lection

product

-

Z-F

axioms

schema

is,

axiom

of r e p l a c e m e n t , The

schema

for

comprehension

a set

V c

U

theory which

such

(t c y 6 V

SU

that

is t r a n s i t i v e

and inclusive a school

a col-

for s e t

of c o m p r e h e n s i o n .

Consider

is the

that

with

collection

implies

item = class

of c l a s s e s

for

= set

a nor-

S U"

A Grothendieck school.

Suppose

axioms

of i n f i n i t y ,

Grothendieck such again

that

so

the that

Z-F

(i.e.,

the e l e m e n t s

x 6 W

that

the

for a n o r m a l To m o t i v a t e

the p o s s i b l e

sizes

of

set

also provide

are

in

transitive satisfy

W

is the

subschool our s t u d y Zermelo

will

universe

there

universes

follows

of items)

that

universe

of

U

satisfies

U

sets

Z-F

collection school

of c o m p l e t e n e s s ,

universes

W

V.

We

sub-

strong which

and i n c l u s i v e

the

the

a normal

are

sets

axioms.

W

It

of c l a s s e s

(and

S Ulet us o b s e r v e use

the u s u a l

156

p o w e r set Re

P(x)

= {YlY c x}.

Within

U

one may define

of all sets of rank less than the o r d i n a l n u m b e r

~

the set b y the

recurs ion R 0 = ~, the l a t t e r for

8

Z e r m e l o universe,

a limit ordinal, Note

set

{~,p~,pZ~,.. 9 }

any

Z-F

> pn ).

contain

then the set

R~+~

is a

t h a t this u n i v e r s e d o e s n o t c o n t a i n the

though the latter set m u s t b e p r e s e n t in

u n i v e r s e , b e c a u s e it can be o b t a i n e d f r o m the s e t

by r e p l a c e m e n t n l

= U Re a U(b i)

s m a l l s u c h t h a t to e v e r y

there e x i s t

i E I

and

g': b i

9 a

This is just F r e y d ' s s o l u t i o n set condition,

w i t h the a d d e d r e q u i r e m e n t s

that

Im(f)

and

Im(b)

are small.

The p r o o f of this t h e o r e m is the s t a n d a r d one, w h i c h w e r e p e a t n o w c h i e f l y to p u t down in p r i n t c e r t a i n s i m p l i f i c a tions w e l l k n o w n in the f o l k l o r e others).

(Lawvere,

Kelly,

F r e y d and

The s o l u t i o n set c o n d i t i o n is c l e a r l y n e c e s s a r y ,

since a functor

F

left a d j o i n t to

U

produces

a solution

161

-

set with x

>

just

one

object

b = F(x)

and

one

(universal)

let the s o l u t i o n

set

condition

map

UF(x) . Conversely,

By

-

completeness,

the

small

product

d = Kb.

be

given.

with

projections

UPi:

Ud

i

Pi" is

d

> bi

exists

a product

defined

in

by

X;

in

C.

there

(UPi) f = f

By

continuity,

is t h e r e f o r e for e a c h

a morphism

i 6 I.

Given

>

Ub i

f: x

>

Ud

any

i

g: x ists small,

> g

Ua

as i n

: d the

is small,

>

a

class

K

g =

set

condition,

then

there

Since

is

locally

(Ug*)f.

of a l l e n d o m o r p h i s m s

{klk:

completeness,

class

with

a n d s o is

K = By

the s o l u t i o n

the

d

the o b j e c t

d

in

C

class

>

there

of

C

ex-

d

in

is

of c o t e r m i n a l

C

with

(Uk) f

e:

an e q u a l i z e r Like

morphisms.

=

f}

dO

>

.

d

for t h i s

any e q u a l i z e r ,

e

is

monic.

Lemma

(Kelly)

If

h:

Proof. the e q u a l i z e r

d0

d

of

Both

(Uk) f =

hence

there

g: x

>

for s o m e

Ua v:

f.

id

U

is

But

exists

m:

do

)

eh

are in

since

e

continuous,

f

is

x

>

we now have

U(eh) f = f,

and

K, e h e = e;

Be cause with

has

another Ud 0

a; to s h o w

m

K;

he =

since

is m o n i c , Ue

equalizes

equalizer

with

g = U(g*)f

then

and

therefore

universal

e

he = all

of t h e s e

f = (Ue)m.

1.

is 1. Uk Uk;

To any g = U(v)m

it r e m a i n s

only

-

to show has

this

unique.

U ( w ) m = g.

v, w:

d0

>

so d o e s As

v

Take By

a.

m.

s:

has The

>

continuity,

U(els)f (monic)

m =

(Ue 1 ) m I

g = ml,

eI

equalizer Since

e

= f

dO

dI

>

> dO

of

Uv,

for s o m e

m I: x

do

e >

the e q u a l i z e r

Uw,

and >

U (d l) .

m I = U (s) f

d.

the

a right

also

a

equalizes

and by

thus h a s

was

w:

one t h e n h a s

e] -

U(eels)f

so

el:

Ue I

s .~ dl

d

= m, so

isomorphism. have

dl,

not,

the e q u a l i z e r

Therefore,

d

-

Suppose

for any s u c h m o r p h i s m

some

1 6 2

But then lemma

v

els

else =

inverse, of

for

so is

and

w,

l.

an we

1

v = w. To e a c h

and a u n i v e r s a l x;

for e a c h a functor

object

morphism

then

F

we

m:

x ~

as u s u a l

left

family

of

thus h a v e Ud 0.

Fx = d O

adjoint

In p l a c e that every

x

to

is

an o b j e c t Choose

the

dO

one

object

of

such

C

d0

function

of

U.

the a x i o m

of c o t e r m i n a l

of c h o i c e , morphisms

one m i g h t has

assume

a chosen equali-

zer. This theorem

shows

"abstract" that

interpretations: G-B set

theory,

universe, model ness

of

the

theorem

usual

Zermelo

theory.

can be

limits.

has

The

spelled

It s e e m s

out

adjoint

several

"small"

another where

set

of the

one, w h e r e

another where

and s t i l l

conditions

specified

this

version

functor

different

"small"

means

means within

sorts

a set

of

of

a Grothendieck

"small"

means within

a chosen

leading

idea

complete-

is

as c o m p l e t e n e s s

probable

that

other

that

for c e r t a i n

theorems

of

-

category

1 6 3

-

theory may be u s e f u l l y r a m i f i e d w h e n o v e r a l l

hypothesis

are r e p l a c e d by c o m p l e t e n e s s

VI

"completeness"

for certain s p e c i f i e d

TYPES A N D I T E R A T E D S C H O O L S

The m a j o r p r o b l e m of the f o u n d a t i o n s t h e o r y remains t h a t of h a n d l i n g s u c c e s s f u l l y

of c a t e g o r y

larger c a t e g o r i e s

"of all s o - a n d - s o ' s " .

This can be done e f f e c t i v e l y b y u s i n g a

succession

like a c u m u l a t i v e

let S

U

of schools,

w i t h i t e m = class = set of

S1

w i t h i t e m = set of

in

U, and w i t h class S

are small;

as

S1

of

and w i t h

Categories within

S2

this p r o c e s s y i e l d s

all

(in

for

n = 2

a sequence

arguments.

U2

is reS

S 2)

the c o l l e c t i o n of all any s u b c o l l e c t i o n of

of schools

~.

C o n t i n u a t i o n of

S c S1 c

S 2 c S 3 c...,

The c a t e g o r i e s w i t h i n

"metacategories"

frequently (a type

may be d e s c r i b e d by a s u i t a b l e

The s t a n d a r d p r o o f that

s i s t e n t l y r e l a t i v e to

S1

Now form a still

Such a scale of schools

theory on top of a set theory) a x i o m system.

Thus

Categories within

m i g h t be c a l l e d ~iant.

are e x a c t l y the

u s e d in i n f o r m a l

U0

(suitably constructed)

e a c h a n o r m a l s u b s c h o o l of the next. Sn

of

are large.

S 1 , with class

r e g a r d e d as a s c h o o l

the o r d e r e d pairs those g i v e n

G-B sets.

w i t h items

of classes

these items,

to

First,

C o n s t r u c t a larger s c h o o l

any s u b c o l l e c t i o n

GB-class

S2

U.

U, w i t h

those w i t h i n

larger school n-tuples

type theory.

be some s e t - t h e o r e t i c a l universe,

lated to

limits.

ZF set theory

G-B set theory is con-

(J. B. R o s s e r a n d Hao Wang,

-

164

-

for Formal Logics,"

15; 113-129,

can probably be translated to show that

(1950))

the consistency

J.

SymboZ~o Log~oj

"Non-standard Models

of the original set theory implies the consis-

tency of a suitable

language for any one

S n.

In this way,

the

use of categories within schools can provide a suitable and consistent foundation

for category theory, by providing an explicit

setting for successively

larger types of metacategories.

-

165

-

ON THE DIMENSION OF OBJECTS AND CATEGORIES III HOCHSCHILD DIMENSION*

by Barry Mitchell**

R. Swan has observed that the situations encountered in [7] and

[8] can be considered as special cases of the

following.

Let

K

be a commutative ring, and let

K-algebra.

(All rings have identities and all ring homomor-

phisms preserve identities.) category and let C(A) on

~: K

> c(A)

A

be a

is an additive

be a ring homomorphism where

is the ring of endomorphisms of the identity functor A.

Let

AA

denote the category of left A-objects in

(that is, ring homomorphisms Let

Suppose that

A

AA~

A ---> HomA(A,A)

denote the full subcategory of

AA

with

A

A 6 A).

consisting of

all those A-objects such that the composition A --> HomA(A,A )

K

by

~.

If

A

is the same as the homomorphism induced

is abelian,

then so is

AA~.

As an example, consider a ring homomorphism ~: K ---> c(A)

*

and let

f(x)

be a polynomial with coefficients

Research supported by National Science Foundation Grant No. GP-6024.

** Dept. of Math., Bowdoin College, Brunswick, Maine.

-

in A

K.

Then the category

satisfying AAr

category Another

f(u) where

example

is another obvious

= 0

Af (see

A

in

the category

and taking

A

A@FA of left A|

of

of

A

AA

A

a lower bound for dim

gl. dim.

AA

F

F-operations

lemma

over K).

1.4 of the for the

the H o c h s c h i l d

The latter

is defined

as a

A e = A* @ A

lemma inversely

in certain

as

to

cases where

it is true that

h.d. for all non-zero

AeAr

~hat

considered

A

and

is known.

Sometimes

category

A

(tensor product

we shall be using this

determine

and

to be the

to give an upper bound

of

F

This is the same as

in the case where

dimension

Actually

d = dim A

commute w i t h

objects

where

is the category

A(FA)

(dim A) is known.

the h o m o l o g i c a l

FA

> c(FA)

K

observed

in

K[x]/(f(x))

with

A

the same K-operations.

dimension

module 9

r

u

is the same as the

which have simultaneous

paper can be used

dimension

w

Then

Swan has further

global

[7,

such that A-operations

and both induce

present

of all endomorphisms

is the K-algebra

ring homomorphism.

operations

-

is had by replacing

K-algebra

of all objects

166

objects A ~

A

(i)

A @K A = d + h.d. A A A

in an abelian

is c o n s i d e r e d

category

as an object

A, where of the

-

This q u e s t i o n form

K(~),

question

is of p a r t i c u l a r ~

denoting

167

-

interest when

a finite p a r t i a l l y

is then r e l a t e d to the e x i s t e n c e

for a finite o r d e r e d set or,

A

is of the

o r d e r e d set.

The

of a "dimension"

in other words,

an integer

d'

such that

gl.dim.

~A = d' + gl.dim.

for all a b e l i a n categories of c o v a r i a n t

functors

we shall show that valid.

However

A.

from

d = d'

to

~A

A.

denotes

For

(1) is always

it is now k n o w n that not every d'.

an o r d e r e d set of 15 elements at the U n i v e r s i t y

the category

d' = 0, i, or 2,

and that e q u a t i o n

o r d e r e d set has a d i m e n s i o n

Spears

Here

A

finite p a r t i a l l y

A counter-example

involving

has b e e n p r o d u c e d by W i l l i a m T.

of Florida.

Details will appear

in

his thesis.

Throughout

this paper

A

will denote

an a b e l i a n

category.

i.

Let

A

AN A D J O I N T

be an

RELATION

(abelian)

c a t e g o r y with

indexed by some cardinal

number

M

then the tensor p r o d u c t

is a right

defined admits

A-module,

for all left A-objects an exact sequence

p.

A

of right

If

in

A

A,

coproducts

is a ring and M |

providing

A-modules

A M

is

168

JA

where

I

and

J

-

> IA

have c a r d i n a l numbers

no b i g g e r

than

When u s i n g the above tensor p r o d u c t we shall always that

M

satisfies

this c o n d i t i o n w i t h o u t

We then have a natural e q u i v a l e n c e

Hom A(M, for

M 6 G A, A 6 AA

denotes

the category If

exact,

M

then

M |

M

is a retract

of abelian

A

(see

M = A

s t a t i n g so.

(l)

A,B) Here

[9,w

G

groups.

and the c o p r o d u c t s

is an exact functor and,

then follows

in

are

A

of the A-object

consequently,

for

M

A.

free

That it is true for all

from the fact that any p r o j e c t i v e

of a free.

i. i

Let A

B 6 A

assume

of trifunctors

~ HornA(M |

of the exact coproducts.

projective

Lemma

and

is p r o j e c t i v e

This is trivial for because

I

HornA(A,B))

always

p.

and

F

~: K

> C(A)

be K-algebras.

be a ring h o m o m o r p h i s m

Then there

is a n a t u r a l

and let

equivalence

of t r i f u n c t o r s

HomAsF, (M, Hom A (A,B)) for

M 6 G A|

, A 6 AA~ ,

and

~ Hom F (M 8 A A,B)

B 6 FA~

.

(2)

-

Proof. on

HomA(A,B) ,

A | F*-module, forgetting

Since

it follows

and using naturality

yields

induce the same K-operators

that the latter is a right (2) makes sense.

we have the natural equivalence

in

on either

M

and

B,

Now, (i),

we see that the

side correspond

to each other.

This

1.2 Relative

to K-algebras

natural equivalence HomA|

and

Replace

the induced

A --~

functor

S: AA~

F

B 6 Z| by

ZA

in 1.1.

be a K-algebra homomorphism.

T: FA~ ~

> FA~

right adjoint the functor R(A) = HomA(F,A)

roles of

A

A,B)

1.3 Let

functor

there is a

of trifunctors

A 6 A|

Proof. Corollary

A, F, and Z,

(M, Hom E (A,B)) ~ Homz| r (M |

M 6 G A|

places)

F

(2).

Corollar~

for

and

-

and so the left side of

F-operators,

F-morphisms

A

1 6 9

AA~

given by

has as left adjoint the S(A) = F |

R: AA~ --~

FA~

A,

and as

given by

(symbolic Hom).

Proof.

The dual of 1.1 yields

A

F

and

Then

so as to get

A

and

(interchanging B

the

in the right

-

HomA| Then,

taking

170

-

(M, Hom A (A,B) ~ Hom A (A, Hom F (M,B))

M = F

(with right A-operators

the given algebra homomorphism)

(3)

.

d e t e r m i n e d by

and combining

(2) and

(3),

we obtain

H~ Since that

|

Hom(F,B) S

A,B) ~ HomA(A,

HomF(F,B))

T(B),

as a left A-object is just

is the required left adjoint.

9

R

That

this shows

is the right

adjoint follows by duality. Remark 1.

The u n d e r l y i n g assumption needed to

obtain the left adjoint have exact coproducts

S (right adjoint R) is that

(products)

the cardinal numbers of

I

and

A

indexed by the m a x i m u m of J,

where

F

admits an

exact sequence of right A-modules JA

If the coproducts A-projective,

(products)

then the left

Remark 2. corollary

If

in

r

A

) 0

are exact and

T: FA~

F

is

(right) adjoint is exact.

A = K,

then

AA

= A.

1.3 gives us left and right adjoints

ful functor Lemma 1.4

~ IA--~

Thus for the forget-

9 A.

(Swan) Let

A

be a K-projective

algebra and let

d = dim A.

171

-

If the c o p r o d u c t s

A

in

are e x a c t ,

then

for e a c h

A

AA~

E

we h a v e

h.d.

A _< d + h.d. A A

.

Consequently

gl.

dim.

Proof.

AA

> --- "--> P1

be a p r o j e c t i v e

resolution

is a p r o j e c t i v e

K-module,

a sequence obtain

in

right

of r i g h t

an e x a c t

A

A

> Po

and,

>

A

> 0

as a A e - m o d u l e .

it f o l l o w s

A-modules.

that

Hence,

Since

A e = A* |

consequently, for

(4)

A

A is

(4) s p l i t s

A 6 AA

,

as

we

sequence

) --- P1 |

Pd |

A

AA- .

It s u f f i c e s

A

> Po |

t h e n to p r o v e

for p r o j e c t i v e

Ae-modules

coproducts

A,

But,

for

A-module

0 ~

in

dim.

Let

0 ---> Pd

a projective

-< d + gl.

P.

that

A

> A

h.d.

> 0

P |

A < h.d. A

In v i e w of the e x a c t n e s s

we need only consider

the c a s e

of

P = Ae.

in this case, w e h a v e

Ae |

where,

A =

by r e m a r k

the f o r g e t f u l coproducts

(A |

A) |

2 following

functor

T:

1.3,

AA

and p r o j e c t i v i t y

A = A |

S

) A. of

A

A = S(A)

is the left a d j o i n t Again,

because

as a K - m o d u l e ,

S

for

of e x a c t is

A

-

exact,

Lemma

and so o u r

conclusion

172

-

follows

from

[7,

1.2].

i. 5

A

Let

be a K - p r o j e c t i v e

that

the c o p r o d u c t s

left

A-module

such

in

A

that

are h.d.

K-algebra,

exact. M = m,

M

If

and

suppose

is a K - p r o j e c t i v e

then

i

h.d.

for all

A 6 A,

M |

where

A < m + h.d. A A

is r e g a r d e d

M @K A

as an o b j e c t

of

AA r Proof.

Consider

a projective

resolution

> ~i

---> M

of

left

A-modules

0

Since

A

and

a n d so w e

It s u f f i c e s projective

are

>''"

then

an e x a c t

for

T:

from

[7, 1.2].

r

But

A @

> A

and

P. K

(5) s p l i t s

> 0

.

(5)

as K - m o d u l e s

sequence

--~ P ~ 8K A --~

to p r o v e

A-modules

> P0

K-projective,

A ---~ ---

P = A. AA

M

obtain

0 --> P m |

case

> Pm

that

h.d.

Again

we

A = S (A) so o n c e

P0 |

) M |

P 8K A ~ h.d. A

need only where

more

A

S

A

consider is the

our conclusion

A ---~ 0

for the

left

adjoint

follows

.

-

Corollary

the h y p o t h e s i s

h.d.

A |

for all A 6 A, where Ae in A~ and d = dim

Proof. so the c o n c l u s i o n

of 1.5 w e h a v e

A < d + h.d. A A --

K

A

-

1.6

Under

by

173

A |

is c o n s i d e r e d

A

as an o b j e c t

A.

Since

A

follows

is K - p r o j e c t i v e , by replacing

A

so is

by

A e,

Ae

and

and M

in 1.5.

2.

is a s m a l l

If w e let

K(~)

morphisms

of

necessarily

FUNCTOR

denote ~. with

the

K(~)

CATEGORIES

category

and

free K - m o d u l e

is a ring,

then

on t h e s e t of

c a n be c o n v e r t e d

identity)

K

into a ring

if m u l t i p l i c a t i o n

(not

is d e f i n e d by

the rule

(kx) (ly) =

(k/) (xy) w h e n

range y = domain

x,

= 0 otherwise.

Here xy

kl

denotes

denotes

is a f i n i t e of o b j e c t s morphisms),

the p r o d u c t

the c o m p o s i t i o n category although

of t w o e l e m e n t s

of t w o m o r p h i s m s

(that is, it m a y h a v e

and if its o b j e c t s

~

has o n l y

i, 2,

in

K,

number

..., n,

and

~.

a finite

an i n f i n i t e are

of

If number of then

K(~)

-

has

an i d e n t i t y ,

more,

K

rings

K(~)

Lemma

namely

-

If,

11 + 12 + ... + i n .

is c o m m u t a t i v e have been

174

then

K (~)

considered

further-

is a K - a l g e b r a .

by Lawvere

Such

[6].

2.1

Let

~

be a f i n i t e

be any r i n g h o m o m o r p h i s m

category

where

T h e n we h a v e an e q u i v a l e n c e

K

~: K ---~ C(A)

is a c o m m u t a t i v e

ring.

of c a t e g o r i e s

T: ~A where

a n d let

> AA

A = K(~).

Proof. (in o t h e r w o r d s ,

Given

a covariant

an o b j e c t

of

functor

D:

~

,

A

~A) w e d e f i n e n

T(D)

=

~

D(i)

,

of

~.

i=l

where ~,

1,2,...,n

are the o b j e c t s

then we define

morphism

x

of

the a c t i o n

T(D)

of

x

on

If T(D)

x: j

> k

in

to be the e n d o -

g i v e n by

U. = 0 1

for

i ~ j ,

uj = u k D(X) n where

ui

denotes

the i th c o p r o d u c t

injection

D(i) .

for i=l

This We

action

leave

respect

determines

uniquely

it to the r e a d e r to m o r p h i s m s .

an o b j e c t

to d e f i n e

of

AA~.

the b e h a v i o r

of

T

with

-

Now define S(A) (i) = I.A 1 A).

If

x(l.A) ] and so

x

a functor

(that is,

x: j ~

k,

=

-

leave

respect

becomes

reader

TS AA

of

ii

on

c ikA

) S(A) (k)

an o b j e c t

of

.

~A.

the b e h a v i o r

A g a i n we of

S

with

to m o r p h i s m s .

equivalent

on

of the a c t i o n

(ikx) A = lk(XA)

to d e f i n e

It is now easy

that

by t a k i n g

a morphism

S(A)

it to the

~A

we h a v e

S(A) (x) : S(A) (j)

In this way,

>

S: AA r

the i m a g e

(xl.) A = 3

induces

1 7 5

to see t h a t

to the i d e n t i t y is n a t u r a l l y

functor

equivalent

ST

on

is n a t u r a l l y

~A.

In s h o w i n g

to the i d e n t i t y

functor

one m a k e s

use of the r e l a t i o n s 1 1 = 0 for i 3 n i ~ j, 12i = i i, and Z i. = 1 in K(~) to see t h a t any i=l i K(~)-object A is the c o p r o d u c t of its s u b - o b j e c t s I.A l r

in

A.

If

~

a commutative

and

1

ring,

using

~i x ~2

of

| K(~2)

is the p r o d u c t

the r e l a t i o n

ing a l g e b r a

are f i n i t e

K(~)

categories

and

K

is

t h e n we h a v e

K(~I) where

~2

K(~)*

~ K(~ 1 x ~2)

,

of c a t e g o r i e s .

= K(~*),

is g i v e n by

we see t h a t

In p a r t i c u l a r , the e n v e l o p -

176

-

-

K(~) e = K(~*

Now consider

an o b j e c t

homomorphism

~: K

as an o b j e c t

in

of c a t e g o r i e s which

one

K(~)eA~,

denote

sees

follows.

> C(A).

that

Then

U

U~(A)

(A).

(i,j)

UT(A)

7" x ~

6 7" x ~.

(that is, x:

we are g i v e n

K(~)

|

A,

Using

If

>

considered

of

the f u n c t o r

of

S

explicitly

as

is g i v e n by = Hom(i, j) A

(x,y) : (i,j)

i'

a ring

the e q u i v a l e n c e

under

can be d e f i n e d

UT(A) (i,j) for

.

in 2.1 to an o b j e c t

by

On o b j e c t s ,

where

corresponds

established

we shall

2.1,

A 6 A,

x ~)

i and y:

(1) >

j

(i',j')

in

~ j' in 7) then w e

have U where

uz

denotes

If w e r e g a r d that

the

UT(A)

its v a l u e

(A) (x,y)

at an o b j e c t

is the left a d j o i n t

Tj:

7*A

be c o n s i d e r e d

be u s e d We

as a f u n c t o r

lemmas.

j q ~

[5, 1.2] with

dimension

to g e t u p p e r leave

of

for the c o p r o d u c t ~(7*A),

is just

of the e v a l u a t i o n

In v i e w of

all h a v e h o m o l o g i c a l will

z th i n j e c t i o n

as an o b j e c t

Sj

> A.

u z = U'yzx

bounds

then w e see Sj(A),

where

functor

it f o l l o w s

7 as d o m a i n equal

(i).

that

UT(A)

and w h o s e

values

This

remark

to h.d. AA.

for h . d . U T ( A ) .

it to the r e a d e r

can

to v e r i f y

the f o l l o w i n g

177

Lemma

2.2

If the o b v i o u s

z

and

T

isomorphism

are small

Lemma

U

~

under

object

(~x~)A

T

is an e q u i v a l e n c e

,

to t h e o b j e c t

U

>

~

the i n d u c e d e q u i v a l e n c e

(A)

If object), ~A

is t a k e n

#

t h e n the > A

the c o p r o d u c t A(#)

x

corresponds

U

(A).

