ALGEBRAS 0F FlNlTE COHOMOLUGICAL
DSMENSION

F29

Thesis fer the Segree a? m. i).

EfiiiI‘H‘éQAW STATE {ENVERSWY

fiUSEP-H PHTHQNY WEHLEN, JR.
1970

        

  

LI BR A R Y
Michigan State
University

This is to certify that the

thesis entitled
ALGE‘BRAS
OF FINI I'd COHOMOLOG I SAL IL‘I MENS I ON
presented by
JOS EPri AN I‘HUNY Nbdiqafiv . JR .

has been accepted towards fulfillment
of the requirements for

ELL—ll... degree in Lia Enema 131 Cs

XKJ J”

H
Majofirofessor

Date Jill’ .l.” .l. 70

0-169

 

 

 

 

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ABSTRACT

AlGEBRAS OF FINITE COHOMOIDGICAL DIMENSION

BY

Joseph Anthony Wehlen, Jr.

Let A be an algebra over a commutative ring R. We
say that R-dim A = l. hd.Ae(A), where Ae = A 8k A*. Our
main theorem extends a result of Endo and Watanabe [Osaka Math
J. fif1967)].

Main Theorem: Let A be a finitely generated, pro-
jective algebra over a commutative ring R. Let n be finite.
The following are equivalent:

(a) R-dim A = n

(b) Rm-dim Am = n for all maximal ideals m of R,
with equality holding at some m.

(c) R/m-dim A/mA = n for all maximal ideals m of
R, with equality holding at some m.

This leads to the following theoremsconcerning commu-
tative R-algebras: (1) If A is a finitely generated, pro—
jective, commutative R-algebra, then R-dim A = 0 or
R-dim A = m.- (2) Let A be a finitely generated, projective,
central Sealgebra, and let S be a faithful,
finitely generated, projective R-algebra. Then R-dim A = n

iff R-dim S = 0 and S-dim A = n. Thus we may consider

 

 

Joseph Anthony Wehlen, Jr.

centr a 1 I1

‘dimensional algebras. We can further show that if

R is a semi-local ring and N is the Jacobson radical of the

finitely generated R—algebra A, then R-dim A/N = 0. If R

is a finite direct sum of local hensel rings this implies that

A = S + N where R-dim S = 0 and S is a subalgebra of A,

using a result of Azumaya [Nagoya Math. J., 211951)].

We define a generalized triangular matrix algebra
Tn(Ri; Mij) and an induced generalized triangular matrix
algebra to be as in M.Harada [Nagoya Math. J., 2_7_(1966)].

Then we show (1) If A is a finitely generated algebra over

a local hensel ring (R,ro) with A/N = 6: Di’ where the Di
are division algebras, and the R-dim A s 1, then A '='
Tn(Ri; Mij) where the Ri are separable R-algebras and

Ri ® R/m ; Di; and (2) If A is a finitely generated algebra
over a local hensel ring (R,m) with R-dim A s 1, then A

is isomorphic to a generalized triangular matrix algebra in-

duced from (1).

 

 

-=-— 3.3:: r-v-v— ’—

 

 

 

 

AlGEBRAS OF FINITE CDHOMOIDGICAL DIMENSION

By

Joseph Anthony Wehlen, Jr.

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Mathematics

1970

 

 

ACKNOWIEDGEMENTS

The author wishes to express his gratitude to Professor
E. C. Ingraham for suggesting the problem and for his helpful
suggestions and patient guidance during the research. He also
wishes to offer special thanks to Dean Sanders and Lawrence
Spence for the helpful conversations he has had with them during
the research, and to Margaret Curley for her encouragement.
Finally, he wishes to thank his wife, Mary Ann, for her special

aid and encouragement.

ii

 

 

 

TABLE OF CONTENTS

INTRODUCTION ...............................................
Chapter

I. PRELIMINARIES ....................................

Section (a): Historical Background ...........

Section (b): Basic Homological Results .......

Section (c): Basic Ring Theoretic Results
Section (d): Properties of Separable
Algebras and the Module J ......
Section (e): Classical Results in the
Cohomology Theory of Algebras

II. EQUIVALENT CONDITIONS FOR AN ALGEBRA TO HAVE
FINITE COHOMOIDGICAL DIMENSION ...................
III. COMMUTATIVE AIGEBRAS OF FINITE COHOMOLOGICAL
DIMENSION ........................................
IV. EXAMPLES AND COUNTEREXAMPIES .....................
Section (a): Generalized Triangular Matrix
Algebras ........................
Section (b): Not Necessarily Projective
Algebras of Dimension One .......
Section (c): Eilenberg's Theorem .............
V. PRELIMINARY STRUCTURAL RESULTS ...................
Section (a): Separability of A/N ............
Section (b): Dimension of Residue Algebras
Section (c): Generalized Triangular Matrix
Algebras ........................
BIBLIOGRAPHY ..........

iii

 

Page

10
14

16

19

23

36

42

42

46
57

61

63
65

67

71

 

LIST OF TABLES

Table

iv

 

 

Page
47
48
52
53

55

 

 

 

'INTRODUCTION

This thesis is a study of finitely generated algebras
(over a commutative ring) of finite cohomological dimension.
It began as a study of those algebras which have cohomological
dimension one. However, it soon became apparent that the most
interesting questions required some knowledge of algebras of
cohomologically finite dimension. Thus the breadth of the
question had to be increased at some expense to the depth.

By an algebra A over a commutative ring R we mean
a ring A together with a ring homomorphism e mapping R
into the center of A. For this paper, all rings have an
identity and all ring homomorphisms carry the identity onto
the identity. By a finitely generated R-algebra or a projec-
tive R-algebra, we will mean that the algebra is finitely gen-
erated or projective as a module over R.

We say that a left module M over a ring R has (left)
homological dimension n (hdR(M) = n) iff

(i) there exists an exact sequence with the Pi
R-projective of the form: 0 a Pn a Pn-l a...» P0 a M a O --
such a sequence is called a projective resolution of M of
length n -- and

(ii) there is no projective resolution of M of

length less than n.

 

 

 

Given an algebra A, we define the enveloping algebra
A8 to be the algebra A 8% A*, where A* is the ring opposite
to A. A is a left Ae-module under the operation
(a ® a')-x = axa'.

Now we define the cohomological dimension of A to be
the homological dimension of A considered as a module over
its enveloping algebra. We denote this dimension by R-dim A.

In Chapter I, we give the historical background of the
problem and the basic results needed in the rest of the thesis.

The main result of the thesis is contained in Chapter

lap-flh‘ .‘.| '. 'fi I I,
. - , _. "J .
, . an-

n. -

“15:"?9'3” '

II and yields many of the results contained in later chapters.
The theorem generalizes to arbitrary cohomological dimension

3 result of Endo and Watanabe.

MAIN THEOREM: Let A be a finitely generated, projective
algebra over a commutative ring R. The following are equivalent:
(a) R-dim A = n
(b) Rm-dim Am 5 n for all maximal ideals to of R
with equality holding for some 'm, where Rm is the localiza-
tion of R at m.

(C) R/m—dim A/ro A s n for all maximal ideals m of

R with equality holding for some m-

Under suitably strong hypotheses, we are able in

Chapter III to reduce the study of finitely generated algebras

0f cohomblogical dimension n to the study of finitely gen-

. . . e O
erated algebras of cohomological dimen51on n over their cent r

 

 

 

 

 

 

We are also able to conclude that there exist no finitely gen-
erattxi, projective, commutative algebras of non-zero finite
cohomological dimension over any commutative ring.

Chapter IV is devoted to examples and counterexamples.

Finally, in Chapter V, we give some preliminary results
on the structure of finitely generated algebras of finite
cohomological dimension. Our best result shows that finitely
generated, one-dimensional algebras over local hensel rings are

essentially a type of upper triangular matrix algebras.

 

 

 

 

-CHAPTER I

PRELIMINARIES

This chapter is devoted to giving the historical back-
ground of the problem and the basic information needed for
understanding the remainder of the thesis. We will prove only
those results whose proofs are not clear in the literature.
References will be given only for those results which are gen-
erally not well-known.

For the sake of clarity, we divide the chapter into
five sections: (a) Historical Background, (b) Basic
Homological Results, (c) Basic Ring Theoretic Results, (d)
Properties of Separable Algebras and the Module J, and (6)
Classical Results in the Cohomology Theory of Algebras. We

now proceed with:

Section (3): Historical Background

In 1945, Hochschild defined a cohomology theory for
associative algebras over a field using what has come to be
known as the "bar resolution." [19] An algebra A over a
field R is called classically separable if A'GR K is
semi-simple for every field extension K of R. Hochschild
was able to show that such algebras were characterized by the

fact that R-dim A = 0.

 

 

 

In their book [9], Cartan and Eilenberg extended the

defidlition of the cohomology theory to associative algebras
over commutative rings by using the functor Extne (A, ). This
theory agrees with thatdefined by the bar resolution in the
case that A. is projective over the ground ring R. Thus
the fact that many of the results we will prove connecting
the dimenSion of an algebra with the dimension of the residue
algebras over the residue fields require the assumption that
the algebra is projective is not surprising. Cartan and
Eilenberg commented that "the method utilized leads to new
connections with the theory of algebras, and deserves further
study."

In 1954, Eilenberg published a paper [11] on algebras
whose cohomological dimension was finite. He was able to char-
acterize completely those finitely generated algebras over a

field which have finite cohomological dimension. The main

theorem which he proved, I have called throughout the thesis:

The Eilenberg Theorem (1.1): Let R be a field and
A be a finitely generated R-algebra with Jacobson radical
N. Then R-dim A = n iff A/N is R-separable and

hdA(N) = n-l.

This has proved to be the best homological characterization

of algebras of finite cohomological dimension over a field.

In fact from this theorem and from a later work of M. Auslander
[1], one is able to deduce that, in the case of a finitely

generated algebra over a field with the algebra modulo its

 

 

 

 

 

radical separable, the cohomological dimension of A is equal

to the global dimension of A, where by g1. dim A we mean the
supremum of the (left) homological dimension of M over all
left A-modules M. Thus, for algebras over a field, the study
of finite algebra dimension essentially reduces to the study

of the global dimension. One would then hope that the algebra
dimension Would be significant in studying algebras over an
arbitrary commutative ring.

The case of infinitely generated algebras over a field
has been studied by Rosenberg and Zelinsky [30]. Their work
provides us with many examples which show the necessity of the
hypothesis of finite generation in many of the theorems proved
in this thesis.

A series of ten papers "On the Dimension of Modules
and Algebras," published in the Nagoya Mathematical Journal
from 1955 through 1958, indicates certain significant results
concerning algebras of cohomological dimension one. Two of
the significant results show that for a finitely generated
algebra A over a field R: (l) the dimension of A being
less than or equal to one implies that the dimension of any
residue algebra is finite [14] and (2) that if A has the pro-
perty that all of its residue algebras have finite dimension,
then A is the image of a finitely generated algebra B of
dimension less than or equal to one with B/J(B)2 fl A/J(.A)2 [24].

