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SOME RESULTS ON SEPARABILITY

AND PURE INSEPARABILITY
FOR ALGEBRAS OVER COMMUTATIVE RINGS

presented by
Linda Lee Deneen
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SOME RESULTS ON SEPARABILITY
AND PURE INSEPARABILITY
FOR ALGEBRAS OVER COMMUTATIVE RINGS

By

Linda Lee Deneen

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1980

ABSTRACT

SOME RESULTS 0N SEPARABILITY
AND PURE INSEPARABILITY
FOR ALGEBRAS OVER COMMUTATIVE RINGS

By

Linda Lee Deneen

Let R be a Noetherian inertial coefficient ring and let A
be a finitely generated R-algebra (that is, finitely generated as an
R-module) with Jacobson radical J(A). Let S be a subalgebra of
A with S + J(A) = A. We show that for every separable subalgebra
T of A there is a unit a of A such that aTa'1 E_S. Moreover,
we show that if S + I = A for a nil ideal I of A, then R can
be taken to be an arbitrary commutative ring, and the conjugacy result

still holds.

If A 3_S are rings, Bogart defined A to be purely inseparable

 

over S if the A-A bimodule map u :AJES A0 -—+ A has small kernel.
For A a finitely generated R-algebra and S a subalgebra of A,

Ingraham defined S to be an inertial subalgebra of A if S is

 

separable over R and S + J(A) = A. If A is commutative and A/J(A)
is separable, it is shown that S is an inertial subalgebra of A if
and only if A is purely inseparable over S and S is separable over
R.

If A/J(A) is not separable, the situation is more complicated.

We show that if A is a finitely generated algebra over a commutative

 

semilocal ring R, then there is a finitely generated, faithfully flat
(in fact free), commutative R-algebra P such that (A<3RP)/(B QRP) is
P-separable. If B is a subalgebra of A for which B<8R P is an
inertial subalgebra of A<8R P for any such P, then extending a

definition of Bogart we define B to be a Wedderburn specter for A

 

over R. If A is commutative, we show that B is a specter for A
over R if and only if A is purely inseparable over B. We conclude

by giving some properties of specters.

ACKNOWLEDGEMENTS

I am deeply grateful to Professor Edward C. Ingraham for his
patience and understanding in guiding my work on this thesis. His
love of mathematics has been a constant inspiration to me. I am
indebted to Professor William C. Brown for his careful reading of
this work and for his kind comments and helpful suggestions. I
would also like to thank the friends, teachers, and family members
whose support over the years has been invaluable. Finally, I greatly

appreciate the excellent typing of Barbara Miller.

ii

TABLE OF CONTENTS

Chapter
I. PRELIMINARIES ......................
§l. Notation and General Results ............
§2. The Concept of Separability ............
§3. Wedderburn Factors and Inertial Subalgebras
II. A THEOREM ON THE LATTICE 0F SUBALGEBRAS OF AN ALGEBRA . .
III. PURE INSEPARABILITY AND NEDDERBURN SPECTERS .......

BIBLIOGRAPHY ......................

CHAPTER I
PRELIMINARIES

In this chapter we present some background results which will be
used frequently in the remaining chapters of this thesis. The concepts

of separability and inertial subalgebras are particularly important.

§l. Notation and General Results

 

All rings will be assumed to be associative and to possess an
identity element l. All subrings will contain the identity of the
overring, and ring homomorphisms will map the identity to the identity.

Suppose A is a ring, R a commutative ring, and 9 is a ring
homomorphism of R into the center of A. Then a induces a natural
R-module structure on A defined by r - a = e (r)a for r e R, a c A,
and we say that A is an R-algebra. If A is a commutative ring as

well, then we call A a commutative R-algebra. An R-algebra A is said

 

to be finitely generated or projective if it is finitely generated or

 

 

projective as a module over R. For all rings R we let J(R) denote
the Jacobson radical (or radical) of R.
This lemma provides a link between J(R), J(A), and the maximal

ideals of R.

Lemma l.l: [l2, Lemma l.l, p. 78] Let A be a finitely generated

R-algebra, and let. r](mA) denote the intersection of the mA as m
runs over all maximal ideals of R.
(a) J(R) - A g J(A).
(b
(c) If A is projective, J(R)~ A gr1(mA).
( )

There exists a positive integer n such that (J(A))n g FlOmA).

v

d If A is separable, J(A) =r7(mA).

The next result gives an important connection between the radical

of an algebra A and the radical of a subalgebra of A.

Proposition l.2: [4, Corollary, p. l26] If A is a finitely

 

generated R-algebra and S is its subalgebra, then J(A)r1 S g J(S).