TX~

of

of an e l e m e n t

S(A)

copies

y E ~.

S

A.

(morphism)

the r i g h t o p e r a t i o n s

of

~

y

x 6 #

=

u

on

y

=

u

yx

where

by

X,

A(~)

functor denotes

the a c t i o n

on

we have

(1)

xy

A(#)

are g i v e n b y

X*u

the

(A).

Denoting

In this c a s e w e h a v e

the left o p e r a t i o n s

,

for the f o r g e t f u l

= A(~)

of

xu

for all

T*•

(that is, a c a t e g o r y w i t h o n e

left a d j o i n t

~

U

)

categories

MONOIDS

is a m o n o i d

is g i v e n by

of s m a l l

~*x~A

i n t o the o b j e c t

3.

T:

(~•

(U (A)) ~

under

2.3

If then,

T

then,

of c a t e g o r i e s

T*•215 the o b j e c t

categories

u

(A) = A(~)

are g i v e n b y

where

(i) a n d

-

Of c o u r s e w e m u s t a s s u m e ~.

If t h e s e

coproducts

consequently,

by

that

a n d so, A,

9

corollary

restriction

A(~)

T:

x = bt

x 6 ~

with

i n d e x e d by

In this

submonoid

of

one w h o s e

dual

B,

h a v e by

then

~.

A right

If

and

has

an e x a c t

B

representation then

if

A

Z(~)

is

has

a n d is e x a c t .

t 6 9

implies

a left partitionin@

is a left p a r t i t i o n i n g

T

a subset

exists

with

in

for the

t 6 ~,

S

Z(~),

coproducts

a unique

then xt 6

A

Z(T) c

has

Consequently, B,

and

(2)

S

~

~artitionin@

submonoid submonoid

of of

has e x a c t p r o d u c t s

right adjoint.

~

is

~.

If

indexed

In this c a s e w e

[7, 1.2]

h.d.

A < h.d. y

for any

.

then

case w e c a l l

is r i g h t p a r t i t i o n i n g by

~,

admits

t h e n the r e l a t i o n

x 6 T.

i n d e x e d by

is e x a c t and,

of a p p r o p r i a t e

b 6 B

exact coproducts

that

of

HA ---> ~A.

Z(T)-module.

1 E B,

S

1.3 g i v e s us a left a d j o i n t

free as a r i g h t

If

then

= h.d. A A

to the e x i s t e n c e

functor

form

has c o p r o d u c t s

[7, 1.2] w e h a v e

such that each element of the

A

is a s u b m o n o i d

subject

-

are e x a c t

h.d.

If

1 7 8

left

T-object

Examples following:

--

A

(3)

7T

A.

of p a r t i t i o n i n g

submonoids

are t h e

- 179

(i)

If

is b o t h

then

(2)

their with

free

~ left

If

T

is a g r o u p and right

and

ending

with

is a left p a r t i t i o n i n g is r i g h t

If

I

then

the p a r t i a l l y

left

and

right

this

form left

If

right

of

T of

with

(5) monoids

1T ~

•

'IT

I'

and

of t y p e

submonoid

see

that

Similarly,

(See

a special

8

and T x 8

case

w

and

and

and ~2

right and

of

left

respectively,

submonoid

of

right

T

taking

9 x 1 X

is a

8o

sub-

~i x T 2

is a

In p a r t i c u l a r ,

submonoid

partitioning

B

of the

partitioning

then

~i x 72.

partitioning

then,

T =

free

(2).

of all p a i r s

see t h a t

are

is a

definition.)

of e x a m p l e

consisting

subsets,

partially

for the

~2

T1

are

(J, J N I')

are m o n o i d s ,

we

s 6 8,

[7,

J

of t h e

If

is a left o

~e6.

submonoid

~I

is a left

~* x ~

we

9,

of

is

together

1

of

partitioning

left p a r t i t i o n i n g T

9 e8

and

to be

is a set a n d

(I, I').

subset

(1,s) and

B

submonoid

free monoid

is j u s t

(4) to be the

is a s u b g r o u p ,

partitioning.

an e l e m e n t

partitioning

of t y p e

Indeed,

9

partitioning.

(3)

monoid

and

are m o n o i d s

e

product , then taking

all w o r d s

-

of

submonoid

~, of

if then

-

Lemma

180

3.1

If

T

of

#,

and if

by

#,

then

is a left and r i g h t p a r t i t i o n i n g A

has

coproducts

h.d. T,xTA(T) for all

the r e l a t i o n x 6 t.

T

Since

tlxt 2 6 T

It f o l l o w s

that

T* x T - o b j e c t s .

(3) if w e r e p l a c e by

and exact products

indexed

__< h . d "#*x~A(~)

~

is left a n d r i g h t p a r t i t i o n i n g ,

with

tl,t 2 6 T

A(~)

is a r e t r a c t

Therefore, by

implies

#* • ~,

of

the result T

by

that

A(~)

follows

T* • T,

as from

and

A

A(~).

Theorem

3.2

If

A

has e x a c t

is a free m o n o i d

or g r o u p

coproducts

for all n o n - z e r o exact products

A 6 A.

as w e l l

= 1 + h.d. AA

~

is p a r t i a l l y

as e x a c t c o p r o d u c t s

#

then

A(~)

If

and

i n d e x e d by

on one g e n e r a t o r ,

h,d.#,x

then

submonoid

A 6 A.

Proof.

left

-

(4)

free and

indexed by

A

has

#,

(4) is valid.

Proof. Then we have

Let

an e x a c t

~

be p a r t i a l l y

sequence

in

~A

free on

I

generators.

-

0

relative yields,

to any

> IA(~)

left

181

8 > A(~)

~-object

in the case w h e r e

-

A

A

~

> A

[7, 2.1].

has e x a c t

In p a r t i c u l a r ,

replacing

~*,

(2), w e o b t a i n

and u s i n g

A

by

h - d . ~ , x ~ A(~) To p r o v e the c a s e w h e r e case

sequence

is a s s i g n e d generator by

[7,

the o t h e r

~

operators.

w

8

over

~A, A by A(~),

and

~

~,

by

-

inequality we consider

(5) is a s e q u e n c e

the m o r p h i s m

(2) this

-

< 1 + h.d. AA

is g e n e r a t e d

identity

Using

coproducts

h , d . A _< 1 + h.d. AA

(5)

> 0

first

by a single

element.

of

objects where

~* x ~

has m o r e

(If

~* x ~

is n o t a

In t h i s A

t h a n one

morphism.)

Now,

we have

h.d. A = 1 + h.d. AA

for all n o n - z e r o cation

A

with

identity

of this y i e l d s

h.d. ,x A = 2 + h.d. AA This,

A second appli-

operators.

together with

the e x a c t

h - d . ~ , x ~ A(~) The g e n e r a l

case t h e n

of a s i n g l e

element

follows

in e x a m p l e

sequence

(5), y i e l d s

> 1 + h.d. AA

f r o m 3.1 (3)).

.

.

(taking

J

to c o n s i s t

-

Corollar~

~

is the d i r e c t p r o d u c t of

or groups on one generator, ~,

-

3.3

If

over

182

and if

A

n

free m o n o i d s

has e x a c t c o p r o d u c t s

then

h.d.n,x~A(~) for all n o n - z e r o

A 6 A.

p r o d u c t s of p a r t i a l l y

The same is true for d i r e c t

free m o n o i d s

to h a v e e x a c t p r o d u c t s over

Proof.

= n + h.d. AA

if

A

is a s s u m e d f u r t h e r

~.

This follows by an easy induction,

using

2.2.

R e m a r k i. infinite.

Corollary

n

p r o d u c t of p a r t i a l l y integer

free m o n o i d s has a k - f o l d

free m o n o i d s

as a d i r e c t f a c t o r for e a c h

k.

R e m a r k 2.

Corollary

3.3 implies the w e l l - k n o w n

result di m K[X 1 ,.o.,X n ] = n .

4.

If AE

A

is

This follows f r o m 3.1, u s i n g the fact t h a t an

i n f i n i t e p r o d u c t of p a r t i a l l y

positive

3.3 is true e v e n if

~

ABELIAN GROUPS

is a f i n i t e g r o u p of o r d e r

is such that

mA

is an i s o m o r p h i s m ,

m, then

and if

183

h.d.

for any n - o b j e c t again we then

finite

abelian

A

suppose

~*x~ A(~) that

g r o u p of rank F

on

Applying

[7, 3.4].

this

see t h a t

Now,

where

(i)

A = h.d.AA

structure

h.d.

abelian

-

r.

G

(2)

.

is a f i n i t e l y

Then we can write

is a free a b e l i a n group.

= h.d. AA

g r o u p of r a n k

Combining

generated G = F x

r,

(2), 3.3,

and

~

is a

a n d 2.2, w e t h e n

obtain:

Proposition

4.1

Let rank of

be a f i n i t e l y

and suppose

r, G

G

m.

is

If

A

Before without

proof

has e x a c t c o u n t a b l e

Theorem

g r o u p of

torsion

subgroup

coproducts,

then

= r + h.d. AA

A 6 A

such t h a t

looking

at b i g g e r

a generalization

t h e o r e m of I. B e r s t e i n

abelian

t h a t t h e o r d e r of t h e

h.d.G• for all n o n - z e r o

generated

[3].

m

A

is an i s o m o r p h i s m .

abelian

groups we

due t o B. O s o f s k y (See a l s o B a l c e r z y k

state

[i0] of a [2].)

4.2

Let

D

for some i n t e g e r

be a d i r e c t e d n _> 0,

set of c a r d i n a l

a n d let

{R i ' ~J} i

and

number {Mi ,

Mn ~J} i

- 184 -

be d i r e c t

systems

of r i n g s

and abelian

groups

such that

M i

is a left

R.-module 1

for e a c h

~J. (rm)

for

r 6 R. 1

limits,

and

i 6 D,

= ~J(r) i

m 6 M.. 1

If

R

and such that

uJ(m) i and

M

are the d i r e c t

then

M. h . d . R M _ r + h . d . A A

(6)

A 6 A.

The a s s u m p t i o n

structure

where

denominators

is a p r o j e c t i v e as a G - o b j e c t k n o w by

(5)

then

Proof. K-object

for

m + h . d . A A

Remark.

~

-

If there is an exact sequence

> Km-i

pi

189

d i a g r a m at

A

in

nG

then

-

190-

0 ----> K 2 ---~ pl

where may

p1

take

is p r o j e c t i v e pl

the left a d j o i n t

and

denotes

For object have

of

is split.

In fact, w e

with

an e p i m o r p h i s m

is the c o n s t a n t

~ Sm(Z) m6M for the e v a l u a t i o n

the set of m i n i m a l

i 6 ~

~A

strictly

K2

> 0

to be the d i a g r a m

denotes M

and

r K1

and A

i

at

and

A

at

than

we

let 0

i.

S. i

functor

elements

r Li(A )

Si(A)

diagram

greater

A 6 A

where

of

~.

L. (A) 1

denote

elsewhere. in

~A

Ti

We then

whose

kernel

o v e r the s e t of e l e m e n t s Combining

this w i t h

the

of

5.2 w e

obtain:

Corollary

5.3

F o r any 0 ~

where

pl

and

p0

i 6 ~

K 2 --~ p1

an e x a c t

) p0

are p r o j e c t i v e

For a finite

d(~)

Lemma

we have

ordered

set

= m a x h.d. i6~

> Li(Z) K2

and

~

D 6 ~A

h.d.

Li(Z)

.

we h a v e

D < d(~)

+ m

--~

,

in

0

is split.

we define

5.4

F o r any

sequence

~G

-

where

m = sup i6~

h.d.D.

~

.

If

Otherwise exact

.

By

~ has let

induction

only

one

sequence

in

K

, D

= D 1

h.d.Lk(D)

sequence

h.d.

(8) w e

sidered the

if

as

see

for

D

element

of

, L

(D k) ~

k i @ k.

is t r i v i a l .

~ , and

form

0

the

Now b y

(8)

.

5.1

(a)

< h . d . AD k + h . d . L k ( Z )

> d(~)

h.d.

a diagram

we

have

+ m,

then

K > m + d(~).

over

~ - {k}

from But

, and

< d(~)

the

K can

so t h i s

+ m

exact be Gon-

contradicts

5.5

If d(~)

= 0, gl.

all

directly 5.3

this

elements

induction.

Corollary

for

then

of

1

= h.d.D k | Lk(Z)

Consequently

number

~A ~ K

K k = 0, a n d

on t h e

element,

k be a minimal

0 Then

-

l

Proof. of

191

nontrivial

A.

i, dim.

dim.~A

Proof.

The

applied

to

5.4

The

5.1(b).

or

3,

then

~A = d(~)

If d(~)

3 + gl.

from

2,

< gl.

right other

> 3,

+ gl. then

dim.hA

hand

dim.

we

have

< d(~)

inequality

inequalities

A

+ gl.

in

are

(9)

dim.A

(9)

follows

consequences

of

.

-

192

-

C o r o l l a r ~ 5.6 For any

A 6 A

we have

U~(A) This

Proof. that U

(A), r e g a r d e d

~ d(~)

follows

+ h.d.AA

from

5.4

in

. view

as an object of ~(~*A),

of

the

has S

fact,

(A) at 1

the i th vertex.

Corollary

5.7

If ~ has an e l e m e n t

i such that h . d . L .

(Z) = d(~) i

and Li(Z)

is strong,

then

g l . dim~ for all A.

In this case

global dimension,

Proof. (b).

other

if K is a c o m m u t a t i v e

ring of finite

= d(~)

The first statement

follows

from 5.4 and

Then t a k i n g A to be the c a t e g o r y of K-modules,

statement

follows

Remark hypothesis

+ glo dim,A

then

dim K(~)

5.1

~A = d(~)

of 5.7

Remark

i.

2.

d(~) This follows

from 5.6 and 1.4.

Any g e n e r a l i z e d m - b r a i d

(See

the

[8,

satisfies

w

For any set ~ we have - i < dim Z(~)

< d(~)

from 5.6 and 1.4 in v i e w of the fact that gl. d i m . G = 1

.

the

-

Theorem

-

5.8

If d(~)

= 0, l, or 2, t h e n

h.d.U~(A) for all n o n t r i v i a l

Proof. a n y case.

for d(~)

= d(~)

Corollary

N o w by c o r o l l a r y < dim

By

5.5 and

lemma

= h.d.U

Therefore

inequality

1.4 w e h a v e

(Z)

for s u c h ~ w e h a v e

(Z) is s t r o n g w h e n d(~)

our conclusion

[8, 4.6]

follows

= 0,

f r o m 5.1.

we know that

(10)

gl. d i m . A > 3 + gl. dim. A if and o n l y ordered

if ~ c o n t a i n s

subset.

(See

[8,

it is e a s y to see t h a t crown,

where

then

n > 2.

same as

it c o n t a i n s

a suspended w

(C2) , o n l y w i d e r .

~(c2)

crown

as an u n c r o s s e d

for the d e f i n i t i o n s . )

if ~ c o n t a i n s

an u n c r o s s e d

Now suspended

o n e of the f o r m

L e t us d e n o t e

(5), and

in

(Z) = d(~) .

5.2 w e t h e n see t h a t U Therefore

5.6 g i v e s us o n e

Z(~)

= 0, i, 2, or 3. h.d.U

l, or 2.

+ h.d.AA

A 6 A.

d(~)

From

1 9 3

is

this

(6)

$

One can write

set b y dd. The

set T(c n)

T h u s c 2 is t h e is s i m i l a r

down a projective

to

resolution

-

194

-

of the form 0

,pS

~p~

,p~

~P0

~U

(Z)

....~0

C n

w h e r e the kernel of The m a t r i c e s

pl

~p0 is split but not projective.

involved are rather complicated,

not give any d e t a i l s

here.

h,d.U c

(A) = 3 + h . d . A A n

for all n o n t r i v i a l A.

It then follows

and we shall

from 5.1 that

-

195

-

REFERENCES

[i]

Balcerzyk,

S., "The Global Dimension of The Group Rings

of Abelian Groups II" [2]

Balcerzyk,

8

Fund. Math

S., "On Projective Dimension

of Modules".

Berstein,

Cartan,

(1966)

of Direct Limit

(1966)9

I., "On the Dimension of Modules

IX", Nagoya Math. J. 13; 83-84 [4]

58; 67-73,

Bull. Acad. Polon. Sci. Set. Sci.

Math. Astron. Phys. 14; 241-244 [3]

9

H. and S. Eilenberg,

and Algebras

(1958).

Homological Algebra.

Princeton University Press,

Princeton,

New Jersey,

1956.

[5]

Eilenberg,

S., A. Rosenberg,

and D. Zelinsky,

Dimension of Modules and Algebras,

Math. J. 12; 71-93, [6]

Lawvere,

Mitchell,

(1957).

B.,

10,

(April,

"On the Dimension

1963). of Objects and Categories,

I". J. of Algebra 9; 314-340,

[8]

Mitchell, II".

Nagoya

F. W., "The Group Ring of A Small Category".

Notice8 A.M.S. [7]

VIII".

"On the

B.,

(1968)9

"On the Dimension of Objects and Categories,

J. of Algebra 9; 341-368,

(1968).

9

-

[9]

196

-

Mitchell, B., "On Characterizing Functors and Categories". Mimeographed, Bowdoin College,

[lO]

(1967).

Osofsky, B., "Upper Bounds on Homological Dimensions". (to appear.)

-

197-

LOCALLY NOETHERIAN CATEGORIES AND GENERALIZED STRICTLY LINEARLY COMPACT RINGS.

APPLICATIONS. by

JAN-ERIK ROOS (Department of Mathematics, University of Lund, SWEDEN.) CONTENTS w O.

Introduction

w i.

Characterizations of locally noetherian categories

w 2.

Locally noetherian and locally coherent categories. The

..............................................

conjugate category w S.

1

.......

3

.......................................

7

Coperfect categories. The conjugate category of a locally noetherian category, grull dimension of G~othendieck categories

...............................................

w 4.

Structure of endomorphism rings of injective objects in

w 5.

Explicit realization of the dual of a locally noetherian

locally noetherian categories

............................

9

12

category .................................................

19

w 6.

TopoloEically coherently completed tensor products

39

w 7.

Explicit study of the dual and the conjugate of the Gabriel

.......

filtration of a locally noetherian category .............. w 8.

42

Generalized triangular matrix rings with a linear topology, and classification of stable extensions of locally noetherian categories

w 9.

46

Change of Krull dimension in stable extensions of locally noetherian categories

w i0.

....................................

....................................

63

Application i: The structure of right perfect, left coherent, stable PinEs ...................................

64

w ii.

"Application" 2: Quasi-Frobenius categories

66

w 12.

Final remarks BIBLIOGRAPHY

w O.

..............

............................................

68

.............................................

70

Introduction. The principal aim of this paper is to pmove (and to give applications

and precisions of) a structure theorem (cf. notably Theorem 6 and its corollaries below) for those categories

~

that satisfy the following two

-

198

-

axioms: A)

C

is an abelian category, that has direct limits that are exact

functors when taken over directed sets (this is the axiom AB 5 of [30] ); B)

C

has a set of generators

object of

C

(i.e.

Na

{Na}

, where each

Na

is a noetherian

satisfies the ascending chain condition on sub-

obj acts ). Following Gabriel [24], [25], we will say that

C

is a locally

noetherian (abelian) category if A) and B) are verified. Here are some examples of locally noetherian categories: i) ring 2)

Mod(A) = the category of unitary left modules over a left noetherian A; The category

prescheme X 3)

Qcoh(X)

of quasi-coherent sheaves over a noetherian

[25_];

The category

Mod(0 M)

of sheaves of 0_x-mOdules, where

noetherian presche e w i t h structure sheaf ~)

The categoz~

Tots(Z)

is a locally

[32];

of abelian torsion groups, and more generally

any closed subcategory [25] of 5)

X

Mod(A),

where

A

is as in 1);

The category of topological discrete G-modules, where

G

is a pro-

finite group [583 ; 6)

Let

be a graded r i n g ,

A = (An)n 9 0

ring, and where eac~ natural left A~

An

where

A~

is a left

is a finitely generated left A~

structume.

Then the category

category

Modgr+(A)

if and only if

A

for its

Modgm-(A)

negatively graded left A-modules is locally noetherian.

noetherian

of

However, the

of positively graded ~modules is locally noetherian is noetherian in the graded sense, and this condition

is strictly stronger than the one mentioned above (example: the Steenrod algebra, which is coherent, but not noetherlan [15~); 7)

The dual category of the category of pro-algebraic commutative g~oup

schemes over an algebraically closed field [4~]. It should be remarked that the cases 2) and 3) are quite different and that in the examples 2)-7) the category

~

is not in general equivalent to

9-

1 9 9

-

a module categor7 (i.e. a category of type i)). We make no attempt to summarize the results of this paper here, and we only remark that the principal ~esult is that the study of locally noetherian categories is entirely equivalent to the study of a certain class of toRological rings with a linear topology that is exRlicitly described.

Thus, problems and results that seem to belong to "pume cate-

gory theory" are in fact equivalent to "purely ring-theoPetlcal" ones~ and this gives new results in both directions.

More details about this are in

particular given in w 5, where it is also explained how our results ape inspired by and related to earlier results of Kaplansky [3~, Matlis [40_], Gmothendieck ~31], SePre [56], Gabriel [2~,

[2~ and Leptin ~(I,

~"

Several ring-theoretical and "categorical" applications as well as several open problems ape scattered throughout the paper.

Among the "categorical"

applications, we wish particularly to stress those given in w 8, where we ape able to classify all "exact sequences of categories"

o

,A

,s

,o

where

D

is a localizing subcategory of

where

D

is supposed to be stable under the fo1~nation of injective

envelopes in

Q

(locally noetherian), and

C.

Some results of this paper have been su~aDized in [54],

and the

pmesent paper will be a pamt of a systematic study of Grothendieck categories, that we hope to publish soon (hopefully together with a solution of at least some of the problems mentioned above). In omdem to facilitate the reading of this paper and in ordem to give some general hackgmound, we will first briefly review and complete some well-known results about locally noetherian categories.

w i.

Characterizations of locally noetherian categories. A categomy satisfying the axiom A) above, as well as the following

weakened form of B:

-

Bg)

~

200

-

has a set of generators

{G ) , a~ I is generally called a Grothendieck category [2~ , and Grothendieck proved in [3ql that such a category has sufficiently many injective objects.

now

~

is also locally noetherian, then we have much mope precise results.

In fact, Matlis proved in [4~, that if A

If

~

is of the form

is a left noetherian ring, then every injective in

~

Mod(A),

where

is isomorphic to

a sum of indecomposable injective objects (which in the commutative case cormespond

to the prime ideals of

isomorphic in a natural way. generally every directed

A), and any two such decompositions are

Furthermore, in this case every sum and more

lim

of injectives in

Mod(A)

is again injective

>

[l~.

These results were extended by Gabriel [23~, C2~,

general locally noetherian case.

[2~I , to the

However, there ape also several con-

verses of these results, which in the module case are due to Bass [4], Papp [45] and Faith-Walker [213 and which together with the Matlis results can be formulated as follows: PROPOSITION I.-

The following conditions on a category

Mod(A)

are

equivalent: (i)

Mod(A)

is locally noetherian (i.e.

(ii)

Ever~ injective object of

Mod(A)

A

is left noethePian);

is isomorphic to a direct sum of

indecomposable in~ectives~ (ill)

Any sum of injectives in

(iv)

Any (directed) direct limit of invectives in

(v)

[Bounded decompositions of injectives. 3

such that every in~ective in each havin K (vl)

~ s

Mod(A)

is injective; Mod(A)

is in~ective;

There exists a cardinal

c__

is isomorphic to a sum of injectives,

Kenerators!

[Strict cogenerator.]

that every, object sum of copies of

Mod(A)

in

There exists an object

Mod(A)

C

of

Mod(A)

such

is isomorphic to a subob~ect of a direct

C.

If we now replace

Mod(A)

by a general Grothendieck category

then the conditions (i)-(vi) above are not all equivalent in general

~,

-

201

-

(as for the general formulation of (v), cf. below), a counter-example to the implications (iii) => (i), (iv) => (i), (vi) => (i) is for example given by a non-discrete spectral category i).

The results of Gabriel are

exactly the implications (i) => (ii), (1) => (iii) and (i) => (iv). Also the implication (ii) => (vi) is universally valid but we do not know for example whether (ii) => (i)

is always true for all Grothendieck categories.

As for the condition (v) as it stands it has no meaning for Grothendieck categories.

However, if

that an A-module HomA(M,.)

M

s

is an infinite cardinal, it is easy to see

has ~

generators if and only if the functor

commutes with c_-directed unions.