These results have recently been generalized by
Abraham Zaks to semi-primary algebras, but the cohomological

dimension has been replaced by global dimension. [32]

 

 

 

 

The most significant structural results about finitely
gEIlerated algebras over a field R of cohomological dimension
one were announced by Harada in a 1966 paper [18]. He defines

a generalized triangular matrix ring

mll m12 mln
r (11.; M..) -= “’22 m in R.; m. in M..;
n 1. ij . ii i j ij
0 m .
nn
i = 1, ,n
to be a set of rings R1 and a set of left Ri’ right Rj-

bimodules Mij together with left Ri-, right Rj- homo-

j ° M ® M «M with the k and the i

ik' ij Rj jk ik ‘Pik ‘Pik
J

ik

morphisms m

. . g L .
isomorphisms such that cp ® idM id ® (pjk, with the

M..
kt 1]

operations in Tn(Ri; Mij) defined by

(mij) + (“1'”) = (mij + m'ij)

. l = j t
(mij) (mjk) (Z (pikonij ® mjk»

j

Using this definition, he has shown the following theorem

which we will call:

Harada's Theorem (1.2): If Tn(Ri; Mi ) is semi-primary,

 

 

j
then g1. dim (T (R.; M. )) s 1 if and only if (a) g1. dim R. s 1
n i ij . j-l i
for every i, (b) Rik are monomorphisms; (c) iszij=t§iMitMtj,
then Mij /Mij is Ri-projective; and (d) Mij NH 6-) Mi,i+lMi+l,j ®

@ Mi,j-1Mj-l,j 6-) 141ij as Ri-modules where Nj is the

Jacobson radical of R .

J

 

8

Recall that one calls a ITng R hereditary iff
$1. dim R s l. .

S. U. Chase in a much earlier article [10] defined a
similar concept of generalized triangular matrix rings.

In addition to the structural results concerning algebras
over a field of cohomological dimension one, it seemsreasonable
to consider the formal properties of algebras (over a commutative
ring) of cohomological dimension n in view of the success
that Auslander and Goldman have had in determining the formal
properties of separable algebras (i.e., algebras of dimension
zero) over a commutative ring in "The Brauer Group of a
Commutative Ring." [3] One of their major reSults is the so-

called

Reduction £2 the Central Case (1.3): Let A be a finitely

generated R-algebra and let S be the center of A. Then A

is R-separable iff A is S-separable and S is R-separable.

One of the most useful results in the theory of
separable algebras allows one to determine the separability
of an algebra A from the separability of the residue algebras
A/mA over the residue fields R/m of the ground ring R.
The result we give is due to Endo and Watanabe [17]. A
slightly weaker version was proved in Auslander and Goldman

[3] .

 

 

 

 

 

9

 

“Ir/hi Lndg-Watanabe Theorem (1.4): Let A be a finitely generated
gralsebra. The following are equivalent:
(a) A is R—separable.
(b) Art, is Rmsseparable, for every maximal ideal m
of R.
(c) A/hA. is R/m—separable, for every maximal ideal

m of R.

It is this result which we generalize to arbitrary n
under the stronger hypothesis of A being R-projective. We
are also able to reduce to the central case for arbitrary n
under similarly strengthened hypotheses.

By the Eilenberg Theorem, we know that the Wedderburn
Principal Theorem holds for finitely generated algebras of
finite cohomological dimension, if the ground ring is a field.
For arbitrary ground rings, work relating to the existence of
inertial subalgebras [322, Chapter V] from the point of View
of the ground ring has been done by Azumaya [4], Ingraham [21,
23], Brown [7], and Brown-Ingraham [8]. We are able to find
some results concerning the existence of inertial subalgebras
of algebras of finite cohomological dimension using their
results. One should note that there is no finitely generated
algebra of finite cohomological dimension in this thesis which

does not contain an inertial subalgebra.

 

 

 

1C)

m 03): Basic Homological Results

 

This section is devoted solely to giving the basic
information about homological algebra--tensor product, pro-
jectives, and resolutions--that will be needed in the remaining
chapters.

A module P is said to be projective over a ring R
(or R-projective) if and only if P is a direct summand of
a direct sum of copies of R.

One of the most important alternate definitions is

given by the so-called

Dual Basis Lemma (1.5): A module P is R-projective iff

 

' 'th f ' d
there exists a set {fa’ xa} W1 0 in HomR(P,R) an
x in P such that E f (x)x = x for every x in P
0' 0’ 0’ oz
where the sum is finitely non-zero.
Trivial applications of this lemma yield the following

- results which will be used without mention throughout the rest

of the thesis:

Proposition 1.6: If A is R-projective and B is S-projective

 

‘where R and S are T-modules, R, S, and T are commutative

rings, then A 8% B is R 8% S-projective.

Proposition 1.7: If A is B-projective and B is R-projective,

then A is R-projective.

 

 

 

1].

Saunders MacLane proves several results about iso-

morphisms of tensor products which we must use several times.

The Pull-Back Lemma (1.8): Let R and S be rings, 9: S a R
be a ring homomorphism, G be a right R-module, and A be a
left R-moduleb Let G and A be S-modules under the structure
induced from 9. Then there exists a natural homomorphism

9#3 G 8% A a G 8R A; and, if e is an epimorphism, e# is an

isomorphism. [26, p. 140]

The Middle Four Interchange (1.9): Suppose R,S,T, and V
are K-algebras. Let A,B,C, and D be K-modules which satisfy
the following properties: A is a right R-S bimodule, B
is a left R- and right T- bimodule, C is a left S- and a right
T-bimodule, and D is a left T-V bimodule. Then we have

the following isomorphism:

(A®RB) ®S®Kv (C®TD)g (A®S C) ®R®KT (B®VD).

[26, p. 194]

In particular, since we must use the enveloping algebra
frequently, we indicate two properties which follow from the
above results. First, however, we must standardize some nota-
tion. Where confusion may arise as to the base ring, we will

tarite A 8B A* 3 (A):. Ae will always mean A 8% A*.

jggoposition 1.10: If S is a commutative R-algebra and A

is an R-algebra, then Ae 8R S g (A 8R 8):.

 

12

Proposition 1,11; If “m is an ideal Of R, then A/m A =
e

e
A sR 11/91 and (A ®R R/QI)R = (A 8R show” .

Next, let us recall that a projective re501ution X

of a left R-module M is an exact sequence

.. a X a X a ... a X a X a M a 0
n o

1

where each Xi is R-projective. We may thus restate the

definition of a left R-module M having (left) homological
dimension equal to n by saying that M has a minimal pro-
jective resolution of length n. We will find the following
results relating homological dimension and projectivity very

useful.

Proposition 1.12: Let M be a left R-module, n 2 0, and let
0 a X 4 Pn-l a ... a P1 4 P0 a A a 0 be an exact sequence
where each Pi is R-projective. If hdR(M) s n, X is also

R-projective. [29, p. 135]

Proposition 1.13: Let 0 a M' ~ P a M a 0 be an exact
sequence of left R-modules, where P is R-projective. If

M is not projective, hdR(M) = l + hdR(M'). [29, p. 137]

We will have occasion to use the so-called torsion
functor, Tor§(N,M). Let X be a projective resolution of
the left R-module M and let N be a right R-module. De-
fine

ker (N 8% Xn ~ N 8k xn-l)

R
Tor (N,M) = .
n 1m (N 8B xn+1 d N 8B xn)

 

213

We can now define the weak homological dimension of a left

R-module M in terms of the torsion functor: we say that a

left R-module M has weak homological dimension equal to n

R

n+1(N,M) 3 O for all

(denoted by w.hdR(M) = n) iff Tor
right R-modules N and there exists an No such that
TorRGN ,M) * 0.
n o
In particular, we say that a left R-module M is flat

iff w.hdR(M) = O. This is equivalent to the following condition.

Proposition 1.14: M is R-flat iff for every R-exact sequence

 

O-aA-oB-oCaO, the sequence O—+A®M-B®M-C®M-O

is also R-exact.

We call a left R-module faithfully flat if the following
condition is satisfied: A a B a C is exact iff
A®M~B®M~C®M is exact.

Finally, we call a module M over R finitely pre-
sented if there exists a finitely generated free R-module F
so that there is a short exact sequence 0 a K a F 4 MI» 0
with K finitely generated. We note that over a noetherian

ring, any finitely generated module is finitely presented.

Proposition 1.15: If a module M is both R-flat and R-

 

finitely presented, then M is also R-projective [6, p. 64]

If we suppose that A is a finitely generated, pro-
jective R-algebra, then every term of an Ae-projective resolu-

tion of A is R-projective; furthermore, we may always choose

 

14-

a resolution.of A to be finitely ngenerated. Lastly, if
g. is an R-projective resolution of A of any length, then
for every R-module ‘M, X_8h M is exact.

In concluding, we quote a categorical result of

Northcott [29, p. 93]:

Theorem 1.16: Let T(A) be an additive exact functor of a
single variable module and let X. be a complex (i.e., a
sequence ... a Xn a Xn_1 « ... a X0 4 X a 0 such that
im.(Xn+1 a Xn) is contained in ker (Xn a Xn_1)). If T is
covariant in A, for each n we have a canonical isomorphism

Hn(T(§)) = T(Hn(X)) which determines a natural equivalence

as functors of X,

Section (c): Basic Ring Theoretic Results

In this thesis, by a ring (or algebra) R being semi-
simple we shall mean that the Jacobson radical j(R) of R
is zero and that R has the descending chain condition on
left ideals. To indicate merely that J(R) is zero, we shall
call R semi-primitive. Finally, a ring R is semi-primary
ifflR/J(R) is semi-simple and 1(a) is nilpotent. This
terminology is standard in papers on the cohomology theory of
algebras.

We will need the following result which Ingraham [21,
p. 78] has proved relating the Jacobson radical of a ring R
‘with the Jacobson radical N of a finitely generated algebra

A. over R:

 

 

 

 

15

Prgposition.l.17: If n denotes the intersection over all
m in the maximal spectrum of R, 0(R), we have the following:
(a) J(R)A C N
(b) There exists a positive integer] n such that
Nn s new).
(c) If A is R-projective, J(R)‘A “DOOM

(d) If A is R-separable, N = n(mA).

Since many of the results we obtain are generalizations
of results in the case of algebras over a field when we assume
that the ground ring is a local hensel ring, we give some basic
results concerning such rings.

A local ring R with maximal ideal m is called a
hensel ring if every monic polynomial f in R[X] whose
image f in R/m[X] factors as f=goho, where g0 and
ho are relatively prime, monic polynomials in R/m[X], has
a factorization f = gh in R[X] with g and h monic,
§=go, and h=ho Where g— and h are the natural images
of g and h in R/m[X].

It is well-known that for every local ring R, there
exists a local, commutative R-algebra R such that:

(a) R is a hensel ring

(b) R is a faithfully flat R-module

(c) mR is the maximal ideal of R with R/mR = R/m-

'The proofs for these results are to be found in [28].

 

 

1.6

The most important property 0f local hensel rings for
our work is that finitely generated algebras over local hensel
rings satisfy an idempotent lifting property. The work on this
property is due to Azumaya [4] and the reSult will be quoted
when it is needed in Chapter II.

Also needed is the concept of localization. Given a
multiplicative subset S of a ring R, we can form the R-
algebra RS which is flat as an R-module. If S is the
complement of a maximal ideal m, we write Rm instead of RS
and we note that Rm is a local ring with the property that
Rm/mRm = R/m. [229' 6]

We close this section with the extremely useful reSult

which has come to be called

Nakayama's Lemma (1.18): Let R be a commutative ring. If
M is a finitely generated R-module and mm = M for every

maximal ideal m of R, then M = 0.

Section (d): Propgrties gf Separable Algebras and the Module J

The cohomological definition of separability over a
field R, namely, that R-dim A = 0, provides one with the
‘extension of that definition to algebras over a commutative
ring. In particular, A is said to be separable over R iff
A is projective as a module over its enveloping algebra Ae.
‘We have already given two major reSults in the historical back-
ground concerning separable algebras: the Reduction to the '

Central Case and the Endo-Watanabe Theorem. Since part of our

 

   

17'

work is directed toward obtaining formal properties of n-dimensional
alg8bras in a similar manner to that used in deriving them for

separable algebras, we find it necessary to state several results

on separable algebras. These results can be found in the notes

of Ingraham and DeMeyer [23].

Proposition 1.19: An algebra A is separable over a field R

iff A is classically separable over R and (AzR) < a.

S and S be commutative R-algebras.

Prgposition 1.20: Let 1 2

Let A1 be any Sl-algebra and A2 be any Sz-algebra such that

A 8> A is separable as an 81 8R Sz—algebra. Then, if A2

1 R 2

is a faithful R-module and contains R as an R-direct summand,

A is S -separable.

l 1

Proposition 1.21: Let A be a separable R-algebra and let I

be a two-sided ideal of A. Then A/I is R-separable.

It would be hoped that R—dim A = n would imply that

lt-dim.A/I s n. It is unfortunate that this result is false

(even in the field case. We will give an example of this in

Cha pter IV .

Proposition 1.22: Let A be an R-algebra which is an R-pro-

generator. Then 7R is an R-direct summand of any subalgebra

of A. (We define a module M over a commutative ring R to

be at! Reprogenerator iff M is finitely generated, projective

and faithful as an R-module.)

 

 

13

92.20 OSition 1.23:‘__If A is a separable R-algebra, every

Armodule which is R-projective is also A-projective.

All of the above results are either needed in the re-
mainder of the thesis or else they are generalized in it.

We let J denote the kernel of the map p, in
Hom e(Ae, A) defined by “(a ® a') = aa'. Since p, is a
left: Ae map, J is a left ideal and one easily verifies
that J is Ae-generated by the elements {x ® 1 - 1 ® x:

x is in A}. Clearly J is finitely Ae-generated by
{xi (8 1 - 1 ® xi} if {xi} is a finite R-generating set
for A.

J plays a special role in both the theory of separable
algebras and one-dimensional algebras. For separable
algebras, A is R-separable iff Ae contains an idempotent
f Such that p,(f) =‘ l and J-f = 0. It is clear that
R-dim A = 1 iff J is Ae-projective, but not a direct summand
of Ae.

One notes that several results require that J be
finitely Ae-presented. We have not been able to determine
whether or not this is always true if A is finitely R-
generated.

Lastly, one notes that A8 5 J 6 A as R-modules for
0 -. J —» A894. A _. 0 is always R-split by the R-homomorphism

p defined by p(a) = a ® 1.

 

l9

seetion (e): classical Results _ijl_ % (1011011101032

Theory pf Algebras

In this section we present or, at least, indicate the
nature of several classical results which have not been stated
elsewhere in this chapter. We give them to complete the View
of what has been done previously on the cohomology theory of
algebras. Some of the results will be used again. Others
will not. Although most of the interesting results are drawn
from the series of papers "On the Dimension of Modules and
Algebras," we begin with two reSults contained in Cartan and

Eilenberg [9, Chapter 9].

Proposition 1.24: Let A be a projective R-algebra and S
a commutative R-algebra. Then R-dim (S 8R A) s R-dim A.
If further the natural mapping R _. S is a monomorphism of

R onto an R-direct factor of S, then equality holds.

Proposition 1.25: Let A and B be R-algebras and A63

their direct sum. Then R-dim (A 63 B) = max(R-dim A, R-dim B).

Maurice Auslander in [l] proves an interesting theorem
which we use immediately to demonstrate more clearly a result

whose proof is sketched in [14].

Theorem 1.26: If A is a semi-primary ring with Jacobson

radical N, then g1. dim A = hdA(A/N) = 1 + hdA(N).

 

     

   

20

EADOSition 1.27: If A is an algebra over a field R with
@611) < as, then (a) R-dim A < 00
implies (b) R-dim (A/N) = 0

implies (c) R-dim A = 1 + hdA(N) = hdA(A/N) = g1. dim A.

That (8) implies (b) follows from the Eilenberg Theorem.
That (b) implies (c) follows from the preceding result together

with the following theorem of Eilenberg [11]:

Theorem 1.28: If R is a field, (AzR) < m, and A/N is

 

separable, then R-dim A = hdA(A/N).
Proposition 1.27 has the following Corollary:

Corollary 1.27.1: If (b) holds for A and I is a two-sided

 

ideal in A, then R-dim(A/I) = g1. dim (A/I).

This follows by noting that A/I modulo its radical is

a homomorphic image of A/N which is separable.

In [2], Auslander has shown Proposition 1.27 under

perhaps slightly weaker conditions:

jProposition 1.29: If A is a semi-primary algebra over a
field R with radical N, R-dim A < co, and (A/NzR) < on,

then g1. dim A = R-dim A.

Certain formal properties of cohomological dimension

have: been proved in [16] using spectral sequences:

floposition 1.30: If A and B are R-algebras and B is

R-f lat, then R-dim A ® B s R-dim A + R-dim B. Furthermore,

¥

 

 

 

21

if ‘B is R-projective and contain5 R as an R-direct summand

’of B, then R-dimA sR-dimA®B.

Proposition 1.31: If A is an S-algebra and S is an R-

algebra with R-dim S = 0, then R-dim A s S-dim A.

The most interesting result contained in that paper is
one concerning upper triangular matrices over a commutative

ring R.

Proposition 1.32: If A = Tn(R) is the n X n upper triangular
matrix algebra over R, then R-dim A = 1 and for every R-

algebra B, R-dim A ® B = R-dim A + R-dim B.

From our main reSUlt, we may obtain rather easily the
fact that R-dim A = l, but the additivity result we do not
obtain.

We may also note that the grOup algebra of a finite
group will not provide us with any useful examples, for
Eilenberg and Nakayama have shown in [13] that a frobenius
algebra A or a quasi-frobenius algebra A satisfies the
property that R-dim A = 0 or R-dim A = a.

In the papers of the series that we have not yet
rner1tioned, rather precise estimates are obtained on the
dimension of certain residue algebras [15], the dimension of
dire<:t limits is discussed [5], and an example is given of

an ailgebra which is right but not left hereditary [25].

 

22

It was Chase, who, in hiS ‘attempt to discover a ring
which was left but not right Egg-hereditary, introduced the
concept of generalized triangular matrix algebras which Harada
used so effectively to delineate the characteristics of

algebras of dimension one.

 

— ._ av _—wi.m _,

   

CHAPTER II

EQUIVALENT CONDITIONS FOR AN AlGEBRA
TO HAVE FINITE COHOMOIOGICAL DIMENSION

Our main objective in this chapter is to obtain con-
ditions under which the cohomological dimension of the residue
algebras at maximal ideals of the ground ring determines the
cohomological dimension of the algebra. This is eSpecially
important since, for algebras over a field, Eilenberg's
Theorem gives necessary and sufficient conditions for an
algebra to have cohomological dimension n.

Certain results will be specialized to algebras of
cohomological dimension one where the hypotheses can in many
cases be somewhat weakened.

For the remainder of the thesis, R and S will be
commutative rings; A and B will be not necessarily
commutative algebras.

The main result of this chapter will be:

lflneorem.2.1: Let A be a finitely generated, projective
R-algebra. The following are equivalent:
(a) R-dim.A = n
(b) Rm-dimAm s n for every m in 0(R). Equality
holds for some m.
(c) R/m-dim A/roA .S n for every m in 0(R). Equality
holds for some m.

23

 

 

21k

The following fact Will be ”sad Without mention in

the remainder of the thesis:

Proposition 2.2: If A is an R-algebra and u is an ideal
of R contained in the annihilator of A, then
R/fl-dim A = R-dim A.

: A* = A A*.
Proof A 8k ®RAH

 

23:“: _ _

Proposition 2.3: (a) ASSume Sl-dim A1 = n and SZ-dim A2 = 0

where A and A are reSpectively 81- and Sz-algebras and

1 2 .'
I.

both S1 and 32 are R—algebras. Assume further that A2 is
is R-flat or that S1 is R-projective and A1 is Sl—projective.
Sl ®R 82-dim A1 ®R A2 s n.

(b) If Sl-dim A1 = 1, then S1 QR SZ-dim A1 (8 A2 5 l.

' —o —0 ... -¢ -' -0 A —-0
Proof. (a) Let 0 Pn P1 P0 1 0 be

a minimal (A1):1-projective resolution of A1. Clearly, every
element is R-projective if (A1): is R-projective and hence
( ) QR A2 is exact for this sequence; if the other hypothesis
holds, tensoring with A2 is exact by definition of being

R-flat. Since A is (A -projective,

e
2 2)S2
—. -o . . . —o —-v —0 A A '9
0 PH GR A2 P1 ®R A2 PO ®R A2 1 ®R 2 0
. e . .
is an (A1 QR A2)Sl ®R 82-projective resolution of A1 ®R A2.

(b) o _. J —. (A1): _. A1 —. o is a minimal projective
1

resolution of A1. Since this sequence is R-split, tensoring

With A2 over R is exact for this sequence.

The following corollaries to Proposition 2.3 are of

spec is 1 interest:

 

 

 

   

25

Mary 2.3.1: Let R-dimA = n: and s be a flat,
commutative R-algebra. Then S-dim A ® 8 s n.
Proof: Set S1 = R, 82 = S = A2 in the proposition

and note that S-dim S = 0.

Corollary 2.3.2: Let R-dim A = n, A be R-projective, and

 

S be any commutative R-algebra. 'Then S—dim A ® S s n.

Corollary 2.3.3: Let R-dim A = l and S be any commutative

 

R-algebra. Then S-dim A ® S s 1.

..g-_n_m_._- ._ _
I ‘ ‘ . u -«

We include the following two corollaries as a case of
Corollary 2.3.3 which has special interest and which will be
used strongly in considering commutative algebras of R-dimension

one .

Corollary 2.3.4: Let R-dim A = l and let A be finitely

 

generated over R. For every m in 0(R), R/ro-dim A/roA s l .
and equality must hold at some maximal ideal.

M: Set S = R/m in Corollary 2.3.3. Equality
follows from the fact that, since A is not separable, there
is a maximal ideal to such that R/m-dim A/mA 2 1 by the

Endo -Watanabe The orem .

Corollary 2.3.5: Let R-dim A = 1 and A be finitely gen-
erated over R. For every m in 0(R), Rm-dim Am s l, and
EQUality must hold at some maximal ideal.

Proof: Again apply the result of Endo-Watanabe.