[Foc more details, and for

relations with the notion of object of (abstract) finite type, we refer the reader to [53J( I Thus, if we say that an object category

s

has

~ ~

generators [~

cardinal, cf. ~33J if the functor

C

in a Grothendieck

infinite cardinal, or the finite

Home(C,')

commutes with c--directed

unions, then (v) makes sense, and we can in fact prove the following THEOREM i.-

The followin~ conditions on a Grothendieck category

C

are

equivalent: (i)

~'Bounded decompositions of injectives.'~ There exists a fixed

cardinal

c

composition (i)'

such that every inject ive

I

I =~Ii,

has

where each

Ii

There exists an injective object

is a direct sum of direct factors of (il)

Eyery sum of injectives in

(iii)

["Strict cogenerator.'~

every object of

~

~

in

I

C

< c --_

admits a direct de~enerators; --

such that ever,j injective .in

I;

is still injective;

There is an ob~gct

D

of

~,

such that

i_.s a subobject of a suitable direct sum of copies of

D.

I)

A spectral cateKo~ is a Gmothendieck category, where every object is

injective;

such a category is locally noetherian if and only if it is

discrete in the sense of [27].

For the general theory of spectral categories,

and examples of non-discrete ones, we refer the reader to ~ i 3 and the literature cited there.

-

2 0 2

-

This result will not be needed in what follows, so we give no proof. Indications of the proof together with the necessary generalizations to Grothendieck categories of results of the Kaplansky type about direct decompositions of modules can be found in [55 . As we mentioned above~ the example of the non-discrete spectral categories shows that the categories satisfying the equivalent conditions of Theorem 1 are not necessarily locally noetherian.

The Theorem 1 suggests

the introduction and study of a continuous variant of locally noetherian categories.

However, several problems remain unsolved.

It is for example

not known (except in special cases) whether the condition (ii) of Theorem 1 implies that (directed) direct limits of injective objects are still injective [53].

In what follows, we will however stick to the (discrete)

locally noetherian case.

This case is easily isolated from the continuous

case, if we require the axiom AB 6 [503 instead of axiom AB 5 (the axiom A above) as the ground axiom on our categories. THEOREM 2.-

We have in fact the following

[Characterizations of locally noetherian categories.]

following conditions on an abelian category

C

The

with a set of generators are

equivalent: (i) (ii)

~

is locally noetherian;

[resp. (ii'~ ~

satisfies the axiom

AB 6,

and every sum (resp.

directed

lim ) of injectives is injective;

(iii)

~

satisfies

AB 6

and has bounded decompositions of injectives;

(iv)

C

satisfies

AB 6

and has a strict cogenerator;

(v)

C

satisfies

AB 6

(AB 5

>

might be sufficient here),

and every

in~ective is a direct sum of indecomposable invectives| (vi)

C

satisfies

AB 6

and evePy direct limit of a direct system of

essential monomorphisms is an essential monomorphism. For indications of the proof of this theorem we refer the reader to ~3] and to ~0] where the condition

AB 6

and its variants are discussed.

This Theorem 2 contains both the Proposition 1 above as well as the theorem of

[43] since a module category, and mope generally a category that is

-

203

-

l o c a l l y of f i n i t e type [~33, both automatically satisfy the axiom AB 6 [507.

w 2.

Locally noetherian and locally coherent categories.

The conjugate

category. Gabriel has observed [24], [25] ~f. also [31], [561~, that if locally noetherian and then and

N(C) C

N(C)

C

the category of noetherian objects of

is C,

is an abelian category, that is equivalent to a small category,

is equivalent to the category

contravariant functors from Conversely, if

N

N(C)

to

Lex(N(C) ~

Ab) i) of left exact

Ab (= the category of abelian groups).

is a noetherian abelian category (i.e. an abelian care-

gory, where every object is noetherian) that is equivalent to a small category, then

Lex(N ~

Ab)

is locally noetherian, and its category of

noetherian objects is naturally equivalent to

(1)

N

~

>

Lex(N ~

N.

Thus the map

Ab)

defines a one-one correspondence between the (equivalence classes of) small abelian noetherian categories and the (equivalence classes of) locally noetherian categories. It is natural to ask, what kind of Grothendieck categories we obtain to the right in (i), when category

D.

N

is replaced by an arbitrary small abelian

To answer this, we first need the definition below.

first that an object

C

in a Grothendieck category

finite type [43] if

HOmc(C,')

DEFINITION i.-

C

that

C

Let

where

C'

is said to be of

commutes with directed unions.

be an object in a Grothendieck c@tegory

is coherent, i~f C

C' ~ f > C,

~

Recall

~. We saZ

is of finite type~ and if for any morphism

is of finite type,

Kerf

is so too.

If

C

family of generators~ formed by coherent objects, then we say t h a t

has a C

is

a locally coherent category. The following result can be found with small modifications in [22] and

l)

If

~

is a category,

C_~

denotes the dual category of

~

[34.

-

204

-

its proof is essentially an adaption to the locally coherent case, of the corresponding proof of Gabriel for the special case of the locally noetherian categories. PROPOSITION 2.-

The map

D

~"

> Lex(DO, Ab)

defines a i-i correspondence between the (equivalence classes of) small abelian categories and the (equivalence classes of) locally coherent Grothendieck categories.

Furthemmore

category of coherent objects of into

Lex(D_~

~

Lex(D ~

is naturally equivalent to th? and

Ab),

A_~b), bj means of the functor

D

is naturally embedded

D P--> HOmD(" , D) = h D.

Examples of locally coherent categories: i)

Mod(.A)

is locally coherent if and only if the ring

A

is left coherent

in the sense of [9], p. 63 (cf. also [13]), i.e. if and only if every finitely generated left ideal of 2)

A

J. Cohen has proved [15] that

algebra, is locally coherent. noetherian. Now let

is finitely presented. ModgT+(A),

A

is the Steenmod

This category is however

not locally

[For the notation, see example 6 in the introduction.] ~

be a locally coherent categoz,y, and let

categomy of coherent objects of

~.

Ab).]

Put

~=

Coh(~)

be the

[Then, as we have seen, Coh(C)

abelian, equivalent to a small category, and Lex(Coh(~) ~

where

Lex(Coh(C), Ab).

~

is equivalent to This category (which is

evidently locally coherent) will be called the conjugate category of what follows.

It is easy to see that

the category of left exact functors lim . FurthePmore -

~

~ ~m>

is equivalent to Ab

is

~

in

L e ~ l i m ( ~, A_b_) =

that cormnute with directed

is natumally equivalent to

C.

Oum aim in the next

>

section is to characterize completely the categories locally noetherian category.

Since every object of

~,

where

Coh(~) ~

~

is a

is artinfan

(i.e. satisfies the descending chain condition on subobjects) if

Coh(~)

is noetherian, one might think that the conjugate categories of locally noetherian ones, would be exactly the locally coherent and locally artinian

9 -

205

categories (locally artinian means for us:

-

there exists a family of

generators that ame artinian - this is different from 0ort's ter~ninology [44]).

However, these categories form only a special case of the class of

conjugates of locally noetherian categories, and

Mod(Z)

is not of this

special form (cf. Example 1 following Corollary g of Theorem 13 in w 8). Thus we must study a descending chain condition, that is weaker than the usual one:

w 3.

Coperfect categories. The conjugate category of a locally noetherian

categoz~.

Krull dimension of Grothendieck categories.

As is well

[25], not only does every object

C

in a

G~othendieck category admit a monomorphism into an injective object, but it also admits a minimal one, that is essentially unique, and that is called the injective envelope of

C.

The dual of this notion is the projective

envelope (also called projective cover). jective envelopes

However the existence of pro-

is a rare phenomenon, even if the category is a module

category (so that we have sufficiently many projectives). a ring

R

Bass [3] called

a right perfect ring if every right R-module has a projective

cover, and he proved the following result: THEOREM OF BASS.

-

R

The followin~ conditions on a ring

are equivalent:

(i)

R

is right perfect;

(ii)

R

satisfies minimum condition on pFincipal left ideals;

(iii)

R/radjR

is an artinian semi-simple ring, and

radjR

is right T-

nilpotent [3] (radj = the Jacobson radical); (iv) e

Every direct limit of projective right R-modules is projective;

oe

Bass left as an open problem, whether (ii) is in fact equivalent to: (ii)'

R

satisfies minimum condition on finitely generated left ideals;

but J.-E. BJORK proved recently [6 ] that (ii) and (ii)' equivalent.

are in fact

This generalizes an earlier result of S.U. CHASE (cf. Appendix

to [20]) which says that (ii)' is verified for semi-primary rings (a special case of the perfect ones).

In view of the BASS-BJ~RK result, it is natural

-

to call a module

M

coperfect, if

finitely generated submodules. DEFINITION 2.-

perfect if ~

206

M

-

satisfies the minimum condition on

More generally:

A Grothendieek category

~

is said to be (locally) co-

is locally of finite type (cf. [4~),

family of generators

{Ga } ' where each

G~

and if

C

has a

is coperfect~ i.e. satisfies..

minimum condition on finitely senerated subob~ects. Remark:

If

~

is locally copemfect, then every object of

~

is in fact

copemfect, so we can and will omit the womd "locally" in what follows. THEOREM 3.-

The map

~

I

>~

(cf. w 2) defines a one-one correspondence

between the (equivalence classes of) locally noetherlan categories and the (equivalence classes of) locally coherent and coperfect categories. PROOF:

Suppose first that

prove that hN

~

~ = Lex(N(~), Ab)

HOmc(N,') , N q N ( ~ )

is locally noetherian. is coperfeet (el. w 2).

hN

is copePfect.

is coherent, every finitely generated subobject of

thus of the form by

h N'.

N ~--> h N, N'

the fact that

N(~)

Since

N(~)o

is a quotient of

N.

copePfect if and only if

R

But since

is coherent,

Thus the result follows from

is a noetherian category.

A module category

hN

~,

is fully and faithfully embedded in

Conversely, given a locally

coherent and coperfect category ~, it follows that cu abelian category, and so D is locally noetherian. Remark and example:

The set of objects

is exactly the set of coherent generators for

and so it is sufficient to prove that every hN

We only have to

Mod(.R)

Coh(D)

is an artin~an

is locally coherent and

is left coherent and riKht perfect.

This class

of rings was first introduced by Chase [13], who proved that it is equiva!~nt to say, that every product of right projective R-modules is again projective. In the commutative case, these Pings coincide with the artin~an ones [IS], but in the general case they need not be either left or rlght artlnian, as the following example (which was communicated to me by L.W. Small) shows: R = (~

~>.

(For more details~ and for a general theory of generalized

triangular matrix Pings, cf. w 8 and w 9.)

-

The correspondence

s

2 0 7

-

wili be studied in more detail below,

using the endomorphism pings of injectlve objects, and we will just conclude this section with a few elementary remarks, that are easy consequences of the theory of Krull dimension of Grothendieck categories [25], that we will first recall briefly: Suppose

~

is a Grothendieck category.

Then a full subcategory

is said to be a localizing subcategory [25] if

~

_L

of

is closed under

formation of subobjects, quotient objects, extensions and

(in ~) > In this paper every subcategory will be a strict subcategory, i.e.

~Note)

lim

the subcategoPy should contain all isomorphic copies (in the big category) of its objects.] If

~

is a localizing subcategory, then it is also a

Grothendieck categor,j, the quotient category

~/~

can be for~ned, and the

.M

natural functor

~

3

> ~/~

is exact, and has a might adjoint

is full and faithful [25]. Thus Now let

~/~

C be the smallest (exists!) localizing subcategory of --o

subobjects of

S)

of

s

Form C/Co ,

localizing subcate6ory of C/C etc... ----o

aM

and l e t

containing those

s

C

x.

example:

~,

and

that S

ape

be the smallest

C ~,

that ape simple or 0 ...C...

fop a limit ordinal is evident), and theme exists

such that

C

= C

--M

= ...

If

said to have a Krull dimension defined, and we put such

0

We get a transfinite f i l t r a t i o n l ) C C ~ l C

(the definition of an ordinal

~,

that

is also a Grcthendieck category.

contains the simple objects (S simple S # 0, and only

in

JM

C

--M

= ~,

then

C

--

is

dim ~ = the smallest

Not every Grothendieck category has a Krull dimension. a continuous (non-discrete) spectral category.

Counter-

One can however

introduce a continuous analogue of Krull dimension, and then this last category has dimension

0.

However,

Mod(A, 2) ~8], [49~,

non-discrete aPchimedean valuation Ping

and

~

where

A

is a

the maximal idealjdoes

not even have a K~ull dimension in this mope general sense. It is easy to see that 0

if and only if every

~

C # 0

has Kmull dimension defined and equal to contains a simple object o A

i) This filtmatlon will be called the Gabriel filtration ef

locally ~.

-

noethcrian category

~

is zero if and only if

category

~

2 0 8 -

always has a Krull dimension and this dimension ~

is locally finite [25]. [The Grothendieck

is said to be locally finite if

~

has a family of generators,

formed by objects of finite length, i.e. objects that are both artinian and noetherian [253.~ Finally, the Krull dimension of

Mod(A),

where

is a commutative noetherian ring, coincides with the dimension of fined by means of chains of prime ideals (Krull) ~5].

A

A de-

This last descrip-

tion is valid in some non-commutative cases too [25]. For another definition of Krull dimension (when it is ~ o Now if ~ dimension

0,

), see ~8].

is arbitrary locally noetherian, then T for if

C ~ 0

has Krull

is an object of _~, then a minimal element

in the family of finitely generated non-zero subobjects of simple object.

Thus in particular

~

see thus that

n the module case R

~ = Mod(R),

is artinian, if and only if

and one can prove that

Mod((~ ~))

has finite Krull dimension for

~

study of a category ~

Mod(R)

is locally finite, More detailed

In particular we will see that R

and that every dimension can occur.] Thus: noetherian category

itself is

considered above, we

has Krull dimension I.

results are to be found in w 8 and w 9. Mod(~)

must be a

is locally noetherian if and only

if it is locally finite, which is the case if and only if ~ locally finite,

C

left coherent and right perfect, The study of a locally

of arbitrary Krull dimension is equivalent to the of Krull dimension

0

of a special type (locally

coherent and coperfect), this last category being locally finite, if and only if ~

is so (i.e. if and only if ~

is of Krull dimension

0). This

gives an indication of the enormous difference between the cate~omies of Krull dimension

w 4.

0

@nd the locally finite categories.

Structure of endomorphism rings of injective objects in locally

noetherian categories. In [40] Marlin pmoved, that if

A

is a commutative noetherian ring,

-

a prime ideal in in

Mod(A),

A,

and

209

E = E(A/~)

then the endomorphism ming of

completion of the local ring

A .

-

the injective envelope of E

in

Mod(A)

noetherian category

I

~,

is exactly the

For complete (noetherian) local rings

there are structure theorems, due to I.S. Cohen ~8~, one to expect, that if

A/~o

[82]. This leads

is an injeetive object in an arbitrary locally then

Homc(I,I)

should have a natural topology

and that there should be some explicit structure theory for this topological ring.

We will show below, that this is indeed the case (to a certain degree)

and that, furthermore, the (topological) endomorphism ring of a big injective (for a definition of that see Theorem 4 below) in determines

C.

zero (i.e.

C

~,

completely

In the special ease, when the Kmull dimension of

~

is

locally finite) the above results are due to Gabriel [24],

[25], who also had some indications about the general case, but as we will see later, quite new phenomena occur when we pass to the complete study of the general case. The following well-known proposition about linearly topologized rings will be useful in what follows. Grothendieck category

~

Recall ti~t a full subcategory

is said to be a closed subcategor,] of

~

of a

~,

if

is closed under formation of subobject, quotient objects and (directed) lim

in

C

(such a

D

is then necessarily also a Grothendieck category)

>

and that a topology on a ringl)R on

R,

is said to be a (left) linear topology

if there is a fundamental system of neighbourhoods of

0

consisting

of left ideals (these ideals are then open). PROPOSITION 8.topology on

A

Let

A

be a ring.

The map Zha~ tc each (left) linear

associates the full subcategory

Dis(A)

of

Mod(A)

whose

objects are the discrete topological A-modules, defines a one-one corres~ondence betweQn the (left) linear topologies on gories of

Mod(A).

T

on

A,

and the closed subcate-

Furthermore, the open left ideals

those left ideals such that topoloK~

A,

A/!~Dis(A),

1

of

A

are exactly

and for a ~iven left linear

the~o~respendin~ s @ t o f open left ideals

J

l) In this paper all topologies considered on rings are supposed to be compatible with the ring structume.

-

210

-

satisfies :

~,

(il)

(iii)

~

~

e JT => O % ( ~ e d y

s JT'

a

;

aFbitraz ~ in

Conversely, glven a set a unique lineam top01ogy

J

A => (@:a) = { x l x a s

e J~.

of left ideals satisfying (i)-(iii), there is T

on

A

such that

J = JT"

(Cf. Gabriel [25], p. 411-412 and Bott~baki [9], exemc. 16, p. 157.) Now, let C,

and

of

C,

C

be a G~othendieck categoz~,

A = Homc(I , I). then

HOmc(C , I)

the composite map C C I

It is clear that if

C

f > I

A,

an injective object of

C

is an ambi%mary object

is in a natural way a left A-module [we write

is a subobjec% of

way a left ideal of

I

~ 9 I

I,

then

as l'f].

Thus in particular, if

I(C) = HOmC(I/C, I)

and the exact sequence

(I

is in a natumal

is injective):

0 - - > HOmC(I/C , I ) - - > Homc(I, I ) - - > HOmc(C, I)--> 0 shows that the quotient module Considem now the ease when left ideals of

A

C

A/I(C)

can be identified with

is locally noethemian.

Homc(C,I).

Then the set

that contain an ideal of the folln I(N),

where

J

of

N

is

a noetherian subobject of I satisfies the conditions (i), (ii) and (iii) of Proposition 3 as is easily seen. i = 1,2,

[We have for example that if

Ni~I,

then l(N1) f~l(N 2) = HOmc(I/(NI+N2) , I), m

where

Nl+N 2

is the image if the natural map

image is clearly noetherian if = HOmc(I , I)

Ni, i = 1,2

NlJ.LN 2 m > a~e so.~

A

(2)

0.

I(N)

A~>

lim

N~I 9

A/I(N)

N noeth.

A =

forth a

Fu~thez~nome, for

is complete (by that we mean Hausdorff too).

to see this we have to prove that the natural map

and this

Thus the ring

has a natural linear topology, for which the

fundamental system of (idft) ideal neighborhoods of this topology

I,

In fact,

-

is an isomorphism.

211

-

But the map (2) can be identified with

Home(I, I) --->

lim

Homc(N , I)

N cI N noeth. and lim HOmc(N , I) = Homc(lim N, I) = Homc(Ii, I) HOmc(N, I) .

- -

(Lemma 1 below) that since

f

HOmc(NI, I),

~

is uniquely

-

par~iculam

A

214

-

is topologically coherent, and therefore the first part of

Theorem 4 is proved modulo the following lemma: LEMMA I.i_~n ~,

Let

~

be a locally noetherian category,

A = Homc(I , I),

and

I

a big in~ective

Homc(N1, I) -~--> HOmc(N2, I)

a map of left

m

A-modules; where

N1

ly) unique morphism

and

N2

ape noetherian.

N2 G > N1

such that

Then there is a (necessar~

HOmc(G,I) = g.

PROOF:

Let ~ I and ~ I be injectives containing N 1 and N 2. sI s2 (Heme we can suppose that sI and s 2 ape finite sets, cf. proof of

TheoPem 4 above.)

Thus we have two exact sequences:

0 - - > N i - - > II I - - > K i - - > 0

(i = 1,2).

s.

1

Take

Homc(. ,

I)

of these two sequences, and considem the diasTam:

m

0 - - > Hom~(K1, I ) - - >

Homc([lI , I) --> HOmc(N1, I ) - - >

0

0---> HOmc(K2, I) --> Homc([i I, I) --> HOmc(N2, I) --> 0 -Since

_

HOmc(!II , I) = j f A -- s I

modules

?

--

is projective, there exists a map of

A-

s1

making the right square of (3) commute, and this map is

necessarily induced by a map using (3) and the fact that zag map from

N2

to

0 - - > NI - > ~ (4)

5

~~ I

r >.I I I. It is now easy to see, s2 s1 I is a cogeneratom, that the composed zig-

in the diagram

I

--> K1 -->

0

si t r 0 --> N2---->LI ~ I s2

is zemo.

s2

---> K2---> 0

Thus fmom (4) we get an induced map

lies immediately that the fact that

I

N2

G > N1 ' and one veri-

HOmc(G , I) = ~ . Uniqueness of

G

follows from

is a cogenerator, and so in pamticulam the lemma is

proved, and thus also the fimst pamt of Theorem 4.

-

w 5.-

215

-

Explicit realization of the dual of a locally noetherian category. Recall that if

T

is an arbitrary small (or equivalent to a small)

category, then the category of pro-objects of T, denoted by defined as follows (cf. ~i], The ~ o f

Pmo(~)

are the inverse systems

Hompro(T)({T e}

, {T;}s& 6I

{T }

)=

-J

lim lira HomT(T, T;) - --> 8

are epimorphisms.

(or equivalent to a s ~ l l ) category, then (cf. w 2).

every pro-object of [31], [44], [561.

I


Ts (a' > s)

to

iS

~6]):

{T } ~ a directed set, and the morphisms ape given by the formula:

A pro-object

PrO(T),

T

T

Pro(z)

Furthermore, if

T

is an abelian small is naturally equivalent

is artinian (abelian) then

is isomorphic to a st1?ict pro-object [24], [25],

We will now prove that if

t

(abelian and artinian)

can be realized as a full subcategory of a module category

Mod(R), stable

under the formation of kernels and cokernels (thus also under the formation of

Im, Coim ...) in

Mod(R) i) , then the category

Lex(~, Ab) ~ = Pro(T)

can be given a much more precise Pealizatlon as a category of lineartopological R-modules.

Examples for the possibility of such realizations

which go back to the topological duality theory for abelian groups and vector spaces (see notably p. 79-80 of [3~ and ~

[2,]. [25]. [31]. THEOREM 5.-

Let

[40]. R

be a pinK,

a full subeatego_~ of

l)

q

can be found in

2). T

an@belian artinian category, that is

Mod(R), that is stable under the foz~_ation of

We will see later that such a realization can always be constructed,

and this even in a canonical way. But there are also other interesting eases, besides the canonical one ...

2)

In this paper we restrict ourselves to the abelian case, but there should

evidently also exist a corresponding theo_~y of locally noetherian tcposes

~0],

-

kernels and cokernels in categor~

Lex(T, Ab)

Mod(R).

2 1 6

-

Then the dual of the locally noethemian

is naturally equivalent t o the categor,] Mod (R), "

T

whose ?b~ects ape the linearly to~ologized~ complete and separated Rmodules

M

(R

nei~hbs

considered as discrete) that have a fundamental system of of

0

formed by submodules

M

such that

whose mor~hisms ape the R-linear continuous maps. cokernels, imaKes, coimages, products,

lim

in

and

Furtherm0re ~ the kerns Mod (R)

--


U/V' --> M/V' --> M / U - - > 0, where

M/V'

and

M/U

are in

T.

Thus

U/V'

is closed under the formation of kernels in Conversely, suppose that module of

M.

Then

U

V

is T-lineamly topologized).

T,

since

T

Mod(R).

is a T-linearly topologized open sub-

contains an open

z-linearly topologized) and (U

U

is also in

V

such that

contains an open

V'

M/V ~

(M

such that

is

U/V'~T

The natumal exact sequence

U/V' --> M / V - - > M / U - - > 0 shows that

M/U

is in

3,

for

T

is closed unoerycoKerne•

in

Mod(R)

and so the Proposition 5 is proved. COROLLARY.-

Let

M

be as in the Proposition 5.

Then the intersection

and sum of two open T-linearly topologized submodules of

M

is again of

the same type. PROOF.

Let

U

and

V

be the submodules in question.

We have an

exact sequence

(5)

0 - - > U/U ~ V - - >

Here

U~V

Thus

U/U ~ V

M/V ~ T . that

M / V - - > M/(U+V) --> 0

is an open submodule of

is a quotient of an object in

Thus since

U/U~V

'~at

is in

T T,

T

and a subobject of

is closed under images in i.e.

U C~V

Mod(R),

we obtain

is a T-linearly topologized module.

Finally, the exact sequence (5) gives that UeV

is T-linearly topologized.

M/(U+V) E 3,

and since

is open, it is T-lineamly topologized too.

PROPOSITION 6.-

Let

T

be as in Proposition 5, and let

b$ two T-lineamly topologized R-modules, and R-linear map.

Suppose also that

M2

M1

f > M2

is Hausdomff.

(directed) intersection of those open submodules of

Then

M1

and

M2

a continuous Ker f

MI, that are

is the

- 218

T-linearly topologlzed and that contain PROOF:

Let

{M2,i}i~ I

-

Ker f.

he the (directed) set of all

topologized open R-submodules of diagram:

M2,

r-linearly

and consider the commutative exact

O

O

f MI/f-I(M2, i) ~

o

(6)

f

> Kern f------> M 1

> Kem f

S2/M2,i

9 > f'l(M2, i)

~

M2

.. >

M2, i

t 0

0

It follows that the first line iS a monomorphism, and since open, it contains an open submodule

V

such that

f-l(M2, i)

MI/VET.

is

Thus we have

two maps MI/V onto> Ml/f-l(M2,1) mono> M2/M2, i where the two extreme objects are in object

is in

T,

and so

f-l(M2, i)

T.