 

26

The next set of corollaries Show that, for R-dim A - 1,
vae ““33’ Sometimes obtain equality USing properties of separable

algEbras.

corollary 2.3.6: Assume the hypotheses of Proposition 2.3(b).

1f A2 is R-faithful and contains R as an R-direct Summand,

then S1 8h SZ-dim A1 8k A2 = 1.

Proof: If the S1 ® SZ-dim A 8 A2 = 0, then A

1
Sl-separable by Proposition 1.20. Contradiction.

is

1

Corollary 2.3.7: Let S be a commutative R-algebra containing

R as an R-direct Summand. If R-dim A = 1, then S-dim A 8R S = 1.
Proof: Apply the preceding corollary with R = 31’

A = A1, and S = A2 = 32'

Corollary 2.3.8: Assume that A1 is an Si-algebra for i = 1,2.

If S1 8h SZ—dim A1 8k A2 = l and S -dim A = 0, then

1 l

S -dim A 2 l.

2 2
Proof: Apply Proposition 1.20.

Our last corollary concerns algebras over a special
type of ground ring. A ring R is called absolutely flat

iff every module over R. is flat.

Corollary 2.3.9: Assume that A is an algebra over an
absolutely flat ring R. Let S be any commutative R-algebra.

R-dim A = n implies S-dim A 8% S s n.

 

 

27

We will now prove three lemmas on weak homological
dimensiOn as a prelude to proving the equivalence of (a) and
(b) in Theorem 2.1. The proofs are similar to those in [29,
P‘ 153 ff.]. Although we aSSume that the algebra in the lemmas
is an enveloping algebra, it is easy to see that they remain

true for any R-algebra.

Lemma 2.4: Let M be a right and N be a left Ae-module,
A an R-algebra. Consider M and N as R--R bimodules in
the uSual way. Let S be a multiplicative Subset of R.
Ae (A e
n - u _ E S .
Then there 18 an isomorphism. [Torn (M,N)]S 'l‘orn 8’ NS)
Proof: We note that MS ® e NS =
(A3)
2
(M q, RS) 8 e (N on RS) (M o e N) ®R (RS ®R RS) by
RS A S
the middle four interchange E (M ® e N) 8R RS. This iso-
A
morphism is a functor equivalence. Let E be a projective
resolution of the Ae-module M, so that .. F1 ~ F0 _. M .. 0

is Ae-exact. Since RS is R-flat, we obtain the (AS)e-exact
sequence _. (F1)S - (FO)S - S —o 0 which is an (As)e-

projective resolution of MS. By the first isomorphism in

the proof, we obtain the functor equivalence of
* _. —. N -.
C ) (Fm)S 8 e NS (le)S 8 e «S

(As) (AS)

a _. —o -O ...
nd (**) . . . (Fn ®Ae N) ®R RS (Fn_1 ®Ae N) ®R RS

Hence, passing to homology yields from (*) and (**):

e
T (AS) _ _ _
o E = .
tn (“8’ NS) Hn(}£®R RS) where E £®Ae N
Again, since RS is R-flat, tensoring with RS over
R is exact. Hence, by Theorem 1.16, we have the isomorphism

 

28

1’1 0: ® 2 "' -
n R RS) HnOL) ®R RS or, equivalently
(A )e e
rt“ S ( N ) E [TorA (M N)
n MS’ S n ’ 18'

The lemma is the basic tool needed to prove:

Lemma 2.5: If A is an R-algebra, m is in 0(R), then for

every left Ae-module N, w. hd N s w. hd N.
e m e
A A
Proof: It is sufficient to aSSume that w. hd e N 8 n

e A
, e
and to show that for every right Am-module M, Tor:21(M,Nm) = 0-

We note that PLO = M as Ari-modules. We apply

A9 A6
' m E =
Lemma 2.4 to obtain Torn+1(Mm, Nm) [Torn+1(M,N)]m 0.

Lemma 2.6: If A is an R-algebra and m is in 0(R), then

Ae (Nm) .

YD
Proof: By the preceding lemma, we need only show

for any left Ae-module N, w. hd e(N) = sup w. hd
A m

that w. hd (N) S SUP W- hd (N ). We may again assume
e m Ae m

I'D
that sup w. hd e(Nm) = n. Let M be any right Ae-module.
A \
Ae .m ,. e ' . > >
Then [Torn+1(M,N)]m = Torn+1(Mm, Nm) = O, for every maximal

e
ideal to of R by Lemma 2.4. Whence Torfi+1(M,N) =0 since

it is zero when it is localized at every maximal ideal.

With this tool we are now able to prove the equivalence

0f (a) and (b) of Theorem 2.1.

_Proposition 2.7: Let A be a finitely generated, projective
R‘éllgebra. The following are equivalent:

(a) R-dim A = n

 

 

29

(b) Rm-dim Am 5 n for every m in 0(R). Equality
holds for some m.

M: We proceed by induction on n. The basis of
the induction is the Endo-Watanabe Theorem. So we may assume
that the proposition is true for all integers less than n.

a im lies b : The inequality follows from Corollary 2.3.1.
ASSume that equality did not hold for any to. By the inductive
hypothesis, R-dim A < n. Contradiction.

b im lies a: Assume that 0 —oKn _. Pn_1-+ - Po-oA-o 0
is an Ae-resolution of A with P0""’Pn-1 projective.

Since A is finitely generated and R-projective, the Pi's
and KH may be chosen so that they are finitely generated

and R-projective [yid_., 29, p. 135].

Now (Kn) is Area-projective for every :9 since

:9
RYo-dim Am s n. Thus w. hd e(Kn) = Sup w. hd emu)“, = 0.
A to m
So K is Ae-flat. But since K is R-projective, Kn is
n n
Ae-finitely presented. Whence Kn is Ae-projective. There-

fore, R-dim A s n. Equality follows from the inductive

hypothes is .

Corollary 2.7.1: If Rm-dim Am S l for every m in 0(R)
and equality holds at one :3 and if A is finitely generated
and J is Ae-finitely presented, R-dim A = 1. Further, J

is Ae-finitely presented if A is R-projective or if R is

“Ce therian .

 

30

We now begin the proofs of several results which will
yield the equivalence of (b) and (c) in Theorem 2.1. As in
the Endo-Watanabe Theorem, we need some basic facts about lift-

iog projectives over local hensel rings.
Lemma 2.8: If R is a local ring with henselization (or

completion) R and A is a finitely generated R-algebra,

then for any left Ae-module B, w. hd e B = w. hdae B, where
A A

M a: M ®R R for any R—module M.

Proof: Let N be any right Ae-module with Ae-pro-

jective resolution ... Fn .. Fn-l —+ —' F0 - M - 0. By

the flatness of R, we obtain that

...—oPnrfinld-u—of‘ —oM-0 is anAe-resolution of M.~

By the middle four interchange, we have the following isomorphism:
A " E A
(Fn®RR)®e “(BQRM (Fn®eB)®RR
A ®R R A
Thus we obtain:

(*) ...—~F o'BaF a B—oF s B—o...
n-l-l Ae n Ae n-l Ae

and

** d A _’ ". A -. o o o
()H' Fn+1®eB®RR Fn®eB®RR Fn—1®eB®RR
A A A -
Ae .
Now Torn (M,B) = 0 iff (*) is exact iff (**) is exact since

A

e
R is R-faithfully flat iff Tor: (M,B) = 0. Whence the lemma

150110915 after noting that if M is an Ae-module, then M = M

under the induced Ae-Structure.

 

 

 

     

31

fro 03 it ion 2.9: Let R be a local ring with henselization

(or completion) R. Let A be a finitely generated, projective

g‘alsebra. Then R-dim A = n iff R-dim A - n.

Proof: We proceed by induction on n. R-dim A = 0
clearly implies that R-dim A = 0. Now R-dim A = 0 implies

that w. hd e A = 0. Now, by Lemma 2.8, w. hd A = w. hd A - o.
. Ae Re

A
e
Whence A is A -flat. But since A is finitely generated

and R-projective, it is also Ae-finitely presented. Therefore

R-dim A = 0.

Now assume that the proposition is true for all integers

less than n.
R-dim A = n implies R-dim.A s n by Corollary 2.3.1.

Equality follows by the inductive hypothesis. .Since A is

a P i» A a 0

R-projective, if R-dim A = n and 0 a Kn a Pn-l q ... 0

is a finitely generated Ae-resolution of A with the Pi

Ae-projective, Kn is Ae-flat by Lemma 2.8 and is clearly

Ae-finitely presented. Whence R-dim A.s n. Equality follows

as usual from the inductive hypothesis.

We now state a theorem of Goro Azumaya [4, p. 138] on

llifting idempotents in algebras over local hensel rings:

$orem 2.10 (Azumaya): Let A be an algebra over a hensel

I be a two-sided ideal in A. Then for any

1fling R and let
SEVEBtem of mutually orthogonal idempotents 51,...,en in A/I,

e1,OOO,e

tliéire is a system of mutually orthogonal idempotents
n

it‘ A. such that each ei maps onto ei under the natural

map

 

 

 

32

We now apply this proposition to obtain a lifting. theorem

for Projective modules:

910 osition 2.11: Let A be an algebra over a local hensel
ring R. Let I be an ideal of A. Let L be a finitely
generated, projective A/I-module. Then there exists a finitely
generated, projective A-module P such that P/IP E L.

gro_of: Let F = Lee M be a free, finitely generated
X (= A/I)-module and F be the free A-module such that
(EA) = (FzA). Let 11: F —o P- be the natural map. Now
H0m2(1‘:,1‘:) and HomA(F,F) are alSo finitely generated R-algebras.

Let e .,en be afixed basis of F. Then any at in

1,..

HomA(F,F) may be represented as (aij) with the or in A.

i3

Define an algebra epimorphism ¢z HomA(F,F) - Hom_(E,E)

A
e =' ".e.thfll'd’aa
by ¢((Olij)) (Oliijd 1) (aij), 1 , e o owmg igrm
is commutative:
(aij)
M

F

6...)
_ ( 13 \_
IF

Since L and M are direct summands of P, there

(*)

 

ex ists a set of orthogonal idempotents in Hom_(E,F), namely,
the projections 5 and .F onto L and M rgspectively.
Then, by Theorem 2.10, these idempotents lift to two
Orthogonal idempotents in HomA(F,F), namely, p and 1‘.
From (*), 51': " 11p. Then, p(F)/Ip(F) = np(F) = 5n(F)

5(5) = L. Similarly T(F)/IT(F) = M.

 

33

Set p(F) = P. Then P is a finitely generated, pro-
jeetive A-module with P/IP '5 L since it is a direct summand

of a finitely generated, free A-module.

Let A be a finitely generated, projective

proposition 2.12:
R-algebra. The following are equivalent:

(a) R-dim A = n.

(b) R/m-dim A/roA s n for every m in 0(R). Equality

holds for some :9.
Proof: We again proceed by induction on n with the

Endo-Watanabe Theorem as the basis for the induction. ASSume

the proposition is true for all integers less than n.
If R-dim A = n, then R/m-dim A/mA s n by Corollary

2.3.2. Equality for some to follows from the inductive

hypothesis.
If (b) holds, by Theorem 2.7 and the fact that

A
A/mA ‘5 5%, we may assume without loss of generality that R

1'0
is a local ring with maximal ideal to and R/m-dim A/mA = n.

By Proposition 2.9, we may further assume that R is a hensel

ring. Now if 0 ... Kn -0 Pn-l _. ... P0 —. A—o 0 is a resolu-

tion of A, all terms finitely generated and R-projective, the
finitely generated, projective Ae/mAe-module Kn/mKn lifts
tO a finitely generated, projective Ae-module P such that

E : P/mP E Kn/mKn. We wish‘to show that Kn “=‘ P.

Consider the following diagram:

 

   

 

 

 

 

1” ~ " > PA”? > 0
'.

| -

| f f

Q,

o e

K“ ) 1(n/mKn , O

I

' l

I

A!

O 0

(a) Since 9 is an epimorphism and P is Ae-projective,
there exists an f in Hom e(P, Kn) such that fn = pf.

A
(b) Since p is an epimorphism, we know that

Im (f) +m Kn = Kn; so that m (Kn/Im(f)) = Kn/lm(f).

But Kn is finitely generated, R is local; hence by
Nakayama's Lemma, Kn/Im(f) = 0. Hence f is an epimorphism.
(c) Since Kn is R-projective, there is a map g in
HomR(Kn, P) so that fg ' idK . Therefore, f is R-Split.
So P = Kn o ker (f) as R-modurles.
Now n(ker(f)) = 0 implies that ker(f) : mP a m(ker(f))
C>mJ. ‘Whence ker(f) = m(ker(f)). Therefore, since ker(f)
is finitely generated-~being an R-direct summand of a finitely
generated, projective R-module P--we may apply Nakayama's

Leumm.to obtain ker(f) = 0. 80 P = Kn' Thus R-dim A s n

arui equality follows immediately from the inductive hypothesis.

This concludes the proof of Theorem 2.1. The theorem
[1&353 proved very useful in constructing examples of algebras
OVer arbitrary ground rings with a given cohomological dimension.

Such examples will be given in Chapter IV.

 

35

‘We close the chapter with a basic theorem on algebras
with infinite cohomological dimension. It follows directly

from Theorem 2.1.

Theorem 2.13: Let A be a finitely generated, projective

 

R-algebra. The following are equivalent:
(a) R-dim A B m.
(b) Where the supremum is taken over all m in
0(R), sup Rm-dim Am 1- on.
(c) Where the supremum is taken over all m in
0(R), sup R/m-dim A/mA = a.
2:22;; We prove (a) implies (b). The rest of the
implications follow in a similar fashion.
Assume (a) is true and (b) is false. Then there
exists an ni< m with sup Rm-dim hm = n. In fact this
must occur at some maximal ideal m. So by Theorem 2.1,

R-dim A = n. Contradiction.

 

 

CHAPTER III

COMMUTATIVE AIGEBRAS
0F FINITE COHOMOIDGICAL DIMENSION

In this chapter, we first study the existence of
commutative algebras of finite cohomological dimension. Then,
with the use of some additional hypotheses, we can reduce the
study of algebras of dimension n to the central case.

The first theorem is essentially contained in [1,

p. 75].

Theorem 3.1: If A is_a finitely generated, commutative
algebra over a field R, then R-dim A = 0 or R-dim A = a.
nggf: ASSume A is a finitely generated, commutative
R-algebra of non-zero, finite cohomological dimension n.
Then A is a semi-local ring with the descending chain con-
dition and hence J(A) dfiN is nilpotent. Therefore, A is
complete in its N-adic topology. By [33, p. 283],
A = A1 ® 9 Am where the Ai are complete local rings.
Now, by Proposition 1.25, we may assume that A is a complete
local ring.
Since A is a finitely generated algebra over a field,
n - dim A = g1. dim A by Proposition 1.27. So by [29, p. 208],
A, being a local ring of finite global dimension, is regular
(ideal-theoretically). But by [33, p. 302], a regular local
ring is a domain.

36

 

 

 

37'

‘Now, by Eilenberg's Theorem, R-dim A = n implies
that R-dim A/N = 0. Therefore N 5‘ o and N is nilpotent.
But this is impossible if A is a domain. Whence R-dim A = O

or R-dim A = co.

This theorem yields two results, one especially strong
result for commutative algebras A with R-dim A = 1 and a
slightly weaker result for arbitrary finite cohomological

dimension.

Theorem 3.2: If S is a finitely generated, commutative
R-algebra, then R-dim S # 1.

2:22;: By Corollary 2.3.4, R-dim S = 1 implies that
R/m-dim S/mS = 1 for some maximal ideal m of R. But this

is impossible by Theorem 3.1.

It is interesting to note that Magid [27] has shown
that any one dimensional commutative algebra over certain
classes of rings is countably generated. It is further
known from Rosenberg and Zelinsky [30] that R[X] has
R-dimension one. So we see that the hypothesis of finite

generation is necessary.

Theorem 3.3: Assume S is a finitely generated, projective,

commutative R-algebra. Either R-dim S = O or R-dim S = a.
‘ngggg Assume that n is finite and non-zero. By

Theorem 2.1, R/m-dim.S/mS = n for some maximal ideal m of

R, contradicting Theorem 3.1.

 

 

 

:38

Again by [30], we know that R[X1, X .., Xn] has

2,
R-dimension n.
is necessary. However, no example has been found to prove

the necessity of the hypothesis that A is R-projective for

Theorems 2.1 and 3.3.

We state the following proposition which is proved in

[16].

Proposition 3.4: If A is a finitely generated, projective,

 

faithful S-algebra and S is a faithful R-algebra, then

R-dim S s R-dim A s R—dim S + S-dim A.

Corollary 3.4.1: If A, S, and R are as in Proposition 3.4

 

and S is contained in the center of A and is finitely gen-
erated over R, then S-dim A = 1 and R-dim S = 0 implies
R-dim A = 1.

Proof: By Proposition 3.4, 0 s R-dim A s 1. By

 

Thus again the hypothesis of finite generation

Reduction to the Central Case, R-dim A = 0 implies S-dim A = 0.

Contradiction.

In fact, the proof yields the stronger corollary:

Corollary 3.4.2: If A, S, and R are as in Corollary 3.4.1

and R-dim s = 0 and S-dim A a‘ 0, then 1 s R-dim A _<. S-dim A.

We are now able to begin a reduction of n-dimensional

algebras to central n-dimensional algebras.

 

 

39

PTOEOSition 3.5: Let S be a faithful R-Subalgebra of the
center of A. If R-dim S = 0 and R-dim A = 1, then

S-dim A = l.

Prggf: S-dim A 2 l by Reduction to the Central Case.

Now 0 4 JR 4 (A): a A a 0 is exact and since S is Se-pro-
jective, ( ) ® e S is exact. Whence

0 4 JR SEE S ”8(A): age S a A ege S a 0 is exact. Now

(A): ege S E (A): by the middle four interchange. Now since
“'3‘ se _. s is an epimorphism, ”s#‘ A ®se s —. A as s E A is
an isomorphism by the pull-back lemma. So we have the

(A)§-projective resolution of A: O a JR 8 e S a (A); a A a 0.
S

Therefore, S-dim A s 1. Hence, S-dim A 3 1.

Proposition 3.6: Let S be a faithful R-algebra and let A
be a finitely generated, projective, faithful S-algebra. If
R-dim S = 0 and R-dim A = n, then S-dim A = n.

nggf: By the argument of Proposition 3.5, S-dim A s n.
Assume S-dim A = k < n. Then by Proposition 3.4,

R-dim A s k < n. But this is impossible. So S-dim A - n.

Theorem 3.7: Let A be a finitely generated, projective
S-algebra and let S be a finitely generated, faithful R-Sub-
algebra of the center of A. R—dim A = 1 iff R-dim S = 0
and S-dim A = l.

nggf: Assume R-dim A'= l; by Proposition 3.4, this
implies that R-dim S s 1. But by Theorem 3.2, R-dim S = 0
since S is finitely generated. Applying Proposition 3.5,

we obtain the desired reSult.

 

 

 

4i)

The other direction is merely Corollary 3.4.1.

Theorem 3.8: Let A be a finitely generated, projective
S-algebra where S is a finitely generated, projective,
faithful R-subalgebra of the center of A. Let n be finite.
R-dim A = n iff R-dim S = 0 and S-dim A = n.

2322f: Assume R-dim A = n. By Proposition 3.4, this
implies that R-dim S s n. Again by Theorem 3.3, R-dim S = 0
since S is finitely generated and projective. Applying
Proposition 3.6, we obtain S-dim A = n.

If, on the other hand, we assume that Rodim S = 0
and S-dim A = n, we obtain that R-dim A s n by Proposition
3.4. Assume that R-dim A = k.< n. Then by Proposition 3.6,

S-dim A = k.< n, which is a contradiction.

We include as a final result the following interesting
proposition concerning the relationship among the finite gen-
eration, projectivity, and n-dimensionality of an algebra A

over its ground ring R and its center S.

Theorem 3.9: Assume A is a central S-algebra and S is a

 

faithful R-algebra. Then any two of the following imply the
third:
(a) A is finitely generated and projective over R
and R-dim A = n.
(b) A is finitely generated and projective over S

and S-dim A = n.

 

 

 

41.

(c) S is finitely generated and projective over R
and R-dim S= 0.

Proof: (b) and (c) imply (a): Theorem 3.8 implies
R-dim.A = n. Finite generation and projectivity are clear.

(a) and (c) imply (b): S is a separable R-algebra.
Since A is R-projective, A is also S-projective by the
lifting property for projectives (Proposition 1.23). Finite
generation is clear. That S-dim A = n follows from
Proposition 3.6.

(a)yand (b) imply (9): By Proposition 1.22, S is
an S-direct summand of A and hence an R-direct summand of
A. But A is R-projective and, therefore, S is R-projective.
Clearly S is finitely generated over R. Then by Proposition

3.4 and Theorem 3.3, R-dim S = O.

 

 

CHAPTER IV

EXAMPLES AND COUNTEREXAMPLES

Harada, in his 1966 paper [18], has shown that a semi-
primary hereditary algebra has the form of a generalized tri-
angular matrix algebra with certain special properties, which
we will discuss later in this chapter. This leads one to an
entire class of algebras of cohomological dimension one.

Using Theorem 2.1, we can show that some classes of
algebras have a given cohomological dimension n. However,
when the algebra is not projective, we may for example, show
that an algebra has cohomological dimension one by showing
that the module J is projective. We will construct
several classes of algebras of this type.

Finally, we will give several examples to show that
Eilenberg's Theorem does not hold for algebras over a

commutative ring.

Section (a): Generalized Triangular Matrix Algebras

We now consider the definition of a generalized tri-

angular matrix algebra over a commutative ring:

Definition 4.1: Let A1,...,A be algebras over a fixed

n

ring R and let Mij be left Ai- and right Aj-bimodulesand Mij= 0 for

igreater than j and Mii = Ai' ASSume that R commutes with

42

 

443

the 11

morphisum which satisfy the following properties:

 

 

 

L
: M —0
‘Pij Mu, ®AL Lk Mik
t e
cPit: it ®At At Mit
i , a
cPit' A1 8A1 Mit Mic
and commutative diagrams:
L
1d..<8 m
k
M o M e M 13 j
i' A ' A k
J j JL L L
j .
mLL<8 idflk
V ‘Plik
MiL 8h ‘MLk- ;
A
Set Tn(Ai; Mij/R) = all mlz m1n
a22 ‘ m2n
O .
nn

(8) (m..) j: (m'..) = (In . +m'..)