It follows that the middle

is T-linearly topologlzed.

pass to the inverse limit of the last row of (6), and use that Hausdorff.

Now we M2

is

This gives the exact sequence:

0 --> Ker f--> f'~I 1 f-l(M2, i) --> 0 i and so the Proposition 6 is proved. From now on we have to suppose that

T

is artlnian too, and our

results depend heavily on this assumption. PROPOSITION 7.-

(7)

0-->

With the notations and the hypotheses of Theorem 5, let

(ca)-->

(D a)

(fa) .... 9 (E a) --> 0

be an exact.sequence of directed inverse systems in that each

Ca

is. in

T.

Mod(R).

Then the sequence of R-modules

Suppose

219

-

0-->

lim C

-->

lim D

obtained from (I) ~

-->

lim,

-

lim ~

--> 0,

is exact.

PROOF:

The only thing to verify is that

lim D
llm E 9 .

is onto.

X~ =




qa~(Xs) C X (X~

+ Let

qaS)

+ C ,

where

For this we will use Bourbaki

We will choose as ~ u

objects of

T.

Mod(R)

in the same way. q~stna

qas(q) = ~ q~8

.

(cf. loc. cir.) the empty

X ~

C8-->

of

of two subobjects in

)

Let us prove (iii). is empty (thus in ~

But then

T. Let

),

(Ca) ,

thus a map between

na ~

T.

Ca

is still in ~ ,

X

= xe + C .

Then

and

Ker q~8

n

with

is in T ,

Thus (iii) is verified.

As

one uses the fact that the

of a map between objects of

can apply theor~me I, loc.cit.,

of

The condition (ii) follows

~:~(n ) = ~ + Ker(q~8),

is a map between objects of

Mod(R)

T

or we have an element

to (iv)~ it is verified in the same way~

that

whose associated linear

C

and then use the artinian hypothesis on

v-l.

[83, Chap. 1,

To verify (i), it is thus sufficient to observe that the

intersection in

PROBLEM:

Then

Let us prove that the conditions (i)-(iv) of loc. cir. %J Observe flrst, that the associated linear map qu8 of

is the transition morphism

image in

q~8"

T.

ame verified.

%2

of

then

Everything will follow, if we can prove that

is non-empty.

variety is in

since

It is cleam that

(D),

be t h e r e s t r i c t i o n

set, and those affine subvarieties of

either

= ~a"

is an inverse system of sets~ and it is even an inverse system

theor~me i, p. 138.

v qu8

f (x)

are the transition maps of

qV 8 : X S - - > X

of affine vamieties. lim X

= x

(8 > ~)

Da

v

X

T

is still in

T.

Thus we

and the Proposition 7 follows.

It is easy to see that Proposition 7 can be formulated as saying

lim(1)Ca = 0 [46].

the same hypotheses?

Is it true that

lim(i)c ~ = 0, i _> i,

under

[This is true at least in some special cases [47l

Now that we have Proposition 7 we can easily continue to develop

-

the theory of

Mod (R). T

2 2 0

-

In all that follows, we suppose that we ape

under the hypotheses of Theorem 5 (last time we recall it). PROPOSITION 8.-

Let

M

be an object of

Mod (R),

M.

T

a filtered

1

decreasing family of open T-linea~l Z topoloKized submodules (they ame then in

Mod (R)). T 0-->

Then the sequence of R-modules

O

Mi

9 M

9

1

9 0

lim M/M i < .

i)

i

is exact PROOF:

The proof is analogous to that of prop. 16, p. 391 of Gabriel

[25], but we have to make

T

appear explicitly ever-jwhePe. Let

{U } s

be the directed (cf. The Comollar T of Proposition 5) family of open Tlinearly topologized submodules of

M.

Considem the commutative exact

diaETam 0

> N

0

>

Us

>

M

>

lim

M/U u

(8)


i,s Here

T,

lim M/M i

9

i

(Mi+Us)/Mi ~-~> U s / M I N U s

this object is in




M

0 ---> lim U/U (~M i ~ >


llm M/M i ~ >

0

'>

lim M/(M.+U)I - - *


M 1

$, MI/U

0

It is clear that

E(MI, i)

is in

T,

for

g(Ml, i) = Ml~i U (~MI, i

and we

can use Proposition 5 and its Corollary. Thus the

g(Ml, i)

MI/U ~ T.

Since

family, say

T

is amtinian, there is a minimum element of this

g(Ml,i,).

(i0)

of

form a filtered decreasing set of subobjects in

Ml,i M

Thus + U = MI, i + U,

i ~i x .

But according to Proposition 9 we have

. / • . (MI,i+U) M

I

> 1

1

= i>~i(Ml,i

)

+U

and this together with (i0) and (9) gives

Ml,iX Thus to

Kem f / U ~ K e r T

+ U = Ker f + U.

f = (Ker f + U)/U = (Ml,i M

+ U)/U = g(Ml,i K)

belongs

and the pmopositlon is proved.

COROLLARY.-

Under the hypothesis of Pmoposition i0, the quotient in

Mod(R), MI/KeP f,

equipped with the quotient topolosy, belongs to

ModT(R). PROOF:

Let

U

be an open submodule of

M1

such that

MI/UET.

by the proof of Proposition i0 we know that rv

(Ker f + U)/U-->

Ker f / V N K e r

f

belongs to

T,

and so the exact

Then

-

2 2 3

-

sequence

(ii)

0 - - > Ker f / V ~ K e r

shows that

f - - > MI/U--> Ml/(Ker f + U) ---> 0

Ml/(Ker f + U) E T,

and so

Ml/Ker f

topologized for the quotient topology.

is T-linearly

If we pass to the inverse limit

in (ll), and use Proposition 7, and the fact that complete i )

M1

and

Ker f

ape

we obtain that

M1/Ker f

is also complete for the quotient

topology , thus an object of

ModT(R),

and the corollax,] follows.

Now we wish to compare

M1/Ker f

with its quotient topology, and

Im f

Mod(R~ with the induced topology from

[image in

M 2.

With the notations and h>-pothesis of Proposition i0, the

PROPOSITION ll.-

natural alsebraic isomorphism isomorphlsm, when

Ml/Ker f

h

MI/KeP f and

> Im f

is a topological

are ~iven the quotient and the

Im f

induced topology, respectively. PROOF:

h

Since the algebraic isomorphism

map, it is clearly continuous. ever7 open submodule of Ml/Im f

and

Im f

submodules of

Ml/Ker f

such that

is also open in Now let

M2/ViqT,

(V i)

Im f,

Im f).

The

V i (~Im f

= (Im f + V.)/V. 1 1 Im f / V i O I m

be the set of open

0

(V i (~Im f) for the induced

Since also

is a subobject of

M2/Vi~T,

Im f/V i 6Aim f

Im f/Vi ~ I m

is

f =

we obtain that

f q T.

Now let such that

T.

(we will identify

are open for the induced topology on

thus open for the quotient topology, and so

the quotient of an object of

Im f

and consider

(this is a fundamental system of neighbourhoods of topology on

continuous

Thus it is sufficient to prove that

algebralcally).

M2,

is induced b y a

P

be an open submodule of

Im f / P C

T

for the quotient topology,

~ecall that by Corollary of Pr.oposition I0, Im f

with the quotient topology belongs to

i) The completeness of

Im f

Kerf

ModT(R ~ , and consider the

follows from Proposition i0.

-

224

-

diagram 0

~> V i N I m f ----> Im f

Im f/P

0

I claim That

k(V i N Im f) ~ T .

k(Vi6]Im f) =

But

Vi6~Im f

We have (V i ~ Im f) + P p

V. ~ I m 1

f

P {~V i f~Im f "

is open z-lineamly topoloEized submodule of

quotient topology, and since

P

is also such a submodule,

Im f

for The

P f~Im f6~V. 1

is SO tOo (Corollary to Proposition 5) and so by PmoposiTion 5. in

T

Since

{k(Vi(~Im f)}i

T

(Vi 6Aim f)/~C~V i /Aim ~ ET

is amTinian, The decreasing family of objects

has a minimum element, say

k(Vi,(~Im f),

i.e.

(Vi" ~AIm f) + P = (VifAIm f) + P, i ~ iN. Now apply the Proposition 9 (quotient Topology).

(ViM~Im

~ he

f) + P = ( ~--~M. (Vi ~ I m i>i

last equality follows fmom the fact That

V.M N Im f ~ P . 1

We obtain

We have thus proved That

P

M2

f)) + P = P

is Hausdorf~

Thus

that is open fop The

quotient Topology, is also open fop the induced Topology, and so These two topologies coincide and the Proposition ii is pmoved. PROPOSITION 12.M,

Let

M ~ M o d T ( R ).

In order that a R-submodule

equipped with The induced Topology, belongs To

N

ModT(R) , it is

necessarTand sufficient that

(12)

N =

1 % NCUCM

t)

(This is automatically a directed intersection, cf. PmoposiTion 5.)

U open, M / U & T PROOF:

I f (12) is verified, then we have an exact sequence

of

-

f>

0 ---> N ---> M

225

-

F. I(pPoduct over the U:s of (12)I %. .L

77MIu U

II M2 If we equip

M2

with the product topology (every factor is given the

discrete topology), then we get an object of

Mod (R)

as is easily seen,

T

and

f

is continuous.

Thus

N = Ker f,

belongs to

Mod (R)

(Proposition

T

i0).

C onvemsely suppose that

Mod (R),

and let

T

Then of

U ~N T.

U

N

with the induced topology belongs to

be an open suhmodule of

is open in

N,

and thus

M

N/U /%N

with quotient in

T.

is a quotient of an object

But the exact sequence

(13)

0---> N / U ~

shows that

N-->

M/U--> M/(U+N)--> 0

N/Ug~N

is a subobject of an object of

T,

M/(U+N)~ T

an object of

and so also

If we pass to the

lim

T

too, thus itself

by (13).

in (13), using the Proposition 7, we get an



0

lira N/U 6~N -->

>

N

lira M/U -->

~>

M

lira M/(U+N) --> 0

I~

M/N

>

where the first two vertical morphisms are isomorphisms.

0

Thus the third

one is so too, and this gives

N=

/ IV NCVcM V open, M / V q T

and so the Proposition 12 is proved. END O F T H E that

PROOF OF THEOREM 5:

Mod(R)

It now follows from what we have done,

is an abelian categor,], where the kernels, cokez~els,

images etc. are the algebraic kernels, cokernels, induced, quotient, has ambitrary

llm, N(~)~

ModT(R).

Dis(R).

C_.~

Lex(N(~) ~

and

Now if we take

then all the conditions of Theorem

easily seen to be verified, and so equivalent to

N(~) ~

Ab) ~

5 a~e

is naturally

However, due to the special form of

T,

this

last category can be intePpreted as the category of topologically coherent, complete R-modules, denoted by topologized R-modules

M,

TC(R),

that are complete, and that have a fundamental

system of open neighbourhoods of M/M

O,

formed by submodules

is coherent considered as an object of

ape the continuous linear map. is an object of of

whose objects ape the linearly

TC(R)

TC(R),

FurthercBore

Dis(R) R

M

and whose morphisms

with its natural topology

considered as a left R-module, and the objects

aloe automatically topological left R-modules, when

this topology that is coarser than the discrete one. HOmc(C, I)

R

is given

Finally every

with its natural linear topology [defined by the

HOmc(C/N , I)

(N~C,

C.~

is given by

TC(R)

noetherian~, belongs to

TC(R)

and the equivalence

C ~--> HOmc(C , I).

In the reasoning above, it is not necessary to start with fact, if

such that

R

~.

In

is an arbitrary left topologically coherent and topologically

coperfect ring (R

not necessarily complete), then

az-tinean abelian category, and so locally noetherian although conjugate category

R

TC(R)~

is not in

Lex(Coh(Dis(R)) ~

Ab)

Coh(Dis(R))

Lex(Coh(Dis(R), Ab) TC(R)

is an is

in this case, and its

is equivalent to Dis(R). Let us

summarize and complete the results obtained: THEOREM 6.in ~,

Let

C

be a locally, noethezian categoPy,

R = Homc(I, I)

the endomorphism ring of

I

I

a bi~ in~ective

with its natural

-

I)

llnear topology

, and

TC(R)

229

-

the category of topologically coherent

complete linear-topological R-modules (they are then automatically topologically coperfect). C_~

~

Then we have a natural functor

Homc(C , I) E T C ( R )

(natural topology on Home(C, I))

and this functor defines an e~ulvalence of categories Furthermore~. the con~u~ate category (w 3) ~

of

C

C_~ ~

TC(R).

is naturally

i

e~uivalent to

Dis(R).

Conversely, given any topolo~ically coherent and topologically coperfect complete ring

R,

then

TC(R) ~

is a locally noetherian cateEory,

and so in particular every projective of product of indecomposable prp~ectives. TC(R), e

TC(R) Further

is (uniquely) a direct R

is projective in

and every indecomposable projective is of the form

is a primitive idempotent, (i.e. e

orthogonal idempotents).

Re,

where

is not the sum of two non-zero,

Finally, if

R

is topologically coherent and

topologically cop erfect (but not necessarily complete) then

Dis(R)

the .conjugate of a locally noetherlan category, thus of the form

is

Dis(R),

where /R is also complete (this sort of completion will be studied below). PROOF:

Everything has been proved except the assertion concerning the

projective objects (R complete).

But in

TC(R) ~

every injective is an

essentially unique sum of indecomposables, and so ~y duality every projective in Further

R

TC(R)

is a unique direct product of indecomposable ones.

is projective in

TC(R)

as is easily seen.

be an indecomposeble projective ~ 0 of submodule of

P

such that

Then the map

P---> P/U

P/U

P.

Since

i) R

P

is indecomposable,

and let

is coherent in

is continuous,

and so the projective envelope of

TC(R),

P/U

Dis(R)

Finally, let U

be an open

and non-zero.

thus an eplmorphism in in

P - - > P/U

TC(R)

TC(R)

is a direct factor of

is the projective envelope

is then complete, topologically coherent and topologically co-

perfect, cf. w ~.

P

-

of

P/U.

But

P/U

is coherent in

230

-

Dis(R),

thus in particular a n finitely generated R-module, and so we have an epimorphism ~ R -->P/U--~0 1 in Mod(R), and this is a continuous map, thus an epimorphism in TC(R). n It follows that P is a direct factor of J_LR (n finite) in TC(R), 1 and since P is indecomposable, it is a direct factor of R (essential uniqueness of the decomposition of R), thus of the form a primitive idempotent.

That

e

Re,

where

e

is

can be arbitrary (primitive) is easily

seen. COROLLARY i.-

Any lineamly topologized topologically coherent and

topqlo~ically coperfect complete qing

R

ca n be represented as the

endomorphism ring with its natumal topology of a big injective in a %ocally noetherian category determined by

~,

whose equivalence class is uniquely

R.

END OF THE PROOF OF THEOREM 4 (w q):

Remark.-

! Apply Corollary I.

Since a big injectlve in a locally noetherian category is not

uniquely determined (a given indecomposable injective can occur an ambitramy number of times), we see that we can have R

being topologically isomorphic to

R'.

We will now study

closely and also determine the degree of choice of Recall that if ~

C~--~> C_~, without

~

and

I

more

R.

is an arbitrar-y Grothendieck categoz-y, ~

the spectral category of

R

P--> Spec(~)

an arbitrary injective object of

~,

then the kernel of the natural map (17)

Hom~(I, I)--> Homspec(~)(P(I) , P(I))

is exactly the Jacobson radical of the ring

Homc(I , I)

~his madical

w

will be denoted by

madj(Homc(I , I)~

[59], [19], [51~. FuPthemmore,

since (17) can he identified with the directed

lim

of the e Dimorphisms

- - >

(I

is injective)

Homc(I , I) --> Homc(V , I), V C I

of

I [51], (17) is an epimorphism.

Homc(I , I)/radj(Homc(I, I))

essential subobject

Thus the quotient ring

is naturally isomorphic to

-

Homspec(c)(P(I), P(I))

2 3 1

-

and thus it is a (von Neumann) reEular right

self-injective Pin E [51]. Now in particular, if

C

is locally noetherian then

discrete l')[27~ i.e. of the form

~

Mod(K i ) where

isomorphy classes of indecomposable injectives in i ~ S,

Ki

is the skew field

Then

I(C)

P(C)

of

C

C,

to get

P(C),

and where for The functor

i E S

occurs in

is

take the injective

Ki, {V i}

where the i ~

K. 1

P

and decompose it into a direc~ sum of indecomposables.

is a set of vector spaces over

dimension ovem

is

is the set of

Hom . . . . (P(i), P(i)). ~pec ~ )

easily made explicit on objects: envelope

S

Spec(C)

of

V. 1

I(C).

s

is the number of times the indecomposable

Thus in particulam

I(C)

is a big injective if

and only if

dimK.V i ~ 0, i 6 S . Let us say (cf. [25]) that I is a 1 sober injective if dimK.V i = i, i ~ S, i.e. if and only if every 1 indecomposable occums once and only once in I(C). From what we just have said it follows that the fact that

endomorphism ring of

I,

I

is sober can be easily seen on the

or more precisely on

for this Pin E is naturally isomorphic to

Homc(I , I)/radj(Homc(I , I))

I~EndK.(Vi). is

i

Since by the Theorem 4 of w W every lineaPly topologized topologically coherent and topologically coperfect, complete Pin E is the endomorphism ring of a big injective in a locally noetherian categor-], we get: COROLLARY 2.-

Let

A

be a linearly topolo~ized, to~gloKically coherent

and topologically coperfect complete rin~.

Then

A/radj(A)

Neumann regular right self-in~ective rin~ of the form

~-~EndK.(Vi), i ~S

where the skew fields

Ki

and the left K.-vectorspaces 1

is a yon

V. 1

i

are un~quel~

determined. Remark I.-

l)

A/radj(A)

More generally

is also left self-injective if and only if

Spec(C_) is discmete if

defined [25], and even more generally, if

C

has K~ull dimension C

is locally coirreducible

[423 (this is even a necessary and sufficient condition for discretenesS.

-

di-KV i
TC(A) ~

defines a one-one coz-~espondence

between the topologlcal isomorphy classes of linearly %opologized, topol0gically coherent, topologically coperfect and complete rin~s that ape sober, and the equivalence classes of locall~ noetherian categories I). If we say that two llneaply topologized topologically coperfect, topologically coherent, complete rings Morlta equivalent" if COROLLARY q.-

A

TC(A)--~> TC(B),

and

are "topologlcally

then we obtain:

Every linearly topolo~ized, topologically coherent,

topologlcall~ coperfect , complete rin~

B

is topolo~ically Morita

equivalent to a uniquely determined such a rin~ B

B

A

that is sober, an__~d

can be explicitly described as a topological n~trix rink over

A as

i_. [38]. [39]. The discrete rings of Corolla1~y ~ ape exactly the left coherent, right perfect rings (cf. R e ~ k

and example following Theorem 3 in w 3).

For these rings, the topological Morlta equivalence is the ordinary Morita equivalence ~],

[25] as is easily seen.

COROLLARY 5.-

A ~-> TC(A) ~

The map

Thus:

defines a one-one correspondence

b e.tween the Momita equivalence classes of left coherent, riEht perfect PinEs and the equivalence classes of locally noetherian categories that have a bi~ noetherian in~ective (i.e. every indecomposable injective is noetherian, and there are only a finite number of isomomphy classes of l)

This generalizes the result of Gabmiel ~4],

~5];which says that the

locally finite categories correspond to the pseudo-compact rin~s [25].

-

2 3 3

-

indecomposahle injectives). As we have remarked befome in w 3, example of a locally noetherian category Corollary 5 that is not locally finite.

ModIQ~)= C

TCI~ ~ I ~

is an

satisfying the conditions of

If however

C

is also a module

i

category, then

C

is locally finite [21].

Finally, let us observe that if ~ injective of ~, R = Homc(I , I), 9C

is locally noetherian,

I

a big

then the anti-equivalence

~--> HOmc(C , I ) ~ T C ( R ) m

tmansfor~ns injeetive objects into pmojective objects in

TC(R).

However,

these latem objects are not necessamily pmojective and not even flat in Mod(R)

in general.

The situation can he completely clamified if we use

the theory of Chase ~3]: THEOREM 7.(18)

With the notations and h2]potheses of Theorem 6, the functom ~ ~C

~

> Homc(C , I) 6 Mod(R)

transforms inject ive objects into pmo~ective (resp. flat) ones if and only if

R

is might cohement and left perfect (resp.

PROOF:

R

is might coherent).

Suppose first that (18) transforms injectives into projective

(mesp. flat) objects.

Then since

R = Homc(I, I),

we have that

~ - ~ R = H o m c ( ~ I , I) as a left R-module, and s i n c e ~ I is injective, K -- K K we have by hypothesis t h a t ~ R is projective (rasp. flat) as a left RK module for any K. Thus by Theorem 3.3. ~esp, the left-right symmetric of Theorem 2.1~ of [13] R

is might coherent and left perfect ~esp. might

coherent (cf. also Boumbaki [9], p. 63 Exercise 12)~. Conversely, suppose that be any injective in ~.

Since

R

is as in the last sentence, and let I

is a big injective,

J

J

is a direct

factor of a suitable s u m ~ I , thus Home(J, I) is as a left R-module a L direct factor o f ~ R , thus projective (mesp. flat) by the theorems of L Chase (-Boumbaki) just cited, and so the Theorem 7 is completely proved.

-

PROBLEM:

234

-

Is a left linearly topologlzed, left topologically coherent,

left topologically coperfect, complete, left perfect, rlght coherent ring R

discrete?

(If we omit the condition left perfect, then

necessarily discrete as is easily seen.)

is no!

If so, then the rings

Theorem 7 would be discrete in the projective case. is not necessarily amtinian on any side:

R

(~)

R

of

Such a discrete

R

is coherent and perfect

on both sides, but not aPtinian on any side. We now pass to a second application of Theorem 5 above. linearly topologized, and of

Mod(A)

T = Art(Dis(A))~Mod(A)

formed by the discrete artinian modules.

Let

A

be

the full subcategory Then

T

is evidently

an amtinian abelian categor~ that is closed under formation of kemnels and cokemnels (it is even closed undem the formation of subobjects and quotient objects in

Mod(A)), and by Theorem 5

MOdA~t(Dis(A))(A)

is the dual of

a locally noetherian categor,] in which dual the kernels, cokemnels, etc. ape the algebraic ones. 1)if

A

with its linear topology, itself belongs

to

A

is exactly what is called a stmictly linearly

MOdAPt(Dis(A))(A ) then

compact rin~ in 503, p. lll-ll2 [i.e.S.l.k. Ris E in [38], [39] and a Leptin ruing in ~4]Jand_ then the category

Lep(A)

MOdAmt(Dis(A))(A)

of Leptin modules over

A

can be identified with (el. Gabmiel [24], [25])

which is thus the dual of a locally noetherian category ~4],

[25~.

However in this case, it is not true in general that ever-] projective indecomposable object in

Lap(A)

is of the form

Ae,

wheme

e

is a

pmimitive idempotent (cf. examples in Gabriel [24J). COROLLARY 6 (of theomem 6).

If

A

iS a Le~tin rin~ (a strictly linearly

compact ming) then theme exists a lineamly topqlogized, topolo~ically cohement, topologically copemfect and complete ring potent

e

pamticulam

i_~n B, A

such that

A = abe,

B,

and an idem-

with its natural topology.

I__nn

is the endomorphism Pine (with its natural topology) of an

in~ective (not necessarily big) of a suitable locally noetherlan category. l)

The topology is the induced ... one.

-

Remark.-

235

-

It is probably not true that ever,] Leptin min E is topologically

coherent (if this were true, then we could of coumse choose e = 1

B = A,

in Comollary 6).

PROBLEM:

If the min E

fop a suitable

R

has a linear topology, so that

T CDis(R)

coperfeet and complete.

(T

as in Theorem 5),

Is the converse true?

then

R E Mod4R) R

is topologically

If not, characterize those

Pings that ame obtainable in this manner.

w 6.

TopoloKically coherently completed tensor products. This section will he used in w 7, notably fop an explicit description

of the Gabriel filtration of

TC(A) ~

when

A

is a linearly topologized,

topologically coherent, topologically copemfect and complete ring.

We

will introduce and study a notion of topologically coherently completed tensor product over

A,

~c ~A

denoted by

"

compact [this is for instance the case if

In the case when A

A

is pseudo-

is commutative - this follows

from the theorem 8.4 of Chase [1S]] then this tensor product coincides with the one introduced by Gabriel in ~ 4 3 i) and denoted by

~A

there,

and it is well-known that this last tensor product generalizes the usual completed tensor product for ~ d u l e s over noetherian local rings, used for instance in ~7] and in ~0].