1] ij iJ - ij
. I _—_ L l
(b) (mu) (m iJ.> (i alumni, a m u”
(C) r(mij) = (rmlj) = ( ijr) = (mlJ)r

We call Tn(Ai; Mi

over R.

 

 

 

ij' We consider a family of left A1- and right Aj-bihomo-

j/R) a generalized triangular matrix algebra

 

 

44

Harada has proved the following proposition in [18,

p. 469]:

Proposition 4.2: Let A be a generalized triangular matrix

 

algebra with the Ai's being semi-simple algebras over a field

.1

mik are isomorphisms, then R-dim A s l.

R. If

We can then obtain from this and Theorem 2.1:

Proposition 4.3: If A .,An are finitely generated, pro-

 

1,..

jective, separable algebras over R and the Mij are finitely

generated, projective R-modules and the ¢ik are isomorphisms,
- ‘ ' 1.

then R dim Tn(Ai’ Mij/R) S

Proof: Clearly Tn(Ai; Mij /R) 9,, R/m °-‘ Tn(Ai oR R/ro;

® R/m/R/m). Since Tn(Ai; Mi /R) is R-projective, Theorem

M.
ij 1
2.1 applies. We also have that Ai ® R/m is semi-simple
since it is separable over a field. The projectivity of the
. k . .
A1 and the MU implies that cpij ® idR/m are isomorphisms.
Hence, by PrOposition 4.2, we have that R/mwdim.Tn(Ai; Mij/R)

® R/m s 1. So, by Theorem 2.1, R-dim Tn(Ai; Mij/R) S 1.

Proposition 4.4: If R(qu) denotes the set of (qu)-matrices
with entries from a commutative ring R with 1, then
R X R X8 3 R Xs under the multi lication ma

(p 0%.,qu (q > (p > p p

as left R(po) and right R(sXs)-bimodules.

Harada in [18, p. 479] proves this result for R a
division algebra; but the proof depends only on the fact that

R has an identity.

 

 

 

4L5

Theorem 4.5: If A is a block triangular matrix algebra, i.e.,
A = Tn(Ai; Mij/R) where A1 is an (ri X ri) matrix with
coefficients in R and Mij is an (ri X r.) matrix with

coefficients in R, then R-dim A = l.
.EEEEE‘ Case (a): If R is a field,.J(A) = Tn(0; Mij/R)
is non-zero. So, by Eilenberg's Theorem, R-dim A # 0. But
by Propositions 4.4 and 4.3 R-dim A s 1.
Case (b): If R is any ring, for every maximal ideal
m of R, R/m-dim A/mA = R/ro-dim Tn(Ai e R/m; Mi]. 8 R/m/R/m) = 1

by (a). The conclusion follows from Theorem 2.1.
We note the following result proved in [16]:

Proposition 4.6: Let Tn(R) denote the upper triangular
(an)-matrix algebra with entries from R. If A = Tn(R) and

B is any R-algebra, then R-dim A 8>B = R-dim B +’1.

We thus obtain the well-known class of examples of

algebras of arbitrary cohomological dimension r over a ring R.

Proposition 4.7: Let R be any ring and let A be the tensor

product over R of Tn(R) with itself r times. Then

R-dim A = r.

Proof: Apply the preceding proposition and induct on r.

Another class of interesting examples is afforded by

extending the following result of [14]:

_§koposition 4.8: let N =.l(Tn(R)), R a field. Then

R-dim T[.l(R)/N2 = n - l.

 

 

l
.‘-"l t ‘ t
. \ 1
. . I are

     

46

Theorem.4.9: If R is semi-primitive, then R-dim Tn+1(R)/'N2 = n
for every n.
2
Proof: Tn+1(R)/N is clearly R-projective. So we

may apply Theorem 2.1.

One final example is generalized from [14] again:

Proposition 4.10: Set A = a b c
(3d e) : a,b,c,d,e are in R}'.
0 a

2
If R is a field, then R-dim A = 2 and R-dim.A/N = a.

 

This yields for us the following result:

Theorem 4.11: If A is as in Proposition 4.10, R is any

 

commutative ring, then R-dim A = 2. If, in addition, R

is semi-primitive, R-dim A/N2 = a.

Further examples may be obtained from [31] by generalizing

to arbitrary ground rings.

Section (b): Not Necessarilnyroiective Algebras gleimension.

 

One

We now proceed to give the Special classes of dimension
one algebras which may not be projective over the ground ring.

31, $2 in R

Theorem 4.12: Set A=T(E,R,§) = {(31 52)
0 é '

and s is in RC}, where R is any commutative ring with
one and R = RAH where m is any ideal of R not equal

to R. Then R-dim A = 1.

    

\
= “ I ‘ i
a —-—" a

   

147

(The algebra A was originally dafined by S.U. Chase in 1960
[10] as an example of an algebra which was left, but not right,
semi-hereditary when R was chosen to be absolutely flat.)
nggfz Set e = 1 +-M 0‘ f = O 0 g = 0 1 +-fl
0 0 7 0 1 0 0

Then the multiplication in A is determined by the following

 

 

 

rule:
TABLE 1.
e f g
e e 0 g
f 0 f 0
g 0 g 0

 

 

 

It is convenient to use both (l,f,g) and (e,f,g) as gen-
erating sets for A as an R-module. Considering A as an
Ae-module, we find that J is Ae-generated by (f ® 1 - 1 ® f,
g ® 1 - 1 ® g). It is convenient to find an R-generating set
for J. One can verify that the following elements of J
constitute an R-generating set for J: e ® f, f ® e, g ® e,
g 8>f - l 8>g, f G>g, and g<8 g .

We wish to show that J is Ae-projective. In order
to construct the necessary dual basis, we must indicate the
action of A8 on the R-generators of J. This can be seen
on Table 2. We then note that in A, sle + s f + 33g = 0

2

a s E 0 modiland = 0.

iff s1 3 $2

 

 

 

 

 

 

 

   

48

TABLEZ
1®f(e®f)=e®f l®f(g®f—l®g)=g®f-1®g
1®g(e®f)=0 1®g(g®f-1®g)=0
f®1(e.®f)=0 f®1(g®f-1®g)=-f®g
f®f(e®f)=0 f®f(g®f-1®g)=-f®g
f®g(e®f)=0 f®g(g®f-1®g)=0
8®1(6®f)=0 g®1(g®f-1®g)=-g®8
g®f(e®f)=0 g®f(g®f-l®g)=-g®g
g®g(e®f)=0 g®g(g®f-1®g)=0
{E'E'EE’JS'IB """" ‘ """ i'é'E'ZE'é'; 132-2”; """"""""
l®g(f®e)=f®g 1®g(f®g)=0
fo1(f®e)=f®e f®1(f®g)=f®g
f®f(f®e)=0 f®f(f®g)=f®g
f®s(f®e)=f®g f®g(f®g)=0
g®1(f®e)=g®e g®1(f®g)=g®g
g®f(f®e)=0 g®f(f®g)=g®g
g®g(f®e)=g®g g®g(f®g)=0
{2?};3232'6 """"""" i'é'E'Eg'é‘J'L'Qé'; """"""""
1®8(g®e)=g®g 1®g(g®g)=0
f®1(g®e)=0 f®1(g®g)=0
f®f(g®e)=0 f®f(g®g)=0
f®8(g®e)=0 f®g(g®g)=0
g®1(g®e)=0 g®1(g®g)=0
g®f(g®e)=0 g®f(g®g)=0
g®s<s®e>=0 g®g(g®g)=0

 

 

 

49

And so sll(e ® 6) + 82(6 3 f) + s3(e @8) + $4“ 9 e)
+ 85(f ® f) + 86“" ®g) +S7(g® e)

+88(g®f)+39(g®g)=0 in A‘3

iff siEOmod‘u for 11‘5 and ss=0.

Now any element of J can be written as:

 

sl(e®f)+sz(f®e)+53(g®e)+s4(g®f -e®g-f®g)

+sgf®m+wdg®m

This element is zero iff s1 5 32 2 s3 — 4 — ($5 - 84)

II
a:
ll

mod 91. Therefore, the element is zero iff si s 0 mod 91;
i = 1,...,6.
We therefore define 9: J ... A: ® A: by extending

following map R- linear 1y:

ee®f>=oe®f,m
ea®e>=u®e,f®m
emse>=m®e.g®m

the

9(g®f-1®g)=(g®f-1®g,e®f-f®g)

e(f®g) (fag.f®g)

e(g®g) (g®g,g®g)

One easily checks from the multiplication tables that 9

an element of Hom (J, A6 69 A8).
A8 f 3
Now define n in Hom em: (43 AZ, J) by
A

11(x1, x2)=x1(f®1- 1®f)+X2(g®1' 1698)

One easily verifies that 119 = id . Whence J is Ae-pro-

J
jective. Thus R-dim A s 1.

is

 

 

5C?

But, if m is contained in the maximal ideal m of
R, then RAm-dim A/mA = l by Theorem 4.5. Therefore, by the

Endo-Watanabe Theorem, R-dim A # 0. So R-dim A = 1.

We note that by varying the ground ring R and the
ideal m, we may obtain algebras which are free, algebras
which are projective but not free, algebras which are flat
but not projective, and even algebras which are not flat.

In addition, we have shown the following algebras

have cohomological dimension one over R:

Theorem 4.13: Let R be a commutative ring and m # R an

ideal of R. Set R=R/QI-

Lat A

I!
a:
a:
a:

088 :S.inRandsinR

s4 s5 s6 : si in R and s in R
o 0 g
C z 81 S2 S3
- g : . - - .
0 s1 2 si in R and si in R
' 5
0 S3 4

Then R-dim A = R-dim B ‘ R-dim.C = 1.
Proof: Since by Theorem 4.5 R/m-dim A/mA = R/m-dim,B/mB
= R/m-dim C/mC = l for any maximal ideal m of R contain-

ing fl, R-dim.A 2 l, R-dim B 2 1, and R-dim C 2 1 by the

 

 

51

End04Watanabe Theorem. Hence we need only show that R-dim A s l,
R-dim B s 1, and R-dim C S 1.

Furthermore, since the technique we will use is
identical with that of Theorem 4.12, we will only give the
multiplication tables for the algebras and define the embedding

maps on the enveloping algebra generators of the respective J's.

The Algebra A: Set e = 1 +9] 0 0

 

1
0 0 0
0 O 0
e2 = 0 0 0 e3 = 0 0 0
O l +-m O O 0 0
0 0 0 0 0 1
= + = 0 0 0
81 0 1 fl 0 82
0 0 O 0 0 1 +“fl
0 0 0 ' 0 0 0

Then the multiplication in A is given by:

 

 

 

TABLE 3
----.--f:--.--fz ..... f?--.--§l ..... €2-.---§§-
e b e1 k 0 0 4 g1 0 g3
---- ------ 1 ------------------- 1 ----- 1 -------

o

63... ...... a--f§--.--9---l--9---.--§§-- --9--
e o 0 e3 0 o o
----, -------------------- 4 ------ q ------ p -----
g o g o o g o
—l~--.-—--—-q---l--fi ------ a ...... q-—-3 --------
$2-- .--? ...... 5’ ...... §z_-,-_‘.’--_.-_‘.’ ...... 9--
§_-_.--9 ...... 9---.--§§ ..... 9---.--9---L--9_-

 

 

 

 

 

 

Define p: S Ae a J by p(xl, X2, X3: X4, X5) =
x1(e2®1-1®e2)+x2(e3®1- l®e3)
+x3v(g1®1- 1®g1) +x4(g2®1- 1®82)

e

Clearly p is in Hom e(5A , J).
We define a fin Hom e(J, 5A8) by:
A
6(62®1- 1®62) = (62®e1+e2®e3+e1®e2, -e3®62,0,0,0)
e(e3®1 - 1®e3) = (-e2®e3, el®e3+e3®e1+e3®e2, 0,0,0)

9(81®1'1®81)=(81®€1+81®e3'92®81:

- 0
33 ® g1, e]. ® 92: 0: )
- = _ 0
6(83 ® 1 - 1 ® 83) = (- 82 ® g3! 33 ® 1 ' 33 ® 833

e1®82, 81®83: 0)
One may check that e is a well-defined Ae-module homomorphism,
after extending the map linearly. Clearly p9 = idJ. So J

is projective.

 

 

The Algebra B:

given by e1 = l +-m
O
0
e3 = 0 0
0 0
0 1
g2 = O O
O 1 +~m
0 0
h = 0 0
1 +-M O 0
0 0

-------

.------

 

-------

.......

.......

 

53

The algebra

o o

o o

o o
g1
g3

TABLE 4
.--‘32--.--§;--
--°-----‘:”1-_

0 0

e3 { O

0 0

82 L 0

0

83 - --

L--‘.’--_ ...?2..

 

 

is given by:

 

 

--------

--------

 

 

B has R-generators

-------‘ ------—----

 

 

54

e
We may now define p: 7 B ... J, p in Hom e(7Be, J) by
B
= - + ..
p(xl: .x7) x1(e1® 1 1®e1) X2(e2® 1 1®e2)
+x3(e3® 1 - 1%83) +x4(g1® l - 1®g1)
+x5(gz®l- 1®g2) +x6(g3®1- 1®g3)

+x7(h®1- 18h).

e
We also define a: J .. 7B , e in Hom e(J, 7Be) on
B

the Be-generators of J by:

6(e1®1 - 1®el) = (el®e2, - e2®e12 -e3®e1, 0, o) 0! g1®e3)
9(e2®1-1®e2)= (-e1®e2,e2®el+e2®e3,
-e3®e2, 0, 0, 0, 0)

e(e3®1 1®e3)=(0,-e2®e3,e3®e1+e3®e2,
0, 0, 0, -g1®e3)
e(g1®1-1®g1)=<0. -e2®gl+gl®el+sl®e3,
-e3®g1, e1®ez, 0, 0, 0)
6(82®1'1®82)=(- e1®82,0,82®e +s2®e2-e

1 ® g2,

0,e2®e

3

3. o, 0)

9(g3®1-1®g3)=(0. -ez®g3,g3®91+g3®ez-e3®g3,

0,0,e1®e 0)

3,

e(h®1-1®h)=(h®e -el®h,0,'e ®h, -h®h,

2 3

o, 0, e2 a e3)

e e
After extending e B -linear1y, e is a well-defined B -homo-

e
morphism with p9 = id]. Hence J is B —projective.

The Algebra C: The algebra C has R-generators:

e=l+9100 e=000

e3 = 0 0 0 v = 0 O 0
0 0 0 0 0 0
0 0 1 0 1 0
r = 0 l +~fl 0 s = 0 0 l +-m
0 0 0 0 0 0
0 0 0 0 0 0

and the multiplication in C is given by:

TABLE 5
e e e r S t V
l 2 3

‘3} ° °
‘32 0
f? V
r O
S r
‘ '2
V

 

We define p: 7Ce a J in the uSual manner, and 9
e
in Hom e(J, 7C ) by extending Ce-linearly the mapping de-
C
fined on the ce-generators by:
e(e1 ® 1 - 1 ® el) = (e1 ® e

+ _ -
2 e1°“? e2®e1’ e38H3

0, O, 0, 0)

 

19

 

 

56

9(e2®1-1®e2)=(- e1®e2,e2®e1+e2®e3,

- e3®e2, 0, 0, 0, 0)

e(e3®1-1®e3)=(- e1®e3, -e2®e3,e3®e1+e3®e2,

0. 0, 0, 0)
- a + _ -
e(r ® 1 1 8 r) (0, r ® e1 r ® e3 e2 8 r, e3 8 r,
e1 8 22, 0, 09 0)
e(s ® 1 - 1 ® 3) = (0, - e2 ® s, s Q e1 + s 8 e2 - e3 8 x,
0, e1 8 e3, 0, 0)

e(t ® 1 - 1 8 t) = (- e1 8 t, O, t 8 e1 + t ® e2 - e3 ® t, 0,

0, e2 ® e 0)

3,
e(v ® 1 - 1 ® v) = (- e1 ® v, - e2 ® v + v ® e1 +'V ® 83,

0, 0, 0, - V G V, 0)

e
One may check that e is an element of Hom e(J, 7C ) and
C
e
that p9 = idJ. So J is C -projective.

This completes the proof of the theorem.

In view of Theorem 4.5 and Theorem 2.1, together with
Theorems 4.12 and 4.13, it is reasonable to conjecture that
R-dim A = 1 whenever A is a block triangular matrix algebra
with entries from R/m in every block but the lower left
block which contains entries from R. It is clearly true by
Theorems 2.1 and 4.5 if R/fl is R-projective.

One might also conjecture the possibility that Theorem
2.1, restricted to the case n = 1, could be proved without

the hypothesis that A is R-projective.

 

 

57

Section (c): Eilenberg's Theorem

We conclude the chapter with some examples which show
that the Eilenberg Theorem specialized to n - 1 does not

hold for algebras over arbitrary commutative rings:

Proposition 4.14: In order that R-dim A = 1, it is not

necessary that .I(A) = N be A-projective

Proof: Set A= {(a b) }
: a,b,c in Z .
0 c 4

Then 24-dim A = l by Theorem 4.5. Now N is not 24-pro-
jective since any projective is free and therefore contains
4n elements. But A is 24-free. So were N to be A-pro-

jective, N would be Z -projective. But N ‘-‘ 1(24) ®J(Z4) (4324.

4 Z4

Whence N is not A-projective.

Proposition 4.15: In order that R-dim A = 1, it is not
sufficient that A/N be R—separable and that N be A-pro-
jective.

Proof: Let R = Z the rational integers localized

(2) ’
at two, and let A = R[i]. Being commutative, we know from
Theorem 3.2 that R—dim A f 1. Now A/2A is a finitely
generated algebra over the field Z/ZZ. Clearly, A/2A modulo
its radical is separable. Then by the End04Watanabe Theorem,

A/N is R-separable. Since A is a dedekind domain, N is

A-projective.

It is still unknown whether R-dim A = 1 implies

R—dim A/N = 0. In the final chapter, some partial reSults

 

 

 

58

indicating the possible truth of this statement will be pre-
sented.

Before proceeding to the next chapter, we note a couple
of interesting examples where all of the conditions of the

preceding proposition hold simultaneously.

Example 4.16: Let A be a block triangular matrix algebra
over the semi-primitive ring R with diagonal blocks of rank
2 2 .

n1,...,nm Wlth n1 2 n2 2 ... 2 nm and n - n1 +...+ nm.

N is A-projective.

Set f((aij)) = ( ) where a(j) E j - nm_1(mod n).

3.
120(j)
Clearly f is an element of HomA(N,A) since the shift moves

elements to the left nm_1 places and hence the image lies

in C. We claim that f((aij))(xij) = (aij) where (xij) is
the matrix with an entry of one along the super-diagonal

beginning with the entry xn-n 1+1, “1+1 and zeroes every-

m-

where else.

Now Henna“) = whoops“)

(08(j)ai,c(j)xd(j)ub).

0 L < n + 1

Now 1

2 a .x =
c(j) i:U(J)U(j)1L
ai’nlw L= n1 +y
But n1 - nm_1 + y = a(j) E j - nm-1(mod n).

Therefore j a n1 + y (mod n).

Hence f((aij))(xij) = (aij)' So f and (xij) form a dual

basis for N over A.

 

 

 

59

Example 4.17: Let A be an upper triangular block diagonal
matrix algebra over a semi-primitive ring R which has block
diagonal form (1X1, 2X2, 2x2). Then N is A-projective.

Let (aij) in N be of the following form

ij
0 O 0 e f
0 0 0 g h
0 O 0 0 0
0 O O 0 0
Then set:
= - =010
f1(aij) (a 0 x1
66 5
== - , = 01
f2(aij) b 0 x2 0
33 ,6
f3(aij) = O c 0 x3 = O O 0
Def? 000
0 g ‘0 ‘6
536
= - = 0
f4(aij) 0 d 0 x4 0 0
0:6 000
0:15 5
5
4
Clearly, Z fr((aij)) xr = (aij)°

r=l

 

 

60

By generalizing the dual basis given in the last
example, one can easily see that any upper triangular block
diagonal matrix algebra over a semi-primitive ring has its

radical projective.

 

 

CHAPTER V

PRELIMINARY STRUCTURAL RESULTS

Chase, in an article in 1961 [10], defined a gen-
eralized triangular matrix algebra to be a semi-primary algebra
A in which there exists a complete set of mutually orthogonal

idempotents e1,...,en ordered in such a way that eiNe = 0

j
if i 2 j. This is shown to be equivalent to gl. dim A/NZ < co
and also equivalent to the fact that every factor algebra A/I
has finite global dimension.

We have already commented that it is well-known that
for a finitely generated algebra A over a field R with
R-dim A s 1, R-dim A/N2 < a. In particular we see that this
implies (a) that every triangular matrix algebra Tn(R) over
a field R is generalized triangular; (b) that every homomorphic
image of a generalized triangular matrix algebra is also a
generalized triangular matrix algebra; (c) that for every n,
there is a generalized triangular matrix algebra having

global dimension n, namely T (R)/N2; and (d) that every

n+1
finitely generated generalized triangular matrix algebra is
the image of a finitely generated semi-primary algebra of
cohomological dimension less than or equal to one and the

algebras modulo the square of their respective radicals are

isomorphic.

61

 

alg

on

 

 

62

Harada, by his theorem, distinguishes those generalized
triangular matrix algebras which have dimension less than or
equal to one.

Our attempt has been to obtain similar results for
algebras over a local hensel ring. The reSults obtained show
only that dimension one algebras over a local hensel ring are
generalized triangular.

Eilenberg's Theorem shows that every finitely generated
algebra A over a field R of finite cohomological dimension,
has the property that A/N is R-separable. Then, applying the
Wedderburn Principal Theorem, A contains a separable Subalgebra
S such that A = S 6 N as R-vector spaces.