As a ~tivating (and - as we will see below

in Theomem 8 and Remark 1 - an exhaustive) exa~le fop the introduction of B

~A'

let

~

and

D

be two locally noetherian categories,

the associated topological mines, and

covariant

left exact funetors from

C

Lex/ ~ -.. - (C,D) --

to

D

A

and

the category of

that commute with directed -.%

lim

.. ( , )

is defined in an analogous way I.

L e ~ lim(~,~) = Lex(N(~),~) >

w

~(C) = the categor~ of noetherian t_ objects of ~'I

and that

i)

We have that (el.

Cf. also ~6] for the commutative case.

2)

236

-

-

LeX/v lim(~,D) ~ ---> Rex/qlim(TC(A) , TC(B)). If

T

T(A)

is a functor belonging to this last category, then not only does belong to

TC(B),

but this

T(A)

also has a natural right A-module

structure, compatible with the left B-module structure: plication by by

T

k ~ A

into a map

T(A)

operation

T(.k)

of

M = T(A)

A

on

B-A-bimodule).

is a continuous map

as

A

T(.k) > T(A) in

.k,

'~ > A

TC(B).

right multi-

that is transformed

We will write this

and we will now prove that this right operation

is continuous (so that

M

is in fact a topological

Using the formula

m.~ - mo'~o : (m - mo)'(~ - ~o ) + mo'(~ - ~o ) + (m - mo )'~o we see that it is sufficient to prove the following three results: I)

The natural map

2)

For each

3)

H x A_._> M

mo~

H,

continuous at

0.

For each

~

O

~ A,

continuous at

is continuous at

the map

A-->

M

defined by

~-->

the map

M ---> H

defined by

m ~ --> m'~

To prove I) and 2), let

directed decreasing set of open left ideals of

Then

M/U

U

be an open B-submodule of is an artinian object of

creasing family of subobjects of

has a minimum element TC(B)

and

M/U

is

is

O

{C~L}

be the

that belong to

TC(A),

such that

H/U

is in

T.

and thus the directed deTC(B):

T(A)/U) M

Im(T(C~H)--> which implies that

in

> a . TaN = T e , e --

lim T(Ot e) = T(lim

A,

M = T(A) TC(B),

T a = Im(T(Ote) --> T(A) ~ >

in

mo "~

0.

We have just seen that 3) is true.

and let

(0,0).

But the directed T(A)/U) = 0

H. Ot

C U,

and this proves both I) and 2) and even

eH

the (apparently i)) stronger result that the map equicontinuous

(cf. w 8).

M x A-->

H

is left

More precisely:

i) That the equicontinuity property is only apparently s t r o n g e r e ~ c a n proved in much the same way as the lemma 0.3.1 on p. 71 of [26] is proved (it is essential that

H g TC(B)

as a left B-module, cf. w 8).

be

237 -

THEOREM 8. -

Let

A

and

B

be as above.

Then the functor

RexN Iim(TC(A), TC(B)) ~ T ~--> T(A) 6 BTCT A < ,

~here

BTCT A

bimodules

is the category whose objects are those topological B-A-

M,

that belong to

TC(B)

are the continuous bimodule map~ M ~ BTCTA, M~AC

--,

as left B-modules, and whose morphisms

is an equivalence of categories.

the corresponding functor

TC(A)--> TC(B)

for the fo!lowin E reason:

For

N ~ TC(A),

Given

will be denoted by we have a natural

equivalence of functors: HOmTC(B)(M~CA N , V) e~> BiltOPB(M, N; V), where

BiltoPB(M , N| V)

define B-linear maps

is the set of continuous maps

M ~AN-->

I

~C

Remark i.-

We have seen in w 5 that

Now let

BiltoPB(M , N~ - ) 6 Lex(Coh(Dis(B)), Ab)

M ~ A N,

for a unique

M

be in

L [ - 6 Coh(Dis(B~.

This

L

defines

and one verifies that the Theorem 8 is true. Of course

BiltoPB(M , N| -) s Lex(Coh(Dis(B)), Ab) A

c

M ~A N

The Theorem 8 implies that There exists a "completion" AC

~C

M ~ A N ~-~> M ~ A N ~C

One verifies that Remark 2.-

BTCTA . Then

and so it is of the form

satisfies weaker conditions, and so we have a

such that

that

9 HOmTC(B)(L , -) s Lez(Coh(Dis(B)), Ab)

is anti-equivalence of categories.

HOmTc(B)(L, -)

M x N § V

V.

INDICATIONS OF THE PR00F OF THEOREM 8:

TC(B) 9 L

V ~ Coh(Dis(B)),

If

e A

(N

= eA

even if

M

in this case too. A

M 6 BTCTA

of

M

can also satisfy weaker conditions ...),

in the pseudo-compact case ~ ,

D = Mod(Z), B = End Z (Q/IQ/Z~), ~ ~

~.

then

ReXNlim(TC(A) , TC(B)) = Dis(A) ~
TC(AN) [~'> PC(AK~,

and if ~

BTCTA--~> TC(AX),

and

and this ~ives

should be obtainable as a suitable topological A~

We have not tried to pursue this

further, nor have we tried to see whether there is a suitable lineamly

-

topologized Pin E

AM

238

-

in the general locally noetherian case (then

AM

is of coRPse not necessarily topologically coperfect and topologically coherent).

w 7.-

Expllcitstudy of the dual and the conjugate of the Gabriel

filtration of a locally .noetherian category. Let

~

be a locally noetherian category.

representation

C~>

w

TC(A) ~

Gabriel filtration of

~,

the case when

C

of Theorem 6 to describe explicitly the

and how

locally finite categories.

We will now use the

~

is built up by "extensions" of

We will have the most complete results in

is stable,

i.e. when the Gabriel filtPatlon is stable

under injective envelopes (cf. w 8 och w 9), but here we start first with some mesults in the genemal case. If ~

is an AB 5M-categomy with a family of cogenemators ~0]

(i.e. the dual of a ~othendieck categoz~/), then a subeategoPy

_F of

E

will be called a coclosed category if it is the dual of a closed subcategory [25] of category of

~

E_. ~

It is equivalent to say that

F

is a full sub-

that is stable undem the formation of subobjects, quotient

objects and

llm. In the same way we introduce the notion of a co< localizing subcategomy, the notion of pmoduct of two coclosed subcategomies

of

E

(cf. [2q, p. 395) etc.

If

[

is a coclosed subcategory of

ix

then the inclusion functor

~

E,

.N

, > E

has a left adjoint

is a colocalizing subcategory, then the quotient category

I EJ[

and if can be

.M

fommed and the natural functor

E--~EJ[

has a left adjoint

J,

that

is a full embedding, and so we have an "exact sequence" .x

o

>s

_z< ix

j, ,"

r.js -----~ o

j~

Now we will see that in case

= TC(A)

~.e.

E

is the dual of a

locally noethemlan categor~, then all these categories and functors can

-

2 3 9

be described in a very explicit manner.

-

Our results generalize and complete

those of Gabriel [25], p. ~00. THEOREM 9. -

Let

ideal0[

A

of

TC(A)

be as before.

that belongs to

The map that to each bilateral as a left A-module, associates

TC(A) i

the full subcate~omy

_F = TC(A/~)

M > TC(A)

(ix

is the natural em-

bedding), defines a one-one correspondence between these ideals ~ L the coclosed subcategories of w 6 we have

F

of

i.4i"(M) = A ~ A C

TC(A). M.

subcate~ories cortes p~ond~ng to 0~ 1 _~

If

and

With the notations above and F1

and

F2

a_nd ~ 2 ' and if

are two coclosed FI'_F2

is the product

subcategor~ [25J, then this coclosed subcategomy, cor~esponds to the bi_~lateral ideal TC(A)

C

~i

"O~t 2 '

and that contains ~ l

i.e. the smallest left ideal of "(~2

F

that is in

(this smallest ideal is bilateral and

it is also the image of the nultiplication map ~ l ~ C O ~ 2 ~artieular

A

-- 9 A).

In

is a eolocalizing subeate~ory if and only if ~ 2c

and in this case the in~redients of the exact sequence

0

9 TC(A/~)
"

.)i

)'.

]

can be made explicit as follows: 9

.M

~

9

2) TC(A),

A

TC(A)/TC(A/oL)

can be fdentified with the full subeategomy of

formed by those

/% c~ 0%% ~ --> M 3)

A

M~TC(A)

is an isomorphism

such that the multiplication map (j,

is then the inclusion functor}~

The natural exact sequence

j,jX(M)-->

M - - > iMiM(M)--> 0

can he identified with the natural tensor product exact sequence

-

~

M ---> O ~ 9 M A

Finally,, if

A/OL

then

A = A/Of !IAe,

then

eAe

240

-

is onto). is the pro~ective envelope of

where

e

is an idem~otent of

A/Or A,

in

TC(A), -r %- c and if O ~ =

is with its natural topology a ~opologically cohement

(topologically coperfect is clear) complete ~ing, and (using 2)) TC(A)/TC(A/oI) functor

is naturally equivalent to

M~--> HomTC(A)(Ae, M).

Also

projective envelopes if and only if (gen. case). and then

TC(eAe),

by means of the

TC(A/oI)C_~TC(A) A/Oh

is stable under

is projective in

~ n this last case, we can choose

e

TC(A)

such that (~i = A(l-e),

(l-e)Ae = 0.]

The proof of this theorem is entirely based on the theory of w 6, .M

and the general functorial properties of

j,, ] .

(Compare also [48] .)

Now that we have Theorem 9 and w 5 - w 6, we can easily tmanslate the definitions and results of [2~, p. 382 ... into oum dual language: Let

COROLLARY I.-

O 1 of

A

A, O ~ T C ( A )

be as before. such that

there is a smallest one ~ o ' =~o'

so that

A/~o

TC(A/(~o)C__~TC(A)

envelope of and

--Fc ~i

is a left pseudo-compact ring~ and

A,~71~

in and

TC(A) ~iC~o

a dec~gasin~ filtration of A

such that

TC(A)

w

There is also a

that the image of

A/~I

i__nn

is pseudo-compact (this can be expressed with the projective

A/(> ~ =~i'

and this

is a pseudo-compact left A-module, -U--~c dh ~ is bilateral and G ~ =

is a qolocalizin~ subcate~ory.

smallest left ideal of TC(A)/TC(A/01 o)

A/Or

Then among the left ideals

A

too.)

This ideal O~ 1

. Contlnuin~ in this manner, we obtain by bilateral, left

=

AO6~oDOLID

is also bilateral

... O 0 % a D

...

TC

ideals

{~}

of

241

-

~f

-

~ has no predecessor, i.e. is a limit ordinal, then we put ~ a =

the biggest left ideal in

TC(A)

such t h a t ~ u C ~ 8

there is a unique smallest ordinal 0~

~ O, ~ < a".

TC(A) ~

This

H

~

such that

8 < u 1).~ Then

~H

= 0

and

coincides with the Kmull dimension of

and TC(A/OLo)~176

...~TC(A/oIs)~

coincides with the Gabriel filtration of

TC(A) ~

COROLLARY 2.-

TC(A) ~

The Gabriel filtration of

envelopes if and only if each O[~ = Ae~, A

,

where

...~TC(A) ~

is stable under in~ective em

is an idempotent of

[then eeA(l-e ~) = ~ .

Before we go over to a more detailed study of the stable case in the next two sections, we will first see how the conjugate categories fit into our picture. 'rHEOI~M 10.~

Let

~

be a l o c a l l y

topological ring and ~ =

Dis(A)

noetherian

category,

its conjugate.

A its

Then the map

associated

F~--> F

defines a one-one correspondence between the closed subcategories of and closed subcategories of ~_ ~nd O~b-@ Dis(A/o7 ~.

_FI ~ ~_ can be identified with

The localizing subcate~0ries cozTesp0nd and ~/[ =

= ~../.[~. In particula~ the Gabriel filtration J'v

C-oC ~ I C ' " ~ C - - u C

(19)

co_tic

...~C_

...c

gives rise to a filtration of _C = Dis(A):

c c

...c

,

which can be identified with Dis(A/O~o) C D i s ( A / O L I ) C

.. .~ Dis(A/6n )~ . . . ~

Dis(A). N

FumthePmore

C

is the smallest localizing subcategory of

"O

l)

By w 5 we have that 0]~ =

~ B<m

01.B

C

that

242

contains the simple and coherent objects

-

l)j N

~i

is the smallest

19calizlng subcategor7 of ~_ that contains those oh~ects whose images in ~/~_~ 9ye coherent and simple etc. ~i+I/~i

From the results above follows that

is locally finlte, thus of the form

PROBLEM i:

Can an

stable case)? PROBLEM 2:

AM

PC(AI") ~

be built up from these

A. x 1

(cf. w 6,Remark 3).

(at least in the

(Cf. w 6, Remark 3.)

Does every Grothendieck category

D

of Krull dimension zero

admit a filtration similar to (19) above?

w 8.

Generalized triangular ma.Trlx rinks with a linea~ topology, and

classification of stable extensions of locally noetherian categories. Let CJD

C

be a Grothendieck category,

D

a localizing subcategory,

the quotient category and .M 3

ix (2o)

o -->

D_

>c

~ cjs

the sequence of natural functors.

-->

o

Recall that here

ix

and

jx

have

!

right adjolnts iX(C)

by

D.

i"

= 0 < => C = i

and (D).

JM

that are full and faithful,

and

We will say that (20) is an extension of

_C/D _

Recall that Gabriel has proved that any locally noetherian category

can be obtained in a canonical way by means of successive extensions of locally finite ones [for more details see [25], Chap. IV (cf. also w 7)~ and that he has explicit results about the structure of the locally finite ones (some of which we have extended her~ to the locally noetherian case).

However, as far as we know, the problem to classify

the extensions (20) of locally finite categories or more generally of locally noetherian categories, has not been dealt with in the literature, and we will here give the rather complete results that we have obtained

1)

The smallest localizin E subcatego~y of objects of z e r o (w 3 ) .

~,

is of course

~

~

that contains all the simple

itself, since

~

is of KPull dimension

-

in the stable case.

2 4 3

-

[We say that (20) is a stable extension, or that

is a stable subcategoz~ of

~

object in ~,

D~

is still in

if the injectlve envelope in

= Qcoh(X),

of every

The stable case occurs frequently "in

practice" : It is for example well-known [25] that if prescheme,

~

X

is a noetherian

the categor,] of quasi~oherent sheaves over

then every localizing subcategory of

~

X,

is stable.

Thus consider now an "exact sequence" jH

i

(21) (D

0---> D

> C

9 E

a localizin E subcateEory of is locally noetherian (then

big injectlve of

~.

where Since

D

and and

~ E

the quotient categoz~), where are so too),

and let

Suppose that the sequence is stable.

a canonical decomposition of

(22)

~,

.9 0

I

I

be a

Then we have

([25], p. 375 cop. 2-3)

I = i (Zo)J_tj (II) I~ is a big injective in ~ and I1 is a big injeetive in ~. N. j 1M = 0, we get HOmc(i (Io) , j,(I1)) = 0, and so from (22) we

obtain a direct decomposition of abellan ~oups

(23)

A

=

(iH

Homc(I , I) = HOmD(Io,Io) tLHomc(JM(ll),i

and

Jx

(Io)) II HOmE(Ii,I1)

are fully faithful).

However, it is possible to rewrite (23) so that the Pin E structure of becomes apparent.

If we put

A ~ = HOmD(Io,Io) , A 1 = HOmE(If,If)

M = Homc(J,(Ii) , iM(Io)) , then

A~

and

A1

ape Pings and

natural way a left Ao-module and a ri_~Al-mOdule are compatible Now, if

(we will denote this by Ao, A1, AoMAI

M

and is in a

and these structures

M = AoMAI ) .

are arbitrary, we will denote by

(Ao Ao MA)~ the ring whose elements are tho Tri~les (ao,m,a I) 0 Ai q suggestively w~itten as (ao m )l ( a o C Ao, m ~ M~ a l ~ A l ) , 0 aI

Lmore where

A

-

244

addition is defined by componentwise addition, and where multiplication is defined by "matrix multiplication" !

(aO 0

m )(a~ m') = (aoa~ aI 0 a~ 0

aom'+m al) aleI

(This has a sense, since M = AoMAI.)

This kind of generalized Triangular matrix ring was first introduced by S.U. CHASE in ~l~ (cf. also [3~), and it is now easy to see that the assertion (23) can he made more precise by saying that we have a natural ring isomorphism

(24)

where

A = HOmc(I , I) "~> --

I HomD(Io, Io) -0

M

1

HomE(I l, II)

M = Homc(JM(I1) , ix(Io)).

Now in (24), to give the rings

A~ = HOmD(Io, Io)

and

(with their natural topology), is the same as to give

A1 = HomE(I I, II) ~

and E,

and thus

all information about The extension (21) should be contained in the Ao-Albimodule

M

(this module also has a linear topology ...). Thus The problem

arises To characterize those bimodules

M

that arise from stable exten-

sions. This problem will be solved completely in what follows, but first we will have to develop several pmeliminamies (some of them perhaps of independent interest) about generalized triangular matrix rings. Let

A = Ii ~

AoMAI~

be an arbitrary generalized triangular matrlx

AI/ ring.

We wish first To determine all the left ideals of

start with some examples of such ideals.

A~ AoMA11 where A 0 0 is a left ideal of left Ao-module

A,

A.

For that we

It is clear that

operates to the left Thl-ough matrix multiplication and this ideal can, in fact, be described as the

AoJ~M (M = AoMAI ) on which

follows that every left Ao-submodule

A

V~AoLIM

operates through

Ao.

It

defines a left ideal of

245

A,

that we will denote by A (0~

write AI,

(0V0)

M.~ C< 1 A : (0 v

[note that with this notation we can

M A IAM 0) = (0~ 0 ~ . On the other hand, if

then the left ideal of

0 (0

).

A

is a left ideal of is exactly

Finally any sum of these two types of left ideals of

0 0 ) + (0

M.C~ 1 o~ )

submodule of

Aoll M

restriction,

CK1 C A1

PROPOSITION 13.-

is evidently a left ideal of

such that and

of the form GtI

M. O~1 C W.

0

A1

(0 V~I),

where

V A1

A, which we will Ao/~ M

is an A o-

Except for this last

can be completely arbitramy: be an arbitrary genel-alized

The left ideals of

is a left ideal of

operates on

W = V + M. 0%1 C

W C Aoli M

Le.t A =

triangular matrix rin~.

where

(~i

0 0 (0 ~ )

generated by

write in the form l) (0 WO~l) , where

A

-

A

are exactly the subsets of

is an A o-submodule of

AoAL M,

such that

and where

M . O ~ C V,

and A

by left matrix multiplication. 1 Compute directly the left ideal of A, generated by a set of

PROOF:

(0 V )

generators. Remark.-

If Ot is a left ideal of

VO[C~ Ao/l M

and

O ~ C A1

A,

then the corresponding

are uniquely determined by O~.

Furthermore

((Ao//- M)/rot) A/or

can be identified

with i)

\

0

AI/~I /

, where

by left matrix multiplication which is well-deflned since

A

operates

M.O~ 1 C Vot.

This gives rise to an exact sequence of left A-modules

(25) where AI/O~ I.

0 A

> (Ao]-tM)/Vo~

operates through

A~

on

> AI/O~ 1

> 0

(Aolt M)/VoL , and through

A1

on

This sequence, which will be very useful below, does not split in

general, not even in the case l)

> A/C~

Vot = 0,

O~1 = 0 (i.e. O r = 0)~

We use here an extension of the notation introduced above.

- 246

COROLLARY.-

-

Wlth the notations above, Ot C A

%eft ideal, if and only jif O~1

and

is a finitely generated

Vo~/M. %

are finitely Kenerated as

left A l- and left Ao-modules Pespectivel Z. We now turn to the study of linear topologies on generalized triangular m a ~ i x rings 1).

Combining Proposition 3 of w 4 with Proposi-

tion 13, we obtain: PROPOSITION 14.A A = (0~ A1

M AI)

To give a (left) linear topology on the ring

is the same as to give a (left) lineam topology on the ring

and a linear topology on the left Ao-modul___.~e Ao/i M

such that the

maps:

(Aoli M) • (AolL M) ---> AoAJ_M , {(ao,m),(a~,m')}~-->

(i)

(aoa~,ao m')

and, (ii)

(Aoll M) x A 1 |

-> Ao/L M,

{(ao,m), al};

> (0,ma l)

.

are cont~nuoust and such that fumthe~nore the map (ii) is left equicontinuous in the sense that to each open Ao-SUbmodule is an open left ideal topoloKy on

A

O~ 1

of

A1

such that

V

of

Ao33_M

(Aoli M).0~IC V.

is then the product of the topologigs on

A1

there The

and on

AILM. o Those triangular matrix rings that ape of interest for us in connection with locally noetherian categories will have split linear topologies in the sense of the followin E definition. DEFINITION ~.-

We say that a left ideal

ideal if 0~ is of the forth ( left ideals,

U CM

i)

U o Oil) ' where

is a left Ao-submodule and

linear topolo~ on the ring A

O~ of

A

A A : (0~ O%iC A i

M AI)

is a split

(i = 0,1)

M.CEIC U.

ape

A left

is said to be a split linear topology if

has a fundamental system of open split left ideals9 Recall that in this paper, all topologies studied on rings ape supposed to be compatible with the ring structume.

-

Remark i.A A = (0~

247

To say that the topology on the linearly topologized ring

M AI)

is a split topology is the same as to say that the left A O-

linear topology on

Aoll M

such topologies on

A~

(Proposition 14) is the product topology of

and

M.

Thus P~oposition 14 implies that to give

a left linear split topology on the ring linear topologies on the rings left Ao-module

M

that the operation A

-

such that

A~ M

and

A

is the same as to give left

A1

and a linear topology on the

becomes a topological Ao-Al-bimodule such

M x A 1 ---> M

is left equieontinuous.

is then the product of the topologies on

in particular complete if and only if Remark 2.-

The discrete topology on

trivial examples are given by: A M THEOREM ii.- Let A = (0~ A1)

Ao, A 1

Ao, A 1 A

and

and M

The topology on M,

and so

system

{0%}

that every PROOF: that

A

is a split linear topology.

be a left linearly topologized ring that

o f split open left ideal neighboumhoods of A/Or

is coherent in

decomposition

(26) But if

C = TC(A) ~

A = Homc(I , I)

= HOmC(I/N , I), where -A/l(N) is coherent in 0M A(l-e) = (0 A I)

Then the

0

in

A

such

Dis(A). is a locally noetherian category,

corresponds in a natural way to a big injective

the topology on

Less

is a split linear topoloKy, and we even have a fundamental

We know by w 5 that A

is

are so.

is topologically coherent, topologically coperfect and complete. topology on

A

I

in

is defined by the left ideals

C,

that

I(N) =

N C I

is a noetherian subobjeet, and that every A 0 i0 Dis(A). Put e = (00) . Then Ae = (0~ 0 ),

and the decomposition I = I o l l I1

of

I

in

A = AelkA(1-e) C.

cozTesponds to a

Furthermore

Homc(I o, I I) = HOmTC(A)(A(I-e) , Ae). f

is an arbitramy left A-lineam map

f(1-e) = Ae,

thus

A(l-e) --> Ae,

f(l-e) = (l-e)f(l-e) = (1-e)%e = 0,

by direct matrix computation.

Thus

f = 0

since

then (l-e)Ae = 0

and a f o r t i o r i the right member

-

of (26) is O, so that Now let

C

248

-

Homc(Io, II) = O.

be the localizing subcategory of

C,

formed by those

C

--0

such that

HOmc(C , I I) = O.

I claim that _C,

tion of injective envelopes in C .

But if

C ~ C

--0

-oC

and that

is stable under the fommaI~

is a big injective in

were such that its injective envelope (in ~)

I(C)

--0

did not belong to factor

I

# 0

injective of

C,

then

I(C)

would have an indecomposable direct

that o c c u m s a s a direct factor of

_C and

I~ ~ -r ).

c

c

(Io/1 I 1

is a big

But the pullback diagram

" l(C)

3 Ca (

I1

C =~! a

gives mise to an object C @ 0 that belongs to C . We have a monoi a -o C the composition map morphism I a . a > I1 ' and since C ~ 6 -o 1 Ca ( > Ia ( ~ > Ii is zero and this gives a contradiction. Thus we have a stable exact sequence

0

> C_.o

'. '> C_

"I

> ~C/C

> 0

t.

where

I~

Now if

N

is a big injective of

-n)C, I = I o l L I1

is a noetherian s u b o b j e c t o f

I,

and

I1 ~-~> jMjM(II ).

then since both

I~

and

I1

are directed unions of noetherian subobjects, we get that there are noetherian subobjects thus

N o ~ I~

I(N)~I(No2INI).

left ideal of

A,

and

N 1 C I1

such that

N CNo/I

NI,

asd

It is clear that this last ideal is a split open

and that

A/I(NoJI N I)

is coherent in

Dis(A)

and so

the Theorem ii is completely proved, as well as the COROLLARY.-

With the notations and hypotheses of Theorem ii,

A~

and

A1

are topologically coherent, topologically coperfect and complete for their natural linear topologies. Remamk,assure that

We will later determine exactly those conditions on A

is topologically coherent.