Ingraham [21] has defined an inertial subalgebra S
of an algebra A over a ring R to be an R-separable sub-
algebra S such that A = S + N. (That the sum is usually
not direct follows from the fact that J(R)-S C S n N. He
also defines an inertial coefficient ring to be a ring R
with the property that every finitely generated algebra A
Such that A/N is R-separable contains an inertial sub-
algebra.

We note that Azumaya [4] and Ingraham-Brown [8] have
shown that a semi-local ring R is an inertial coefficient
ring iff R is the direct sum of local hensel rings. It is
further known that a dedekind domain is an inertial coefficient
ring iff it is a hensel ring or has zero radical [21].

We will show that if A is a finitely generated, pro-

jective R-algebra with R a semi-local ring and R-dim A = n < a,

 

 

63

then A/N is R-separable. Hence, we have immediately that,
if R is a finite direct Sum of local hensel rings, A contains
an inertial subalgebra.

Since one of the distinguishing properties of a one-
dimensional finitely generated algebra over a field is that
the dimension of all of the factor algebras is finite, one
would like to see if the same is true of algebras over a
commutative ring. We can only show this when the factor
algebra is projective over R/annihR.

Many of the first results are clear, and little proof

will be offered.

Section (a): Separability pf A/N

Proposition 5.1: Let A be a finitely generated projective
R-algebra. The following are equivalent:

(a) R—dim A = n

(b) 11/] (R) -dim A/J (R)A = n

Proof: Theorem 2.1.

Corollary 5.1.1: The proposition remains true if J(R) is

replaced by any ideal m contained in J(R).

Corollary 5.1.2: If n = 1, then (8) implies (b) without the
projective hypothesis.

Proof: The Endo¥Watanabe Theorem and Corollary 2.3.4.

Proposition 5.2: ASSume A is an R-algebra and B is an

S-algebra with R-dim A = n and S-dim B = m. Then

 

R G S-dim A 9 B = max {n,m].

Proof: Assume without loss of generality that n 2 m,
Endlet O—.X-o...—.X ~0A-00 and O-OY-o...-°Y 43-40
n 0 m 0
be minimal projective resolutions of A and B respectively
* *- ,
as A @R A and B ®S B modules
‘0 O —’ ... "0 x -O ... —o a _.
Then 0 Xn® “1er XO®Y0 A®B 0
is a projective resolution of A 9 B as an (A 69 B);®.
Whence R 63 S-dim A e B s n.
0n the other hand, assume that

0
resolution with r < n. Then, noting that ( ) ®R@ R is

I e _
0—.zr-. -oZ «AeB-‘O, is an (AeMm projective

exact,we have that Oazr®REBR-....—.ZO®R@R-A—oo

is an A ®R A*-projective resolution of A of length less than

n. Contradiction. From this, the equality follows.

Proposition 5.3: let A be a finitely generated faithful

R-algebra, where R is a semi-local ring.

(a) If R-dim A = 1, then R-dim A/N = o.

(b) If A is R-projective and R-dim A = n < no, then
R-dim A/N = 0.

M: We will prove (b); the proof of (a) is
analogous. By Proposition 5.1 and the Chinese Remainder
Theorem, we may assume that R = G :31 R1, where the Ri are
fields and A is finitely generated and projective over R.
Then A = Q) % Ai where each A1 is the algebra generated

by the idempotent of R Set Ni = J(Ai). Then A/N =

i'

m
6 Z Ai/Ni. By the previous proposition, Ri-dim Ai s n < co.
1 .

 

65

Hence, by Eilenberg's Theorem, Ri-dim Ai/Ni = 0. Again,

applying Proposition 5.2, we have that R-dim A/N = 0.

Proposition 5.4: Let A = T (A,; M,,/R) where A, and M,
n i ij i ij
are finitely generated and R-dim A1 = 0. Then
N = T (N,; M ./R), and R-dim A/N = 0.
n i 13
Proof: A/N = T (A./N,; O/R) and R-dim A,/N_ = o
-——-- n i i i 1

since they are factor algebras of separable algebras. Whence

R-dim A/N = 0.

Proposition 5.5: Let A = T (A ; M../R) where A. and the
n i ij i
Mij are finitely generated over R. Let R-dim Ai s 0.

Then A contains an inertial subalgebra S = Tn(Ai; O/R).

It is interesting to note that we have found no
example of any algebra A of cohomological dimension one
which does not contain an inertial subalgebra or such that

A/N is not separable.
Section (b): Dimension pf Residue Algebras
It has been showu in [14] that the following reSult

holds true for algebras over a field:

Theorem 5.6: If A is finitely generated over a field R
and R-dim A s 1, then for every two-sided ideal I of A,

R-dim A/I < so.

We have already obtained the following obvious exten-

sion in Theorem 2.1 and Proposition 2.2:

 

 

 

 

66

Proposition 5.7: If A is a finitely generated faithful
algebra over a ring R, R-dim A = 1 and I is a two-sided
ideal of A such that I contains some maximal ideal of R,

then R-dim A/I < a.

We have also noted the following results which are of

the same type. The proofs are omitted.

Proposition 5.8: let A be a finitely generated algebra
over a commutative ring R and let R-dim A = l.

(a) If u is contained only in maximal ideals m having
the property that R/m-dim A/mA = 0, then Rflu-dim A/uA = 0.
(b) If m is contained in at least one maximal ideal m
such that R/m-dim A/mA = 1, then R/m-dim A/uA = 1.

Proof: The EndoJWatanabe Theorem and Corollary 2.3.4.

Proposition 5.9: Let R be a semi-local ring, A a finitely
generated, faithful R-algebra. Let I be any two-sided
ideal of A Such that A/I is R/fl-projective where
9.1 = I n R. If R-dim A = 1, then R-dim A/I < no.

2522;: Without loss of generality, we may assume that
R = R/fl and A = A/flA. Now, for every maximal ideal m of
R, R/m-dim A/mA .s 1. But (A/I)/m(A/I) ‘=‘ A/(roA + I) E
(A/mA)/(mA + I/mA). We know that R/m-dim [A/I]/m[A/I] < a
by Theorem 5.6. Since R is semi-local, set
max {R/m-dim [(A/I)/nKA/I)]} = n < a. Since A/I is R-pro-
YO

jective, R-dim A/I = n < m, by Theorem 2.1.

 

 

 

67

It should be noted that, in a generalized triangular
matrix algebra, there are many Such ideals I which satisfy

the hypotheses of the proposition.

Section (c): Generalized Triangular Matrix Algebras

Proposition 5.10: Let A be a finitely generated algebra
over a local hensel ring R with R-dim A = 1. Then there
exists a complete set of mutually orthogonal idempotents

e .,e such that e.Ne. C mA for i 2 j-
n 1 J

1,..
ngpfz By [10, p. 21] there exists a complete set

of orthogonal idempotents 61,...,en in A/mA Which are

lifted from A/N. These idempotents satisfy the condition

that 31(N/mA)éj = 0 for i 2 j. Since R is a local hensel

ring, there exists idempotents e1,...,en in A which form

a complete set of orthogonal idempotents. Clearly

eiNej c mA for all i 2 j since aim/tomej = 0 = n(eiNej)

where n is the natural map of A onto A/mA.

Theorem 5.11: Let A be a finitely generated, faithful
algebra over a local hensel ring R such that A/N is a
direct Sum of divisipn algebras Di over R/m. R-dim A - 1
implies that A is a generalized triangular matrix algebra.

ngpfz There exist idempotents e1,...,en in A/mA
(lifted from A/ND which form a complete set of orthogonal
idempotents with the properties:

(a) ei(N/mA)é i o for i 2 j [18, p. 471];

j
(b) ei(A/mA)éi = Di by [18, p. 471];

 

68

(c) ei(A/mA)ej = o for i >j by [13, p. 472].
Since R is a hensel ring, by Proposition 5.9, there exist

e ,en, a complete set of orthogonal idempotents such that

1,...
eiAej : mA for i 2 j and .31 -= 11(ei) and the following hold:
(a') A = oz: eiAe.
Rid
l = . .
(b ) eiAej 0 for i > 3.

(c') eiAei is separable over R.

e Ae <3

I j . A _. ‘ .
and (d ) wiL' i j ejfiej ej 8L eiAeL, defined by

multiplication, are eiAei - ejAe bimodule homomorphisms.

J
n
That (a') holds is clear since 1 = 2 e1 and the e1
1
are orthogonal. By (b), ei(A/mA)ej = 0 for i > j; whence

eiAej C mA = Q 2 meiAej . Hence eiAeJ. = meiAej . Therefore,
by Nakayama's Lemma, eiAej = 0 for i > j. So (b') holds.
Now (e,Ae,)/e,mAe, = e,(A/mA)e, = D, [18, p. 471]
i i i i i i i
which is R/m-separable. So eiAei is R-separable, which

proves (c'). (d') is clear. Setting M1 = eiAej and

j
A. = e,Ae,, we have that A E T (A.; M../R) where a a (6.86.)-
i i i n 1 ij i 3

We note that if A is also R-projective, since

.A =<$ 2 eiAe., we have that eiAej is R-projective.
R 1,:

We now proceed to define with Harada [18, p. 472]
an induced generalized triangular matrix algebra.
Let A = Tn(Ai; Mij/R) be a generalized triangular

matrix algebra over R. Let mij =

 

 

 

 

 

 

69

= Mij(si x sj). Each figj can naturally be made into a left
(Ai)s and a right (Aj)s -bimodu1e, where (A)s denotes the
i .

sXs-matrices with entries from A.

Define ¢ikz ((xt,P)) ®(Aj)s
J'

to be the induced bilinear mappings. Then B = Tn((Ai)sij Wk

((yr,q)) ~ ((2 ¢j:k(xr,p a yM»
P

j/R)

is called the generalized triangular matrix algebra induced

from A.

Theorem 5.12: Let A be a finitely generated, faithful algebra
over a local hensel ring R. If R-dim A = 1, then A is
isomorphic to a generalized triangular matrix algebra induced
from eAe, where e is the sum of a single representative
from each isomorphism class of primitive orthogonal idempotents.
Proof: Suppose A/mA = ® 2 )3 Z Z 5, (A/mA); . Then
—— . . lj Rt
1 j k L
these lift to idempotents Such that A =13 Z Z Z 2 e. Ae
, ij kt
1 J k 4.
since R is hensel. We index the idempotents so that the
first index denotes the idempotent class, whence e = 2 e11.
i
Then eAe has the property that eAe/eNe is isomorphic to
a direct Sum of division algebras. Whence eAe = Tn(Ai; Mij/R)’

by Theorem 5.11.

Notice that because the idempotents are isomorphic,

BA =
We have that Aek$ ekm and eijA eirA' Hence
eijAekL = eirAekm = eiAek where ej = ejl' As above set Mij
= Ae,. S t = X = .
ei J e ”Uj Mij(si sj), and (Ai)Si (eiAei)si

One can readily verify A = Tn((Ai)si; mij/R)'

 

 

70

The one major question left unanswered in this section
is whether or not there can be given any ring theoretic char-
acterization of when a generalized triangular matrix algebra

over a local hensel ring is necessarily one dimensional.

 

 

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10.

11.

12.

13.

 

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