M

that

-

249

-

We will say that a linearly topologized module

M

is topologically

artinian (r~sp. topologically noetherian, resp. topologically of finite length, resp. topologically coperfect) if {U}

of submodule neighbourhoods of

0,

M

has a fundamental system

such that every module

M/U

is

artinian (resp. noetherian, resp. of finite length, resp. coperfect) i). A M THEOREM 12.- Let A = (0~ AI) be a (left) line~ly topologized rin~. Then

A

is (left) topologically artlnian (resp. topologically noetherian,

resp. topologically of finite length) if and only if

A1

and

AoAIM

at?

so for their natural (left) linear topologies. PROOF:

Let O~ be a left open ideal of

where

Gt I

and

A.

Vot are open submodules of

Then ~ = A1

and

V (0 ~ ) ( M ' O ~ C Vc~): 1 A o ~ I M respectively.

Then by (25) above we have an exact sequence of A-modules

0 ---> (AoJIM)/Vot

(27)

where

A

operates through

> A/O~

A~

on

> A1/OC 1

(AolLM)/Vot

> 0

and through

A1

on

A1/0( 1 9 Since in an exact sequcnce (in any abelian category) 0 - - > CI---> C2--> C3--> 0 C2

is noetherian (resp. artinian~ resp. of finite length) if and only if

C1

and

A/Of

C3

are noetherlan (mesp .... ),

it follows easily from (27) that

is artinian (resp. noetherian~ resp. of finite length) as a left A-

module if and only if module and

AI/O~ 1

(AolL M)/V~t

is artinian (resp .... ) as a left A o-

is artinian (resp .... ) as a left Al-mOdule ~ and this

proves the Theorem 12. COROLLARY 1.-

A (0~

A split linearly %opologlzed rin~

M A1)

is(left)

topologically artini~an (r~sp. topologically noetherlan, resp. topologically of finite length) ~ if and only if

Ao, A 1

and

M

are so for their natural

1) These conditions are then verified for all open submodules

U.

-

250

-

(left) linear topologies. In the discrete case we obtain: A M COROLLARY 2.- A rings A = (0~ AI) is left artinian (resp. left noetherian) if and only if the rink and

Ao, A 1

are left artinian (resp. left noetherian)

is an artinian (resp. a finitely generated) left Ao-modtLl_____.~eel).

M

COROLLARY 3.-

Consider a stable extension of locally noetherian categories .M

0-->

s

9 c_

9 c_/~

9 0

Suppose that the endomorphism rink of a big injective I~/~

in

C/D

ID

the left

Example 1.-

and

iM(I~)ILj,(Ic_./~) in

Leptin rink if and only if for every noethePian subob~ect ~,

D

is a Leptin rink 2) (i.e. is topologically artinian).

the endomorphism rink of the bi~ in~ective

in

in

HomD(ID, ID)-module

Let

ToPs(Z)

HOmc(N , iM(ID))

N

C

Then is a

o_ff jx(I~/~)

is artinian.

he the category of torsion abelian groups, and

consider the exact sequence 0 Here

> Tors(~ZZ)

Tors(ZJ

~/Z and

~

[where P

and

> Mod(Z)

> 0

Mod(QQ) are locally finite, and have big injectives

respectively.

The endomorphism rings of these ape 7--[ Z ps p ~ P is the set of prime numbers, and ~p Z is the ring of p-adic

integers~ and Q~v respectively. rings.

> Mod(Q)~

Both these rings are of course Leptin

But the endomomphism Ping of Q IIQ/Z

in

Mod(Z)

ring, for ~Z is a noetherian subobject of ~Q' and

l)

is no__~taLeptin

Homz (ZZ, ~I~) = QIZ

is

According to Hopkins [16], a left artinian ring is left noetherian. If we combine this with Corollary i, we get that if artinian rings and module, then

2)

'

M

M

Ao, A1

are left

an Ao-Al-bimodule that is a left artinian Ao-

is finitely generated as a left Ao-mOdule!

If the endomorphism ring of one big injective in a locally noetherian category

[

is a Leptin ping, then the endomorphism ring of any

injective (big or not) in [

is a Leptin ring (cf. ~8~,[3~ ).

251

-

no_~tan artinian

Endz(Q/Z)-module , for it has the following strictly

decreasing chain of

Q/z= I I

-

--

p e

z

P "P|

Endz(Q/Z_)-submodules:

~,[I

z

p>__3--P|

~ 113 L~ I I z

p>_5 P ~ p>_? P|

p-C P Here

Z

-

pE

P

D II

z

D...

p>ll --P~

p~ P

p~

is the indecomposable injective of type

p

P in

Mod(Z),

cf.

[237. Example 2.p.

Let

~(p)

he the localized ring of

Z

at the prime number

This ring is noetherlan and of K~ull dimension i, and one vemifies

using Corollary 3, that here the endomorphism ring of a big injective i_~sa Leptin ring

A.

This ring

A

I

can not he topologically noetherian

too, for then it would be pseudo-compact, which is impossible, since the Krull dimension of

Mod(~(p))

is i.

Thus

Dis(A)

is a locally amtinian

(and also locally coherent) category that is not lopallygoetheria q.

The

theorem of Hopkins

[16_] cited above can be formulated as saying that if

R

Mod(R)

is a ring, then

locally noetherian.

locally artinian implies that

The example

Dis(A)

Mod(R)

is

shows that this theorem can not

be extended to Grothendieck categories, at least not in this formulation. A M THEOREM 13.- The (left)linearly topologized ring A = (0~ A1) i_~s topologically copemfect, if and only if Aoll M and Vo1 START OF THE PROOF: If 0~ = (0 01I) (M. O~1 C Vot) of

A,

A1

is an open left ideal

then the exact sequence (27) above shows that if

cally coperfect, then

AolIM

and

A1

are so too.

are so.

A

is topologi-

To prove the converse

we need the following generalization of the Corollar~ of Proposition 13: LEMMA l.-

With the notations above, the finitely generated left A-sub-

modules of

A ,u mo u o

0 (T0 + ~ ) '

where

A1/~l / T C (Aoli M)/V~t

A -submodule, and where O

~C

AI/~ 1

is an arbitrary, fiqitely Kenerated .is an arbitrary finitely generated

252

-

Al-SUbmodule

(M.~

M • AI/O~ 1

> (Ao/L M)/VG~ , which is well-deflned, since

PROOF:

is defined by the natural product mapping M.~lC

VO~).

Direct computation. By Lemma i, a decreasing sequence of

END OF THE PROOF OF THEOREM 13: finitely generated submodules of

A/O~

is necessarily of the fern:

"2

D

...

D

D

...

where (29)

Al/Ot I D ~ i ~ %

~ "'" ~ ~ n ~ ''' ~

is a decreasing sequence of finitely generated Al-SUbmodules and where the

Ti

are finitely generated Ao-sUbmodules of

(Aol/ M)/Vot

evem do not neoessamily form a decreasing sequence.

Since

A1

that howis supposed

to be topologically coperfect, the sequence (29)must become stationary from a certain index

n M on, and so in partlculam we get from (28) a

decreasing sequence

(30)

T M .M.)S N DT +~.,,'e N mT +M.'6 M _9... n n nM+l n nM+2 n

Now let

~f i ~ t [ n"+l ) i=l

f i = ti + nM+l nR ~i'

be an A -basis for o where 9~i

C M . ~ x, n

T

C (AolL M)VoL.

T'

and clearly

nN+l

T'

+ M.~,

nM+l

Then evez 7

nX+Z and

t i 6 T M" nM n

generate a (finitely generated!) submodule of by

D

= T

n

Now these

T M, which we will denote n + M.~,. Continuing in

nM+l

n

this manner, we see that the decreasing sequence (30) can be wmitten

(3l)

T n

.M.~

roT, n

nX+l

*M.Z.DT' n

.,.-4" nX+2

where

(32)

t iM n

(Ao/.[. M)/V0[ ~ Tnx 39 TnX+l/D T'n~+2~ ... m

~... n

-

253

-

is a decreasing sequence of finitely generated submodules of But

(Ao/LM)/Vo~

is coperfect

[AolL M

(AolIM)Vot-

is topologically coperfect3 .

It

follows that (32) must become stationary and thus also (31) is stationamy. Since (31) is just another way of writing (80), we finally see that (28) is stationary and so the Theorem 18 is proved. A M COROLLARY i.- If A = (0~ A1) is linearly topolo~ized with a split topology, then

A

is topologically copemfect if and only if

A~

and

A1

are so for their natural linear topologies. We have in fact that if object of

are

is topologically coperfect, then every

o

Dis(A o)

COROLLARY 2.A1

A

is coperfect. A M The ring (0~ A1)

is right perfect if and only if

A~

and

so.

PROBLEM:

Is it true that the topology on a linearly topologized, A M topologically coperfect, complete ring (0~ AI) is a split topology? We will not pass to the topologically coherent case, which by far is the most complicated one.

For our purposes (cf. Theorem ll) it is

sufficient to determine the necessary and sufficient conditions for A M A = (0~ AI) to be topologically coherent for a linear topology such that A

has a fundamental system

O,

such that every

A/O~

{Ol}

of split open ideal neighbourhoods of

is coherent in

Dis(A).

We will say that

A

is ss-topologically coherent if this is true [we do not know if this is the same as saying that the linear topology on

A

is split and that

A

topologically coherent (this could be called s-topological coherence~. In order to express our results in a convenient form we will first need some generalities about linearly topologized (hi-)modules and their tensor products. PROPOSITION 15.bimodule. that

M

Le_~t A ~

Suppose that

and A~

and

A1

be rin~s and let M

M

he an Ao-A I-

are left linearl 7 topolo~ized and

is a topological Ao-Al-bimodule when

A1

is ~iven the discrete

is

254

topology i). {fi};

Le__~ F

-

be a finitely ~enerated left

be a fixed finite set of ~eneratcrs for

submodules of

M $~A 1 F

A 1 -module~ and let F.

Consider the

A~ -

of the form t

f

f

t = Im (h_[u 1

V = VUI' "''' ~

where

U

a left

is an open A~

Ao-sUbmodule of

-linear topoloF~ on

of the finite set of ~enerators a topolosical left

A~

(F" of finite type),

( ) ~)Alfi --~i ~

t --->I_~M 1 M.

1 F,

M~A

{fi}t1 of

These submodules

:

V

that is independent s F.

Furthe,Pmome

-module and for ever,/ A 1 I~A?

M ~SAIF)

M ~AIF" ~

-linear ~ p M~AIF

define th9 choice ....

M ~ A 1 F is F" ~

F

is a continuous

A -lineammap. o PROOF: every

Let

{gi} s he another set of generators for F. I claim that 1 v~l' "''' gs contains a submodule of the form V Ufl' "''' ft

We have a factorization t z().f

t (33)

1

x

Al

> F

>

0

l s

1

( )'~i s

Al

0

1 Here

#

must be given by riEht multiplication with a matrix

elements in

I) This means that

>

M

~ ai~ij i=l

M

A1

j =i

is a left topological A-module, and that for any

the right multiplication map

in particular if for which

cf

AI:

{at} 1 '

a I E A1

(@lj)

M

9a 1

> M

is continuous [this is true

has some linear topology (not necessarily discrete)

is a topological right Al-mOdule3 .

255

-

By hypothesis there exists an open ~.~ijCU.

-

Ao--submodule

of

U

M

such that

Now if we combine this with (33) we set fl' "''' ft V~ ~_

El' "''' gs VU .

This shows that the linear topoloi~y on choice of the basis of

F.

M ~AIF

is independent of the

The other results of the Proposition

15

are

proved in the same way. DEFINITION 5.

-Let

finite type.

C

be a Grothendieck category that is locally of

We say that

p. 52) if fop every, map is so too.

If

A~

C"

C ~ f

A O -module! then

coherent if

M

O,

> C,

M

M

C"

is of finite type, K e r f M

a left linearly

is said to be topolozicall 7 psgudo-

has a fundamental system such that every

We say that

where

is a left lineamly topolo~ized rinF,

t0polo~ized

hoods of

is ~seudo-coherent (el. Bourbaki [9],

M/U

U of

A ~ -submodule nei~hboum-

is pseudorcohp.rent

[in

DiS(Ao) ].

i_~sstron~ly topolo~ical&y pseudo-cohgren ~ if in addition

there is an open submodule

W

the indus

M/W

topolo~ (then

of

M

that is topoloKically coherent fo~

is automatically pseudo-coherent, and it

follows that theme exists a fundamental system of such W:s). We can now formulate the main mesult of this section: THEOREM i~.

)

-The followin~ conditions of the left split linearly tops A~

~-Ing A = [ 0 \

M A1

l)

A

2)

(i) The rinks

ame equivalent:

is (left) ss-topolo~icall 7 coherent. A~

and

A1

are topolo~ically coheren T for rheim natural

left linear topolo~ies~ (ii)

For ever~ coherent

CE Dis(A1),

M~AIC

is Strongly topolo~ically

pseudo-cohere~ fop its natural topglop~3 , and fom every C"

@> C

i_~n Coh(DiS(Al)) ,

(I~

~ )-I(v) A1

monomorphigm

is tpDologically coherent

-

(induced topology) if

V CM

~AIC

256

-

is an open submodule that i9

topolo~isally cohe.r~nt (it even suffices to require this for one such V). Remamk i. -In the course of the proof of Theorem 14 we will see that (ii) can be replaced by the (apparently) weaker condition: (ii)" There is a fundamental system of

0

i_n.n AI,

a)

AI/OL 1

{OfI } of open left ideal neishboumh99d9

such that:

is coherent in

DiS(Al) ;

b) M ~ A 1 AI/G~ 1 is strongly topolgKically pseudo-coherent;

c)

FoP all finitely ~enerated

Al/Otl

FI, Fl~----> Al/Otl, M|

to~ olo~ically~p seudo- cohe rent, and

1 _ F1

~olo~icall~ of is topo

(IdM~AIi)'I(v)

finlte %3~e_for at least one open submodule

i~s

V C M~AIAI/Ot I

that is

topologically coherent. Remark 2. -In the discrete case Theorem i~ has a nice formulation in terms of Tot: COROLLARY 1. -The followin~ conditions on th9 arbitramy~enePalized triangular matrix rln~

A =

(:o

A1

~

l)

A

2)

(i) A 0

are equivalent:

is left coherent. and

A 1 am9 left coherent min~s;

(il) Fop every coherent pseudo-coherent for

C ~Mod(Al) , th9 lef-t Ao-module

i = 0

and co_herent for

A1 TotI (M, ~)

is

i > i.

Here we also have an (apparently) weaker formulation of (ii): (ii)~or. 1 is left a left

For all finitely ~ene.rated left ideals

A ~ -pseudo-coherent, and A 0 -module.

A1 Tot I (M, Al/~t I)

Ol1

of

A1, M ~AIAI/Otl

is of finite t~pe as

-

Remark S.

257

-

-There should also exist a formulation of the part (ii) of

condition 2) of Theorem 14 in terms of linearly topologized Tot:s, a formulation that reduces to (ii) in Corollary 1 in the discrete case. We hope

to retur~ to this later.

Remark ~.

-If

A1

is discrete, then Theorem 14 is simpler, and if

A1

is even a skew-field then we have: COROLLARY 2. -The followln~ conditions on the left split llnea~ly topologized ring i)

A

2)

Ao

A =

(o) 0

A1

(AI = skew-fleld) are equivalent.:

is left ss-topologically coherent. is left topologically coherent, and

as a left topological PROOF OF THEOREM 14:

A ~ -modul_.~e. Suppose that

A =

M

is 9tronKly pseudo-coherent

(:o,)

has a split linear

A1

~

topology, let (7[ =

o 011

be an open split left ideal of A

(MOtlC U)

and let us try to analyse the condition that

A/(A

is coherent in

Dis(A).

We have that

CAoo M~ 1

(34)

AI/OI 1

0

where

A

operates to the left by matrix multiplication

is well-defined since

M 0 1 1 C U~.

direct sum decomposition in

(ss)

where

A/01

A

(36)

A/0L

Dis(A):

A~

on the riEht factor of (35).

coherent in

" > M/U

It is clear that (34) gives rise to a

= Ao/Oto II ( 0

operates through

[M_ x AI/O~ 1

on

AI/OII M/U 1

Ao/Olo

and by left matrix multiplication

Thus:

Dis(A)

l

AolOl o 0 MIU

is coherent in

Dis(A).

8) (0 AIlO~1"

is coherent in

Dis(A).

a)

-

It is clear that Gt=

(M Ok

0 (t

~)

1~

258 -

is equivalent to:

of

U)

A

For each open split left ideal

and for each left

A-module map

#

~i

arbitrary finite) :

t

Ker ~

t IA I ~

is of finite type.

MIU

But to give

1

r

~--> A~176

is the same as to give elements

9 t

{aoI}

of

Ao/~o

such that

~o,a'io : 0(i : i, ..., t)

in

Ao/~o:

The

1 , a-i

multiplication map

Ao

,,

,o,>

passes then to the quotient

Aol~ o

-i ca o

and

#

is then given by:

t < Ao/01c 1 [ i

AolO%o

>

ao/Ot o

M/5

t vi.-i

I

o

. ~

i=l

AI/C~ z

0

ao a o E

Ao/Oi o

aI =i

and so t z ( ).~i

(37)

1

Ker r --

, Ao/ o) I_I

Ao/

1

0

t

%

I_! AzI~ I 1 which thus has to be of finite type. Now let us turn to the condition is equivalent to: left

8)

of (36)~

For each open split left ideal ~ I

A-module map (t

arbitrary finite):

It is clear that of

A

8)

and for each

-

_I

00

1

.}. o.

is of finite type.

Kem u

(:,"

-

t):

O 1 o ,~i

AIIOI 1 u is the same as to give elements such that

0

These

= 0 (in

-

--'--'>

But to give

i=l

(i = i, .9

(38)

AIIO~ i/

259

A1101 I )

t

relations

M/U),

~U.a~ "

( 1 C:;I Ou~

U

o

=0

0~].

can be wmitten:

= 0

(in

M/U), o~i I .-i aI

= 0 (in

AZ/mZ)

(i = i, ..., 1:) the middle 9 is defined by restrictlon of the map

M x All01 1 § M/U]

and

(38) can be expPesse~by sayln E that the natumal multiplication maps -i ^.. "m ~ M/U Ao/O~ o

-i "M A~

(39)

-" MIU

~:> M/u

M

-i -aI 9 S/u

M/u

define quotient maps

-1

-i

A1/~A 1

9a 1

> AI/~ 1

J

(i = 1, ..., t) and

u is then given by

[we use the notations in the formulas to the

right of (39)] :

vl -i.h t fA

lj~0

o

l&

o

r~l~ Al/~l

~

a

o

0

m V 1.

al

o 1

~ >

--i

0

tZ vi al.a-i1 1

and so

+ m .al)1

260

-

-

t

(40) Ker ~' --

9

t

~1 Ker(~ tj AI/OI

0

~ i

.-i

r ( ) a1 i=l > AI/O~I )

]

which thus has to be of finite type. In order to analyse (37) and (q0), we now need the following Eeneral lemma, which Eeneralizes Lemma 1 above.

LEMMA 2. -Let A =

I Ao 0

M ) A1 he an arbitraz.I Keneralized trlan~ulammatrix

V

ring and

an arbitrar?/, left ideal o f

(MOIIC V)

,%,

t_he, finitely ~enerated

A-submodules of ~

(Acll M)/V

)

A.

The__~n

are exactly_ t h o

A -submodules of the for~:

ro * Im(M |

9 >

/_L(Ao~M)/v 1

e~

0

F1

t

llCAo,, 1

where

F1 C

t I IAI/~ 1 1

defined by PROOF:

is an arbltrary

Ac-submodule of finite type, and

is an arbitrarT. Al-.Submodule of finite t~l]~e [here

t t ~ ~ M x ~ [ A1/ol I _-t_"> ,I I (AomM)/V ( M . @ I C 1 1

9

is

~7 V)J,

Direct computation.

Now the Lemma 2 implles that to say that type is the same as to say that

Ker %

of (37) is of finite

-

261

-

t

z ( ).~i

t

Z~r([

1

o

I Aol~ o i

is of finite type as a left Th.us. ~)

[cf.(36)]

9 Aolmo)

Ao-module.

is equivalent tO:

The same Lemma 2 implies that

KeP u

Ao/Ot o

is coherent, in

DiS(Ao).

of (40) is of finite type if and

only if:

t A)

t F 1 = KeP(J_IAII~I 1

-i

Z ( ).a 1 1 , > AIIO~ I)

iS of finite type as a left

Al-mOdule.

+( )'a~

(AoI(~o.LL MI~)

KeP

B)

L1

MlVl is of finite

%

Im(M ~A}I

" 9 ~IIM/~)

type as a left Al-mOdule. The universal validity of A) is clearly equivalent to: cohe~nt in

A1/Oh

i__~

DiS(Al) , but the analysis of B) is much mope difficult:

t F1 = Im(j_iAl/O~l 1

t -i Z( ).a 1 1 ~ AIICtl).

Then the exact sequence t

o

9

9 IIAI/~ 1 ~ q

9

1 gives rise to an exact sequence t (41)

M

1

1

9 M|

A

Consider the commutative exact diagram the diagram) :

> o

1%

(Ro. R 1

and

f

are defined by

Put

262

-

0

-

0

(o,~f)(~1)

f > -I-L%t 8~o 1

T

(42)

E

0

>

t

R

; o

0

Heme

p

,~

1

I ,~j~~

,>

~ ,

1

T'

al/~ l

is defined by mes%Tiction of the map

defined by the onto map

M/U

1

., M|

RI

t

t

,,~,I.~%/.~0

T

>

T

M x AllOt I

,> s % 1 q

M x AIIOI1

" > MI~(M~I~)

" > M/U, ~

' and

~I

is

is the

restriction of ~ . If we compare (41) and (42) we see that the quotient of B) is the same as particulam that

Ro/(0,~I)(RI) . The analysis of A

ss-topologically coherent => A ~

for its natumal topology.

a)

above shows in

topologically coherent

Thus since we are intemested in provlng the

equivalence of i) and 2) in Theorem 14 we can and will suppose fmom now on that A~

is %opologlcally coherent for its natural topology, and it is then

pemmisslble %0 suppose that

Ao/61.o is cohement in

DiS(Ao).

Under this

t

hypothesis I I,Ao/ Oto ~ l

is coherent in

DIs(Ao)

of finite %~zpe as an Ao-mOdule if and only if ape of finite type.

and so Im f and

R l(00~ I)(~) o

is

Kem fief, (42)]

But it follows fmom (42) that t

(43)

Im f = Ker

t Ao/~o I/~ll

1 ~ Z( ).~z

M/U t

Im and that t t Z( ) . ~ Kem(//M/~l~ 1 Ker f = P(Rl)

M/U)

M/~

_ 1

~ MIU

=

263

=

Thus using the topology of Proposition 15 it follows from (43) that the univePsal validity of: (45)

f

finitely genePated is equivalent to:

FoP all finitely generated submodules

A1/~ A of

Im

M@A

1

FI~AI/~I,

the quotient m

by the natural (open) image of

F1

U

(u ~ u

~l)

is

To analyse (44), let us consider the commutative

pseudo-cohement.

exact

diagTam

t 0

=~

> R1

(46)

1

t

9 Ker(llMIU 1

I

M

| j_tAi i/c~i

9- i

- 9

~

z( ) az

t

0

,>

t

t

~ M/U) § I I MfO

Im(llM/~

~

0 wheme

M

|149

t

9- i

E( ) a 1 !

9 M/U)+ 0

0

%/ p is obtained by the natural faotorization of

p.

An application

of the snake le~na (191, Chap. i, w i, n ~ 4) to (46) gives an exact sequence

Ker p

*

~ > Ker p - - >

Coker

)

9 0

~t But

Coke~ (Pl)

Ker f

= Kerf

and

Ker

so that to say that

p = __J-~U/M(~%I , 1

is of finite type is equivalent to say that

t

t

(47)

',

1

is of finite type (the maps are the natural ones).

Note that in (47) 9

F1

is any finitely genemated subohjeet of

AI/O~,

t

~l } 1

is any finite set of genemators for

submodule of

M

such t h a t

FI,

and

~

is any open A o

~.-i aI = O(in M/U)(i=I, ..., t).

o

-

264

-

Consider now the exact sequence

| (48)

M ~AIFI

'

> M ~AIAI/cI.1

A1/ot1 il '

>M|

~

0o

From (48) it follows that with the notations of Proposition 15, (47) can

just be of

reforl,ulated by saying that the open submodule

M ~AIFI

is topologically of finite type.

we see in particular that

( I ~ Q A l i ) - l ( ~ U)

If we choose

F 1 = AI/~I,

is topologically of finite type.

Vu

Now from all that has been proved it follows easily that with the notations and hypotheses of Theorem 14 we have the implication l) => 2), where in 2) we have replaced (ii) by the weaker (ii)'.

ThaZ (i) + (ii)" =, i)

under the same hypotheses is also clear, and so we now have Theorem 14 with (ii) replaced by (ii)',

and this implies in particular that Corollary

1 [with the weakened (il)'Cor. l ~

and that Corollary 2 are valid.

Now the passage from (ii)" to (ii) [we suppose that (i) is satisfied] is an exercise in the use of topological

pseudo-coherence and exact

sequences, that we will only do here in the discrete case, where we will prove the more precise result requlred by Corollary i: M~A

1 A1/Ol I

A1 Tot I (M, AI/OI l)

is pseudo-coherent and that

type for all left ideals

Ol I C

of finite type,

AI

induction on the number of generators of coherent for all coherent

C:

if

C

We suppose that

C,

has

that

is of finite

Let us first prove by M ~AI

C

is pseudo-

~enerators, then we can



same as

such that every

lim A/grLa.~?
A1

M ~ A 1N

> Ao

and

such that the diagmams

IdM~)~

IdNC_~~

. lA1 l~91dM

I~

Ao ~ A oM - - 2

M

1

AI~AI N

are commutative, and where multiplication in multiplication (using ~

~91dM

and

#).

A

II > N

is defined by ma%Tix

This seems to be a formidable algebraic

problem, and we might try to make an attack on this at a later occasion. It also remains to make a detailed study of the associated topological

-

272

-

rings in the examples 1)-7) of the introduction.

The case 2) seems to be

particularly rewarding, since any locallzing subcategory of

Qcoh(X)

is

stable, thus in particular the Gabriel filtration is stable under injective envelopes, and the bimodules that describe the cor~-espondinE extensions should be of interest. etc .... .]

[We should make a duality theory for

Qcoh(X)

It should be remarked, that in the examples 5) and 7), parts

of the associated topological rings have already been studied explicitly with nice applications in [I/] I), [29] and [2;]. Finally, so~e stl,ucTume ~heorems for some classes of strictly linearly compact rings, can be found in Ill, [2], [18] and [32].

1)

Brumer [11] u s e s a more r e s t T i c t l v e t h a n t h e u s u a l one - he r e q u i r e s o f open b i l a t e r a l

definition

of pseudo-colpact rings

the existence of a fundamental system

ideal neighbourhoods of

0 ...

-

273

-

BIBLIOGRAPHY

[1]

I.

KR. AMDAL

-

F. RINGDAL, Categories unis~rielles, Comptes rendus,

s~rie A, 267, 1968, p. 85 - 87 and p. 247 - 249.

[2]

B. BALLET, Struetume des anneaux strictement lin~airement compacts qommutatifs, Comptes rendus, s~rie A, 266, 1968, p. ill3 - ii16.

[a]

H. BASS, Finistic dimension and a homoloKical generalization of semi-primar~ PinKs, Trans. Amer. Math. Soc., 95, 1960, p. 466 - 488.

[4]

H. BASS, In~ective dimension in noetherian minks, Trans. Amer. Math. Soc., 102, 1962, p. 18 - 29.

Is]

H. BASS, Lectumes on Topics in Algebraic K-theory, Tara Institute of Fundamental Research, Bombay, 1967. J.-E. BJ~RK, RinKs satisfying a minimum condition on principal ideals (to appeam in J. reine ang. Math.).

[71

J.-E. BJ~RK, On perfect minks, QF-rinKs and rings of endomorphisms of injective modules (to appear).

[8]

N. BOURBAKI, Topologie K~n~rale, Chap. i-2, 3e 6d., Hermann, Paris, 1961.

[9]

N. BOURBAKI, A1g~bre commutative, Chap. 1-2, Her~ann, Pamis, 1961.

[10]

N. BOURBAKI, Alg~bre commutative, Chap. 3-q, Hermann, Paris, 1961.

[11]

A.

BRUMER, Pseudo-compact algebras, profinlte groups and class

formations, J. Algebra, 4, 1966, p. 442 - W70. H. CARTAN - S. EILENBERG, Homologlcal Algebra, Princeton University P~ess, Princeton, 1956. S. U. CHASE, Direct prqduets of modules, Trans. Amer. Math. Soc., 97, 1960, p. 457 - 473. [14]

S.U. CHASE, A generalization of the rink of %-~iangular matFices, Nagoya Math. J., 18, 1961, p. 13 - 25.

[lS]

J. M. COHEN, Coherent Eraded rings and the non-existence of spaces of finite stable homotopy type (to appear in Comm. Math. Helv.).

-

[16]

-

C. W. CURTIS - I. REINER, Representation T h e o ~ o f and AssoclativeAlgebras,

[17]

274

Finite G~oups

Wiley, New York, 1962.

P. DELIGNE, Cohrmologie ~ support propre et constpuctlon du !

foneteur f', Appendix ~D [34]. [18]

J. DIEUDONNE, Topics in Local Al~ebra, Notre Dame Math. Lectures, n

[19]

O

I0, 1967.

C. FAITH - Y. UTUMI, quas!-inlective modules and their endomorphism Pings, Arch. der Math., 15, 1964, p. 166 - 174.

[20]

C. FAITH, Rings with rscendin ~ condition on annihilators, Nagoya Math. J., 27, 1966, p. 179 - 191.

[21]

C. FAITH - E. A. WALER, Direct-sum representations of injective modules, J. Algebra, 5, 1967, p. 203 - 221.

[22]

P. FREYD, Functor Categories an_d ths

Application to Relative

Hrmology (mimeographed notes), Columbia Univemsity, 1962. [23]

P. GABRIEL, Objets injeetifs dans les categories ab~liennes, S~minaire Dubreil-Pisot, 12, 1958 - 1959, Expos~ 17.

[2~]

P. GABRIEL, SuP les categories ab~liennes loealement noeth~riennes et leuPs applications aux_alg~bmes ~tudi~es pap Dieudonn~, S~mlnaime J.-P. Serre, Coll~ge de France, 1959 - 1960.

[25]

P. GABRIEL, Des cat~omies ab~liennes, Bull. Soc. Math. Ft., 90, 1962, p. 323 - 4q8.

[26]

P. GABRIEL, C~oupes formels, S~minaire Demazure-Grothendieck, I.H.E.S., 1963 - 196~, Fasc. 2 b, Expos~ VII B (cf. Math. Rev. 35, 1968,~4222).

[27]

P. GABRIEL - U. OBERST, Spektralkategomien und regul~me Ringe im yon Neumannschen Sinn, Math. Z., 92, 1966, p. 389 - 395.

[28]

P. GABRIEL - R. RENTSCHLER, SuP la dimension des anneaux et ensembles qrdonn~s, Comptes rendus, s~mie A, 265, 1967, p. 712 715.

-

[29]

275

-

v v E. S. GOLOD - I. R. SAFAREVIC, 0 ba~ne pole d klassov, Izv. Akad. Nauk SSSR, set. mat., 28, 1964, p. 261 - 272. Engllsh translation in: American Mathematical Society TPanslations~ Set. 2, 48, 1965, p. 91 - 102.

[30]

A. GROTHENDIECK, SuP guelques points d'alK~bre homoloKique, Tohoku Math. J., 9, 1957, p. 119 - 221.

[31]

A. GROTHENDIECK, Technique de descente et th~or~mes d'existence en K~om~trie alg~brique, II: Le th~or~me d'existence en th~orie for~nelle des modules, S~minalre Boumbaki, 12, 1959 - 1960, Expos~ 195. (Has also been published by Benjamin, New York, 1966.).

~2]

A. GROTHENDIECK (and J. DIEUDON~), El~ments de g~om~tTie alK~bri~ue, Publ. Math. I.H.E.S., n ~ 20, 196~.

[33]

M. HARADA, On a special

type

of hereditary abellan categories,

Osaka J. Math., 4, 1967, p. 243 - 256.

[3.]

R. HARTSHORNE, Residues and Duality, Springer, Berlin, 1966.

[3s]

I. KAPLANSKY, Dual modules over a valuation ring, Proc. Amer. Math. Soc., 4, 1953, p. 213 - 219.

[36]

I. KAPLANSKY, Infinite Abelian Groups, University of Michigan Press, Ann A~bor, 1954.

[37]

F. W. LAWVERE, The category of categories as a foundation for mathematlcs, Proceedings of the Conference on Categorical AiEebra, La Jolla 1965, Springer, Berlin, 1966, p. 1 - 20.

[38]

H. LEPTIN, Linear kompakte Moduln und Ringe, I., Math. Z., 62, 1955, p. 241 - 267.

[39]

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[.0]

E. MATLIS, Injective modules ov@r noetherian rings, Pacific J. Math., 8, 1958, p. 511 - 528.

[.1]

B. MITCHELL, Theoz7 of Categories, Academic Press, New York, 1965.

[421

C. N~STASESCU - N. POPESCU, qgelques observations sur les topos ab~liens, Rev. Roum. Math. Pumes et Appl., 12, 1967, p. 553 - 563.

-

2 7 6

-

Y. NOUAZE, Cat6~ories loealement de.type fini e t categories localement noeth~riennes, Comptes rendus, 257, 1963, p. 823 - 82,.

[.q

F. OORT, Commutative Group Schemes, Springer, Berlin, 1966.

[.5]

Z. PAPP, On algebraically closed modules, Puhl. Math. Debrecen, 6, 1959, p. 811 - 327.

[.6]

J.-E. ROOS, Sur les foncteurs d~riv~s de rendus, 252, 1961, p. 3702 - 370,.

["7]

lim.
EjP

iums

The

~-L,Ej

are e x a c t

denotes

(t)

of

Ej

the e p i m o r p h i s m

~(to) ,

~(t)

9 (toM| M6M

etc. 6

> t

if in

resolution.

> t"

> O

~(t)> t

Using

[20]

(9)

a short exact

> 0

in

[M,A]

sequence --~ EjP

(t")

> O

long e x a c t h o m o l o g y into an a b s o l u t e

in

L.Ej(-)

is t h a t the func-

functors

projective

0 ---> t'

also

and ~ - L n E j

~ , one can s h o w t h a t

rise to a s h o r t e x a c t

0 ---> EjP

with

AB4)

If

homology.

--->

t I = ker

L.Ej

we denote

derived

9 (tlM| M6M

and

[~,A]

Grothendieck's

Ln EJ

to p r o v e

t 6 [M,A]

rise to a r e l a t i v e

9

where

thing

are the a b s o l u t e

is AB4).

(i) gives

it the A n d r ~

category

exist.

functors

In the f o l l o w i n g

and call

Proof.

case the

>

Ej

or s a t i s f i e s

a x i o m AB4), 5 t h e n the a b s o l u t e In the

be the i n c l u s i o n

of c h a i n

sequence

exact

com-

associated

connected

A.

the n o n - a u g m e n t e d

complex,

i.e. w i t h o u t

t.

- 283 -

sequence

of functors.

Since

~-LnE J

vanishes

for

n > o

on sums

9 (tM | [M,-]), it follows, by s t a n d a r d homoloM gical algebra, that ~ - L , E j is left universal. In other words,

the functors

functors

(4)

LnE J

~-LnE J

are the

of the Kan e x t e n s i o n

A c o m p a r i s o n w i t h Andre's

H(,-)

:

[M_,A_]

[C,A_]

in

[i] p.

agrees with the Kan e x t e n s i o n

it follows Since

Since both

Hn(,-)

: [M,A]

sums of r e p r e s e n t a b l e

15

Ej

Ho(,-)

from the exactness 9

of

functors

is valid. because

This h o w e v e r functor

3-5, shows

is not so.

t : M

and defines

H

> A n

(-,t)

H,

Ej

a complex

in Ho(,-)

are right exact,

for

H

( ,-)

n > o

on

H,(,-)

are the left

_~ A , ( , - )

= L,Ej(-)

is "by chance"

in an e n t i r e l y

Recall

that,

on sums of r e p r e s e n t a b l e

It may seem at first that this

Andr~ c o n s t r u c t s

[~,A].

it follows by s t a n d a r d homo-

derived

Hence

9

theory

vanishes

algebra that the functors Ej.

: [M,A]

(3) that they coincide.

[~,A]

functors,

left d e r i v e d

[i], the functor

and

logical

of

Ej

homology

view of his second a x i o m on page

functors. 7

(absolute)

d i f f e r e n t way.

that he associates

with every

of functors

: C

to be the n-th h o m o l o g y

C

(t) of

C

(t)

> A (cf.

[I] p. 3).

7 In the n o t a t i o n of Andre, C should be r e p l a c e d by N. Note that in view of [20] (~) a "foncteur ~ l ~ m e n t a i r e ~ of Andr~ is the Kan e x t e n s i o n of a sum of r e p r e s e n t a b l e functors M----> A.

-

It is not difficult on

M

of

Cn(t) .J

an objectwise construction

Namely,

choose

an

[1] prop.

Ej(S,.J) of

A,(,-)

of

1.5 because Ej-acyclic

is valid. 9 which,

the Kan extension

is

Thus Andre's

procedure

in homo-

functors

of

of

t; apply

Ej

is true for his computational the restriction resolution

of

of the complex S.J

and

generalize

(2),

results

of Andr~

(3) and the nice b e h a v i o u r

on representable

functors.

Theorem (a)

For every Ap(J-,t)

Ej.

We now list some of the properties

in part, of

Cn(t)

C,(t) .J

t.

the left derived

The same

They are consequences

(5)

Moreover,

resolution

of

and that the Kan

Ej-acyclic 8 resolution

is an = S,

Cn(t ).

to compute

and take homology.

M

functors

turns out to be the standard

algebra

on

is

split exact

logical

device

-

to show that the restriction

is a sum of representable

extension

S

284

functor : M ---> ~

t

the composite > ~

8 A functor is called Ej-acyclic vanishes on it for n > o.

if

is zero for

p

>

O.

[s

LnE J : [M,A]

9 In [20] we show that this computational closely related with acyclic models.

method

is

[1]. of

-

(b)

Assume

that

an o b j e c t (M,C) p

(c)

>

A

in

C

Ap(-,t)

sketch

Then

Ap(C,t)

Assume moreover

that for every

functor

: C ---~ S

[JM,-]

C

be

vanishes

for

index categories.

A,(-,t)

> A

: C

As f o r

M

it follows

vanishes

this

it

direct

from

is

a s s u m i n g that

of

limits

such direct

assumption

are finitely

(a) and

on a r b i t r a r y

applications,

this w i t h o u t

of

the hom-

Then

also preserves

(In most examples

sums,

M 6 M

preserves

over d i r e c t e d

: _C ---> _A

establish

and let

o~

has finite

Mv 6 M.

category

such that the comma category

fied if the objects

M

-

is an AB5)

is directed. I0

limits.

If

285

sums

great A

is satis-

generated.)

(c) that @ M~,

where

importance

is AB5).

to

We will

later how this can be done.

The p r o p e r t i e s sequences

of

(3),

that in an AB5) categories

(a) -

(2), footnote

category

direct

(c) are s t r a i g h t f o r w a r d 15),

[20]

con-

(9) and the fact

limits over d i r e c t e d

index

are exact.

i0 A category of objects morphisms morphisms

D is called d i r e c t e d if for every pair D,D' i~ D there is a D" 6 D t o g e t h e r with D ---> D", D' ~ D" and--if for a pair of y,l: D0---~ D 1 there is a m o r p h i s m u: D 1 ---> D 2

such that uy = ~l. (M,C) are m o r p h i s m s

Recall that objects of the category M > C, where M 6 M.

286

-

(6)

A change

sequence small J'

(cf.

[I] prop.

full sub-category

: M' --~ C

A'

and

A"

J = J'.J", composite Ej,

and

it follows of

>

of

containing

~

: M

> M'

Ej

2.4.1). objects

A

is AB4)

A

into E j , - a c y c l i c

representable The

A

and by

Since is the

homological a spectral

algebra

sequence

(-,t) P+q

has enough that

objects.

projectives

(cf.

[13]

Ej,, takes E j - a c y c l i c

But this is obvious

the Kan e x t e n s i o n

from

of a r e p r e s e n t a b l e

The same holds

for p r o j e c t i v e

functors. II "Hochschild-Serre"

[i] p. 33 can be obtained Likewise, to a u n i v e r s a l

Denote by

with

t : M --~ A

is again representable.

be a

the inclusions

[M',~]

One only has to verify

(9), because

functor

or

M.

Thus by standard

functor

M'

: [M,A] ---> [~,A]

A' (-,A" (-,t) ) " ~ P q

provided

let

Andr4 homologies.

that

[~,~].

is for every

rise to a spectral

For this,

Ej,, : [M,~] --~

(7)

[20]

J"

gives

8.1).

the associated

: [M',A]

there

of models

-

sequence

of Andr~

in the same way.

a composed

coefficient

spectral

coefficient

spectral

functor

gives

rise

sequence.

ii The a s s u m p t i o n that M' is small can be replaced by the following. The Kan e ~ t e n s i o n Ej, : [M',A] --~ [~,~] and its left derived functors L,Ej, exist and LnE J, vanishes on sums of representable functors for n > o.

-

(8)

Theorem.

Let

t : M

functors,

where

and

A'

A

have enough projectives. functors

L,F

F

: A --~ A'

are either AB4)

categories

Assume

and

be or

that the left derived

F

has a right adjoint.

Then

sequence

L F.A

P

for

-

> A

exist and that

there is a spectral

provided

2 8 7

(-,t)

;

A

q

the values

(-,F-t) P+q

of

t

are F - a c y c l i c

(i.e.

LqF (tM) = 0

q > o ).

(9)

Corollary.

~(-,t)

: ~

A,(C,F.t) product

> ~

If for every

vanishes

on an object

~-- L,F(Ao(C,t)).

formula,

preserving. the property

provided

For,

let

%(C

p > o

This gives Ao(-,t)

C = @ C

,t) = 0

for

the functor C 6 ~,

then

rise to an infinite

: ~

> A

is sum-

be an arbitrary p > o.

co-

sum with

Then the canonical

map =.

(i0)

@ A,(C~,F.t)

is an isomorphism

because

L,F(O Ao(C~,F.t))

> A,(@ C~,F.t)

9 L,F(Ao(C~,F.t))

~ L,F(Ao( ~ C~,F.t))

holds. 12

12 This applies to the categories of groups and semi-groups and yields infinite coproduct formulas for homology and cohomology of groups and semi-groups w i t h o u t conditions. For it can be shown that they coincide with A, and A* if A' = Ab.Gr., A = C-modules, t = Diff and F is tensoring With ~ ~-~mmi~g into some C-module (cf. [2] Ch. 1 and Ch. X).

- 288 -

(ii)

Cgrol!ary.

Then every gives

finite

: C ---> A' --

satisfied. if

i

sub-sum

9 C~i i

Proof EC

: [C,A]

> A

evaluation tions

on

of

F

and

t

>

for

the c o n d i t i o n

in

(5c)

is

,F.t) --~ 9 A . ( C v , F . t )

(C~i,t) 9 C~

(8) and E6

of f u n c t o r s

: ~

~').

is v a l i d

for e v e r y

*

of

and

(but not

A,(-,t)

formula

A,(e C

--~ 9 A

is AB5) for

coproduct

In o t h e r w o r d s , A , ( e C~i,t)

A

formula

provided

8

i

finite

that

coproduct

rise to an i n f i n i t e

A,(-,F.t)

holds

Assume

.13

(ii).

(Sketch)

: [C,A']

--~ A'

associated

imply

with

By we d e n o t e

C 6 C .

the

The a s s u m p -

t h a t the d i a g r a m

_A

/E'(F.-) [M,A] J >

is c o m m u t a t i v e .

The d e r i v e d

can be i d e n t i f i e d

with

spectral

arises

sequence

E c'.EJ'(F.-)

into

formula

A

of

for

[C,A'] '

Ec.E J

(-,t)

: C

~" u

functors

t ~

A

) A'

of

(C,F.t)

E 6 . E ~ ( f . - ) :[M,A] ---~ A' .

As a b o v e

f r o m the d e c o m p o s i t i o n F.

and >

A

The

infinite

in

(6) the

of coproduct

can be e s t a b l i s h e d

by m e a n s

(5c).

13 This a p p l i e s to all f i n i t e c o p r o d u c t t h e o r e m s e s t a b l i s h e d in B a r r - B e c k [2] Ch. 7 and A n d r ~ [i] w i t h _A, _A' , t and F as in f o o t n o t e 12.

- 289 -

Thus (8).

it also holds

for the E 2 - t e r m of the spectral

One can show that the d i r e c t

sum d e c o m p o s i t i o n

E 2 - t e r m is c o m p a t i b l e w i t h the d i f f e r e n t i a l s iated f i l t r a t i o n one obtains

A.(-,F.t)

of the spectral

an infinite

An abelian

resolutions

and the assocIn this way,

formula

interpretation

and n e i g h b o r h o o d s

paper of U. O b e r s t version

of the

for

A'

:

(12)

sequence.

coproduct

sequence

[18].

is c o n t a i n e d

remained

this we b r i e f l y

review

non-abelian

in a f o r t h c o m i n g

We include here a s o m e w h a t

of ~his i n t e r p r e t a t i o n

problem which

of Andre's

improved

and use it to solve a central

open in B a r r - B e c k

[2] Ch. X.

the tensor p r o d u c t

|

For

between

func-

tors w h i c h was

i n v e s t i g a t e d by D. B u c h s b a u m

[5], J. F i s h e r

[10], P. Freyd

[ll], D. Kan

[17]

C. Watts product

[24], N. Y o n e d a

[13], U. O b e r s t

[25], and the author.

The tensor

is a b i f u n c t o r

(13)

8_ : [M_opP,Ab.Gr.]

x [M,A]

d e f i n e d by the f o l l o w i n g u n i v e r s a l s 6 [s_~

t 6 [M,~]

and

(14)

[s | t,A] ~

in

s,t

and

A.

> A

property. A 6 ~

isomorphism

natural

[18],

[s,[t-,A]]

For every

there

is an

-

It can be c o n s t r u c t e d A-modules,

namely = A

where

A

choose

a presentation

X @ Y

to b e the c o - k e r n e l

where

Y~ = Y = Yu

of c o n t r a v a r i a n t Z |

[-,M]

Y

"

---> 9 A

=

free)

X

---> 0

X

and d e f i n e

9 Y~---> 9u Y u ,

is p l a y e d b y the f a m i l y

functors where

M 6 M

and

Z

denotes

(13)

follows

8 t = tM

The u n i v e r s a l

property

(2) in the

of

following

bifunctors

(L s |

--~ L

that the d e r i v e d and

sense

s @

(8 t)(s),

and the v a l u e s groups.

--~ [Z 8

of

Under

Tor

(s,t)

makes

Tot

(s,t)

can be c o m p u t e d

s

these

and has

way: [-,M],[t-,A] ]

(15) and the c l a s s i c a l

> ~

abelian

A

[tM,A] --~ [T,[tM,A]]

: [M~

AB5))

of

[-,M])

[20]

One can s h o w by m e a n s

(resp.

~

module

of the i n d u c e d m a p

The role

lemma

[-,M] | t,A]

property

(e A v) | Y = 9 Yv,

9 A

as above.

f r o m the Y o n e d a

t

2)

3) for an a r b i t r a r y

(Z |

about balanced

A 8 Y = Y;

between

Thus we d e f i n e

(15)

[w 8

product

= Y;

---~ A b . G r . ,

the integers. 14

and c o n t i n u e

l)

representable

: M~

-

like the t e n s o r

stepwise:

and

2 9 0

argument

functors

: [M,A]

> A

provided

A

are free conditions

its u s u a l

by p r o j e c t i v e

of h a v e the

is AB4)

(resp.

torsion

the n o t i o n

properties,

e.g.

or f l a t r e s o l u t i o n s

14 N o t e t h a t the f u n c t o r s Z @ [-,M], M 6 M are p r o j e c t i v e and f o r m a g e n e r a t i n g f a m i l y in [Mopp,~b.Gr.]. This f o l l o w s e a s i l y f r o m the Y o n e d a l e m m a [ 2 ~ ~ ) and (5).

-

in either variable. representable

291

-

We remark w i t h o u t

functor

A |

[M,-]

: M

proof that every >

A

is flat.

should be noted that for this and the following one cannot

replace

the condition

AB4)

It

(until

(22))

by the assumption

that

has enough projectives.

(16)

U. Oberst

exact sequences objectwise of

s @

~(L,s and

@ t

@)(t)

@

abelian

He shows

C 6 C

: M~

to ~

@ t)(s)

Thus the notion

[J-,C]

He establishes

an isomorphism

(17)

A,(C,t)

t : M---~

abelian

of

is a relative Andre's

projective

result that

complex

C

(t)

C

: C

A

p. 3) or a n o n - a b e l i a n special

(C,t)

variable.

of which

~

C

[J-,C],

of

Z |

of

at

C C

With

are free

t)

out that a non[i] p. 17

[J-,C].

Thus

either by the

(cf.

(4) and

[1]

turns out to be a

fact t h a t ~ T o r , ( - , - )

projective

s

is the inclusion.)

can be computed

evaluated

on

a functor

and points

resolution

by a relative

are

functors

sense.

in the sense of Andr4

case of the well known

be computed

|

resolution

> A

makes

J : M

~

which

any conditions

the values

~_~Tor,(Z

for every functor

of short

have the property

is a s s o c i a t e d

(Recall that

resolution

[M,A]

~-Tor,(s,t)

there

~

that the derived

without

---> Ab.Gr.

groups.

the class and

relative

~ ~(L,

every object

considers

[M_~

split exact.

and

t.

in

[18]

resolution

can

of either

- 292

U. Oberst also observes that a n e i g h b o r h o o d of

C

(cf.

of

Z | [J-,C].

("voisinage")

[1] p. 38) gives rise to a relative projective Therefore,

it is obvious that

A,(C,t)

can

also be computed by means of neighborhoods. (18)

The notion of relative

difficult to handle in practice. sequences

(7) and

(8)

~-Tor,(-,-)

is somewhat

For instance,

the spectral

and the coproduct formulas

(i0) and

(ii) cannot be obtained with it because of the m i s b e h a v i o u r of the Kan extension on relative projective Moreover, also

Andre's computational m e t h o d

[i] prop.

t

Call a natural transformation

a ~-natural

transformation

9 imp(M)

> O

and

The above difficulties [M,A]

>

[~,A]

and

not preserve ~ - n a t u r a l

if for every

O ---> imp(M)

M 6 M

---> t'M

9 t'

the morphisms are split.

arise because the Kan extensions [M~

--~

transformations.

(17) etc. of the relative ~ - T o r

Oberst

| [J-,C],t)

~ : t

[c~

do

Our notion of

absolute Tor, does not have this disadvantage.

similarly

The proper-

can be e s t a b l i s h e d

for the absolute Tor, using the techniques

[18].

of

U.

We now sketch a different way to obtain these.

The fundamental Kan extension

(cf.

in question need not be relative

projective.

ties

1.8

(4)) cannot be e x p l a i n e d by means of ~ - T o r , ~

because the resolution of

tM

resolutions.

relationship between

0

and the

Ej : [M,A] ---~ [~,~] is given by the equation

-

(19)

(Z |

where

t

and

C

respectively.

[J-,C])

293

|

t--~ E

are arbitrary

To see this,

-

(t)(C)

J

objects [J-,C]

let

[M,A]

of

= lim

and

[-,M ]

C be

>

the canonical direct

representation

limit of c o n t r a v a r i a n t

index category

for this

[J-,C]

of

representation

(M,C).

and it follows

(15) and Kan's

~Z |

[J-,C])

8 t = lim t M --

the homology

(20) Thus

(3) of

A A,(C,t)

tions of

Z |

= E

(t)(C).

t,

Since

[J-,C],

A

(-,t)

is

(t).J

t)

either by p r o j e c t i v e

(e.g. n o n - a b e l i a n

C

[-,M~]

.

(Z |

can be computed

resolutions

= lim Z | --->

c o n s t r u c t i o n 15 that

neighborhoods 16) or flat resolutions or Andre's

[J-,C]

with

E P (t), where P, (t) is the J 9 it follows from (19) that

(C,t) --~ Tor

[J-,C]

is isomorphic

Z |

as a

Note that the

J

of the complex

flat resolution

Hence

__>

> S

hom-functors.

the comma category from

: M~

and

of

resolu-

resolutions t

S ,

(e.g. cf.

and P, (t)

(4)).

15 Kan [13] 14 constructed Ej : [M,A] > [C,A] objectwise. He showed that Ej(t) (C) is the direct limit of (M,C) ---> A, (S > C)~ tM. 16 Further examples are provided by the simplicial resolutions of Barr-Beck [2] Ch. 5, the projective simplicial resolutions of type (X,o) of Dold-Puppe [8] and the pseudosimplicial resolutions of T i e r n e y - V o g e l [19]. A corollary \ of this is that the Andr~ (co)homology coincides with the theories d e v e l o p e d by Barr-Beck [2], D o l d - P u p p e [8], and T i e r n e y - V o g e l [19] w h e n C and M are defined appropriately. Note that there ar~ more pr--ojective resolutions of Z | [J-,C] than the ones described so far (for instance, the resolutions used in the proof of (21) below).

-

2 9 4

-

The above m e t h o d s prove v e r y u s e f u l in e s t a b l i s h ing the t h e o r e m b e l o w w h i c h is b a s i c for m a n y

(21)

Theorem.

a category However,

C

Let

M

be a full small s u b - c a t e g o r y of

w h i c h has sums.

if a s u m

@ Mi 6 C

M

n e e d not h a v e finite sums.

is a l r e a d y

a s s u m e d t h a t e v e r y s u b s u m of

@ Mi

ing t o

M,

t : M

> A

M ---> 9 M~

where

M

6 M.

p > o

v a n i s h e s on a r b i t r a r y

Proof. resolution

M,

M 6 M

it is M.

Moreover,

and

e M~

6 C

factors t h r o u g h a s u b s u m b e l o n g Let

a sum-preserving

Then for

in

is also in

assume t h a t for e v e r y p a i r of o b j e c t s every morphism

applications.

A

be an AB4)

functor, i7

the f u n c t o r 9 M~,

sums

c a t e g o r y and

%(-,t)

where

M

: C

9

A

6 M. I8

The idea is to c o n s t r u c t a p r o j e c t i v e

N, (@ M~,M~)

of

w |

[j-, ~ M~]

: M_Opp

; Ab.Gr.

w h i c h remains e x a c t w h e n t e n s o r e d w i t h | t : [M~

> A

.

W i t h e v e r y p a i r of o b j e c t s and e v e r y full small s u b - c a t e g o r y

17 The m e a n i n g is that e x i s t in M.

t

M

C 6 C of

Mc

and

M 6 M

C ,

has to p r e s e r v e the sums w h i c h

i8 I am i n d e b t e d to M i c h e l A n d r 4 for an i m p r o v e m e n t of an e a r l i e r v e r s i o n of the theorem. He also f o u n d a d i f f e r e n t p r o o f b a s e d on the m e t h o d s he d e v e l o p e d in [i], a s s u m i n g that M has finite sums and that e v e r y m o r p h i s m M > 9 M factors t h r o u g h a finite subsum.

-

Andr~

associated

augmentation gives ~

a s.s.

M

(M,C)

denotes

full

complex

M~

which

defined

for e v e r y

Z |

n

to

M.

of a s u m

ascending

arrows

Hi

where > Mi-i

morphisms.

M~

contraction

Sn(M)

6 M,

on

of t h o s e

resolution

of

Z

| M,(-,e

~n(M,C). M

be the of

is

(~ Mv,~)

M~)

[-,~]).

where

subsums

N

and c o n s i s t s

More

of

6 M_

o _< i _< n

for

Mo

9 M

> 9 M

complex

> Ab.Gr.

precisely,

there

is o b j e c t w i s e

a sum

the c a n o n i c a l

(e M ,M)

N

is a s u m m a n d

and the d o u b l e

denote

: N n(s S ,~) (M) ~ For

It

,

and let

--> 9 M v

and

as follows.

38).

of s u b s u m s

The a u g m e n t e d

of f u n c t o r s

defined

Mi

M~

(Z |

an

Ab.Gr.

group

The

S

chain

Mn_ 1 [-,~n],

to

consisting

M,

[i] p.

M~

where

to be a s u b c o m p l e x

in d i m e n s i o n

Mn

of

belong

from

with

of f u n c t o r s

the free a b e l i a n

C = (9 M~,

sub-category

together

(cf.

[JM,C],

| M C-,C) ---> ~, | [J-,C]

Let

> ~ |

[J-,e M ]

s p l i t exact.

Nn+ I(s S ,~) (S)

~ M

and a s u b s u m

The is

9 M

i ~i S~ 6 M. 9 M -----> ~ M the c a n o n i c a l m o r p h i s m , i ~i ~ T h e n t h e r e is f : M > 8 M~ be a m o r p h i s m , M~ 6 M.

denote Let

by

a unique Mf

decomposition

~ 9 M

tains

the

assigns > Mf

M

> Mf=>

is the s m a l l e s t

"image"

The m o r p h i s m

M

;

-

(M,C)

set

rise to an a u g m e n t e d

Z | M--n(M,C)

2 9 5

of

S_l(M)

f. : Z |

to a b a s e e l e m e n t in the c o m p o n e n t

9 M

subsum

Obviously [JM, ~ M f : M Z |

]

such

that

which

con-

is an o b j e c t > 9

> ~ M

[M~M f]

M

of Mf

f

of

(~ | its

indexed

of

~.

[M,Mo])

factorization by M f

> ~ M

+

296 Likewise,

Sn(M)

: Nn( ~ M ~ , ~ ) ( M )

to a b a s e e l e m e n t indexed

by

M

M

Mf

> Mn

subsums check

s

,M)(M)

natural

in

t i o n of

Z |

.)

Z |

....> e~ M~.

Since

[JM, S M

Hence

is

].

~ |

[M,M n]

indexed Mf

and

by ~i

are

it is n o t d i f f i c u l t

to

of (However

N , ( ~ M~,~)

s

(M)

is n o t

is a p r o j e c t i v e

and the p - t h h o m o l o g y By

Ap(

assigns

factorization

[M,M f]

o < i < n,

[J-, 9 Mu]

M ,M) | t

its

is a c o n t r a c t i o n

> Z |

M

> 9 M ~

o

where

(M)

of the s u m m a n d

component

"'" ~ o ,

> N n + l ( ~ My, ~) (M)

> Mn

... M

in the

9 M

that

N,(~ M

>

~"

of

f : M

n

> Mf

-

(15) the

resolu-

of the c o m p l e x

latter

consists

of a s u m

tMn >

in d i m e n s i o n homology

except

a certain present

n.

s.s.

- -

Mn

hn

-"'" M ~ by

t% > %-1

M

~ > 9 M

~

~> %

~ ~I~i >

complex

I originally

M. A n d r ~

is r e d u n d a n t

n

has t r i v i a l assumed

the Kan c o n d i t i o n

a contraction

h,

which

then pointed

because

is a s u b s u m

maps

indexed

"'" M O

as follows.

chain

9 M i ~i

the s u m m a n d

of

tM~i

9 M ~

.

The morphism of the com-

by > 9 M~

out

the a b o v e

is d e f i n e d

of an a s c e n d i n g

that

(which is

t h e c a n o n i c a l morphism.

> 9 t/qn+ 1 =

zero,

I know).

condition

Mui

: 9 t/qn

ponent %

has

t h a t this

in d i m e n s i o n

t h a t the o b j e c t >

Denote

To p r o v e

in all e x a m p l e s

subcomplex

o

set s a t i s f i e s

to me that this

Recall

M

identically

on the

-

2 9 7

-

m

component M~i

of

> %

to c h e c k A

tM

(e M p u

>

that

,t)

>

n-i

h

= 0

9 tMn+ 1

i n d e x e d by

"'"

M

> 9 M ~ v "

o

is a c o n t r a c t i o n . for

This

It is r o u t i n e

shows

that

p > o. Q.E.D.

(22)

Corollary.

consisting

L,E

Ej,

be the f u l l s u b - c a t e g o r y

: [M',A] ~

= A' ,

j,

representable sequence

M'

of sums of o b j e c t s

Kan e x t e n s i o n functors

Let

in

M.

[~,A]

exist and that

functors

(7) c o l l a p s e s

for

n > o.

Assume a n d its

LnEj, Then by

and one o b t a i n s

of

t h a t the left derived

vanishes

on

(21) the s p e c t r a l

an i s o m o r p h i s m

A, (-,t) --~ > A,' (-,A"o (-,t)) where

t : M

) A

is a f u n c t o r

: M'

> A

is its K a n e x t e n s i o n

A"(-,t) O

~

: ~

) A

with a certain (cf.

(33)

realized the

(21) a n d on

M'. 19

~

The v a l u e A'

as in

of

(22)

lies

in the f a c t t h a t

can be i d e n t i f i e d w i t h cotriple

(34)).

C

(the m o d e l

induced

associated cotriple

In this w a y e v e r y A n d r ~ h o m o l o g y

as a c o t r i p l e

latter carries

in

the h o m o l o g y

homology

and all information

o v e r to the f o r m e r

c a n be about

and vice versa.

19 If A is AB5), t h e n the t h e o r e m is a l s o t r u e if M' an a r b i t r a r y full s u b - c a t e g o r y of ~, s u c h t h a t fur every M' 6 M' the a s s o c i a t e d c o m m a c a t e g o r y (M,M') directed. T~is f o l l o w s e a s i l y f r o m (5b) a n d (7).

is is

-

2 9 8

-

It has b e c o m e a p p a r e n t in s e v e r a l places s m a l l n e s s of

M

be removed.

that the

is an u n p l e a s a n t r e s t r i c t i o n w h i c h s h o u l d

A n d r 4 did this by r e q u i r i n g t h a t e v e r y

has a n e i g h b o r h o o d in

M.

C 6

A n o t h e r w a y of e x p r e s s i n g the

same c o n d i t i o n is to a s s u m e t h a t e v e r y f u n c t o r Z |

[J-,C]

: M~

) Ab.Gr.

admits a p r o j e c t i v e

resolution.

It is then c l e a r f r o m the a b o v e t h a t t h e r e is an e x a c t c o n n e c t e d s e q u e n c e of f u n c t o r s properties n > o.

A

Since

o

(,-) M

= E

J

A,(,-)

and

A

n

: [M,A]

(-,A |

[-,M]) = O

is not small, not every f u n c t o r

n e e d be a q u o t i e n t of a sum of r e p r e s e n t a b l e

L,Ej = A,.

ples this is h o w e v e r the case, e.g.

if

G-projectives

C.

of a c o t r i p l e

G

in

M

[~,~] w i t h the for t : M

functors

c a n n o t a u t o m a t i c a l l y c o n c l u d e that

(23)

,)

~

and one

In m a n y e x a m -

c o n s i s t s of the

So far w e h a v e o n l y d e a l t w i t h A n d r 4 h o m o l o g y w i t h

r e s p e c t to the i n c l u s i o n

J : M

homology associated with a cotriple

> C G

and not w i t h the in

C

d e f i n i t i o n of a c o t r i p l e we refer to B a r r - B e c k

(for the [2] intro.).

One r e a s o n for this is t h a t the c o r r e s p o n d i n g m o d e l c a t e g o r y is not small.

A n o t h e r is that the p r e s e n c e of a c o t r i p l e

is a m o r e s p e c i a l s i t u a t i o n in w h i c h t h e o r e m s o f t e n h o l d under weaker conditions

and proofs are easier.

The a d d i t i o n a l

i n f o r m a t i o n is due to the simple b e h a v i o u r of the Kan e x t e n sion on f u n c t o r s of the f o r m

M-I~

M

the r e s t r i c t i o n of the c o t r i p l e on

M.

our a p p r o a c h w o r k s

~

~,

where

G

We now o u t l i n e h o w

for c o t r i p l e h o m o l o g y .

is

-

(24)

Let

G

be a c o t r i p l e

any f u l l s u b - c a t e g o r y G-projectives

(X)

: GX

of

~,

and include

t h a t an o b j e c t

2 9 9

X 6 C

> X

every

admits

With every

efficient involves

>

t.

the v a l u e s

H,(-,t) G

called

t

M

are

C 6 C.

where

(Recall

if

c : G

: ~

) id

GC

C

are

>

Barr-Beck

functors cotriple

Their construction

of

is a l s o w e l l

where

t

derived

A , also

functor

of w h i c h

The objects

functor

the c o t r i p l e

: ~

and denote by

GC,

a section,

called

H,(-,t) G

C

is c a l l e d G - p r o j e c t i v e

of the c o t r i p l e .

[2] a s s o c i a t e

in

the o b j e c t s

is the c o - u n i t free.)

-

homology of

H,(-,t) G

on the f r e e o b j e c t s

defined when

t

with co-

of

only

C.

Thus

is o n l y d e f i n e d

on

M.

(25)

Theorem.

The Kan e x t e n s i o n

exists.

It a s s i g n s

cotriple

derived

Ej(t.G) AB4)

= t-G

functor

H

is valid.

t : M

(-,t)

o

G

: [M,~]

~

: C --

(Note t h a t

A

the

> A. --

~

)

[~,~]

zeroth

In p a r t i c u l a r

n e e d n o t be AB3)

or

for this.)

Proof. extension,

According

we have

restriction We

to a f u n c t o r

Ej

map

to s h o w t h a t

[Ho(-,t)G,S]

limit ourselves

and leave

to g i v i n g

it to the r e a d e r

e a c h other.

A natural

rise to a d i a g r a m

to the d e f i n i t i o n for e v e r y

~

of a K a n

S : C

[t,S.J]

A

the

is a b i j e c t i o n .

a m a p in the o p p o s i t e

to c h e c k

>

direction

t h a t t h e y are i n v e r s e

transformation

~ : t

>

S.J

to

gives

-

tE (GC)

tG2C

s~ (GC} i

.

-

i

for every

The top row is by construction

a coequalizer. SC

functors. t : M tG

----> t

: [~'~]

>

> A

is an

[~,A]

in [2] Ch. I

imply that

is an exact connected sequence of

(objectwise split)

'-)G ~

we obtain

algebra the following

>

[~,~]

is valid.

exist and

Moreover ~-L,Ej(-)

~ H,(

'-)G

denotes the class of short exact sequences

which are objectwise

w i t h o u t proof that for of representable

epimorphism,

The left derived functors of the Kan

Ej : [M,~]

holds, where [M,A]

established

H (-,t.G) = 0 for every functor n and since the canonical natural transformation

L.Ej(-) ~ H,(

in

transformation

Since

Theorem.

extension

In

> S.

by standard h o m o l o g i c a l (26)

Thus there is a unique m o r p h i s m

a natural

The properties '-)G

of

w h i c h makes the d i a g r a m commutative.

this way one obtains Ho (-,t) G

SC

-

C 6 C.

~

s~ (c)

SGC

i

S G E (C)

H,(

Ho (C,t) G

, (GC)

SG2C

Ho(C,t) G

-

>> tGC

, (G2C)

Ho(C,t) G

3 0 0

split exact.

n > o , Hn(

functors.

'-)G

We remark

vanishes

on sums

-

(27)

C o r 0 1 1 a r ~.

in

C

jectives

-

The cotriple

the G-projectives. 20 G'

3 0 1

homology

In particular,

only on

two cotriples

give rise to the same homology coincide.

depends

G

and

if their pro-

One can show that the converse

is also

true.

(28)

It is obvious

Barr-Beck

[2] Ch.

acyclicity

III for

criterion

resolution

of

tG,

in h o m o l o g i c a l tG,

on

M

: C

is an Ej-acyclic

(29)

It also

(-,t)

(for

follows

[2] by means

resolution G,

and Andr~ homology

see

As in

(-,t)

(4) the

of the s~ procedure

the restriction of

t

[2] Chap.

of

and because I).

(26) that cotriple A

property

the standard

is b e c a u s e

from

G

the models deduce

holds

the u n i v e r s a l

is actually

This

of

are the usual

of functors.

in

) A

algebra.

= tG,

*

sequence

H,(-,t) G

Ej(tG,.J)

H

H,(-,-) G

for e s t a b l i s h i n g

of an exact connected constructions

from the above that the axioms

coincide,

homology provided

*

for the latter

are

M.

this from the first half of

can only be obtained set theoretical

One might be tempted

to

(26), but apparently

it

from the second half.

difficulty.

For details

The reason

we refer to

is a

[23].

20 In many cases the cotriple homology depends only on the finitely generated G-projectives. An object X 6 C is called finitely generated if the h o m - f u n c t o r [X,-] : C ~ S preserves filtered unions of subobjects.--

-

F r o m this previously

it is obvious

established

homology.

fications

which

result

(3O)

The a s s u m p t i o n when

following: t : M

M

In adjoint.

(8)-(11)

It suffices

sub-category

(31)

M

derived and

G : C

directed

instead

M

is as in functors

[M,A]

~

(24), and

: [M_~ is defined [M~

Tor,(Z

|

~

M

direct F

that

limits. need not have a right

F

is right exact. 11 we obtain

a spectral

for a

sequence

> Ap+q(-,t)

~,

in

and the functor

and

A

denote [~,A]

the left >

[M,A]

product , ~ , sxt ~

(14) but may not exist However

[J-,C],

of these properties.

respectively.

• [M,A] as in

some useful modi-

is seldom p r e s e n t

of the Kan extensions [~,A]

carry over

can be replaced by the

the functor

of

The tensor

hold.

(5c), which

Hp(-,Aq(-,t))G

where

s s

in

(26) , (7), and footnote

From small

We list below

is not small,

preserVe

that the properties

from direct proofs

The cotriple

> A

-

for the Andr~ homology

to the cotriple

examples

3 0 2

(19) and

s @ t for every

(32)

t) ~ H,(C,t) G ~ - T o r , ( Z

| [J-,C], t)

-

The first i s o m o r p h i s m

shows

by either a p r o j e c t i v e

3 0 3

-

that

H,(C,t) G

resolution

of

Z |

can be c o m p u t e d [J-,C]

or an

E -acyclic r e s o l u t i o n of t. The former is a g e n e r a l i z a t i o n J of the main result in [2] 5.1, b e c a u s e a G - r e s o l u t i o n is a projective

resolution

the acyclic model

(33) there

of

Z |

argument

[J-,C];

in

[4]

(cf. also

W i t h every small s u b - c a t e g o r y is a s s o c i a t e d

cotriple G : ~

(cf. > ~

a cotriple

[2] Ch. X). assigns

G,

w h i c h belongs

The co-unit

M.

r e s t r i c t e d on a s u m m a n d (26) and A,(-,-)

(25)).

of a category

c a l l e d the model

to an object f : df

to

M

[20]

induced

C 6 ~

> C,

df

is

the sum

the domain

~(C)

: 9 df

f : df

9 df, f df of ) C

) C.

The theorems

(22) enable us to compare the Andr~ h o m o l o g y of the inclusion

the model

M ~

induced cotriple

of objects

M

6 M

G

~ in

w i t h the h o m o l o g y C.

Since every

is G-projective,

Theorem.

satisfies

the c o n d i t i o n s

preserving

functor

A

Assume

we obtain the

t : M

(-,t)

that the i n c l u s i o n

in

(21).

~

A

is an isomorphism,

A

> C sum-

the c a n o n i c a l map

(-,t)) o

provided

M

Then for every

-- > H , ( - , A

*

is an AB4)

of

sum

following:

(34 )

C

Recall that its functor part

indexed by m o r p h i s m s

9 M

the latter g e n e r a l i z e s

G

category.

-

If Droduces,

t

likewise

A*(-,t)

A

-

is c o n t r a v a r i a n t

we obtain

provided

304

H * (-, A O (-, t) )

that Andr4

be r e a l i z e d under rather weak conditions even of a m o d e l

in B a r r - B e c k

formulas),

(homology sequence

9.1,

9.2

(Mayer Vietoris)

Moreover,

(co)homology as

[2] 7.1,

7.2

(coproduct

over to Andr4

and

[2]

(co)homology.

tends

to classify

extensions

(cf.

illustrate

the use of this r e a l i z a t i o n with some examples.

(35)

carries

of

In this way,

of a map)

cohomology

can

(co)homology

also apply to Andr4

the fact that cotriple [3])

G

induced cotriple.

the c o n s i d e r a t i o n s [2] 8.1

~n~o

category.

The t h e o r e m asserts

a cotriple,

sums

for c o h o m o l o g y

~,

where

on less than

of u n i v e r -

(cf. L a w v e r e

(27) one can show that

of the free c o t r i p l e (co)homology

Recall

[15].

is a c a t e g o r y

in the c l a s s i c a l

the A n d r e

clusion

, then

By means

(co)homology with

in the sense of L i n t o n

= M0

sal algebras

of algebras w i t h

in

~

coincides

associated with

the objects ~

of

generators.

~

inare

-

b)

Let

C = Ab.Gr.

of finitely

305

-

and let

generated

M

abelian

be the s u b - c a t e g o r y

groups.

one can show that the first Andr~ AI(c,[-,Y]) extensions

is i s o m o r p h i c of

C 6 Ab.Gr.

in the sense of Harrison. is a category

of

Using

(34),

cohomology

group

to the group of pure w i t h kernel

Y 6 Ab.Gr.

The same holds

A-modules,

where

A

if

C

is a ring

w i t h unit.

c)

Let

C = A-algebras

category Let

C

of finitely

and let

generated

be a A-algebra

and

Y

M

be the sub-

tensor algebras. be a A-bimodule.

1 Then E

A

(C,[-,Y])

> C

classifies

w i t h kernel

A-module e x t e n s i o n Harrison

The cases of s i m i l a r examples,

Y

is pure

singular e x t e n s i o n s

such that the u n d e r l y i n g in the sense of

(cf. b)). 21

(b) and which

of pure group e x t e n s i o n s

(c) set the tone for a long list indicate

that Harrison's

can be c o n s i d e r a b l y

2 1 1 am indebted to M. Barr for c o r r e c t i n g made in this example.

theory

generalized.

an error

I had

-

3 0 6

-

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[2]

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Offsetdruck: Julius Beltz, Weinheim/Bergstr.