For a finite-dimensional algebra over a field, the radical of a

direct sum of ideals is nice.

Proposition l.3: [l, Corollary, p.29] Let A be a finite-

 

dimensional R-algebra, where R is a field, and suppose A can be

expressed as a direct sum of ideals A],A2,...,An, say

A = A133A29-“3An. Then JCA) = J(_A])63J(A2)EE wEBJCAn).

The next result is probably proven somewhere in the literature.

We sketch a proof here for completeness.

Proposition 1.4: Let A be a finitely generated, commutative

 

algebra over the commutative ring R, and let m be any maximal ideal

of R. ‘Then (J(A))m g J(Am), where for any R-module M, Mm denotes

the localization of M at m.

Proof; Every ideal of Am is an extended ideal, and the prime
ideals of Am are in one-to-one correspondence with the prime ideals of
A which do not intersect R-m. One can show, using the "Going Up
Theorem" [2, Theorem 5.ll, p. 62] that every maximal ideal of Am is
extended from a maximal ideal of A which does not intersect R-rn.
Thus,

J(A ) = fl (0

Q maximal in A

QO(R-m) = 0

> 2 (mom 2 <J<A>> .

m m

Next we state the well-known Nakayama's Lemma along with a useful

corollary.

Proposition l.5: (Nakayama's Lemma) [2, Proposition 2.6, p. 2l]

 

Let M be a finitely generated R-module and I be an ideal of R
contained in J(R). Then Io M = M implies M==0.

Corollary l.6: [2, Corollary 2.7, p. 22] Let M be a finitely

 

generated R-module, N be a submodule of M, and I g J(R) be an

ideal. If M = I- M + N, then M==N.
We will also need the following result on localizations.

Proposition l.7: [7, Proposition 4.4, p. 29] Let M be an R-module

 

such that Mm==0 for every maximal ideal m of R. Then M==O.

§2. The Concept of Separability

 

The theory of separable field extensions is well known. From the
work of Wedderburn, Dickson, Albert, and others during the early l900's
arose a generalization for algebras over fields. If R is a field and
A an R-algebra, then A is said to be separable over R if A¢8R F
is semisimple for every field extension F of R [l, p. 44]. This

has since come to be known as classical soparability. Albert showed in

 

[l, p. 44] that if A is a field extension of R, then (classical)
separability is equivalent to the usual field theoretic definition of

separability. He also gave the following useful result.

Theorem l.8: [l, Theorem 2l, p. 44] Let A be a finite-dimensional

 

algebra over the field R. Then A is (classically) separable over R
if and only if the center of each simple component of A is a separable

field extension of R.

In l960, Auslander and Goldman [3] extended the definition of
separability further to algebras over commutative rings. If A is an
R-algebra and if A0 denotes the R-algebra opposite to A, then we can

form the following short exact sequence of AI®R A°-modules:

o-+J—->A®RA°—E—»A——»o.

Here u is simply ”multiplication": u(a®b) = a'b. u will be used

throughout this thesis to signify the multiplication map, with further
notation added to avoid confusion when more than one multiplication map
occurs. J is the kernel of u, and J is the left ideal of A 8R A°

generated by all elements of the form a®l -l Ba, a e A.

Proposition l.9: [7, Proposition l.l, p. 40] The following

 

conditions on an R-algebra A are equivalent:
i) A is projective as a left A'BR A°-module under the u~structure.
ii) 0 -—+ J ——+-A<®R Ao —E+ A -—+ 0 splits as a sequence of left
A‘9R A°-modules.

iii) A<8R Ao contains an element e such that u(e) = l and

Je =0. (e is an idempotent called a separability idempotent

 

for A.)

Definition of Separability: [7, p. 40] An R-algebra A is called

 

separable if it satisfies the equivalent conditions of Proposition l.9.

If A is separable over R, then from Proposition l.9 we see that
J = ker h is a direct summand of A®R A°; i.e., there is a left ideal H

of AébR A0 such that HEEJ = A®R A°. In contrast we will examine in

Chapter 3 a generalization of pure inseparability developed by

Sweedler [l6] and Bogart [5] in which there is no left ideal H of At8R A0

for which H + ker u = A®R A°.

We list two examples of separable algebras.

Example l.lO: [7, Example II, p. 4l] If Mn(R) denotes the n><n

 

matrices with entries from R, then Mn(R) is separable over R.

Example l.ll: [7, Theorem 2.5, p. 50] If R is a field, then A

 

is separable over R if and only if A is finite-dimensional over R

and A is classically separable.

The following properties of separable algebras will be used

frequently in Chapters II and III. We list them here for reference.

Property l.l2: [7, p. 45] If a is an ideal of R and A is

 

an ha-algebra, then A is also an R-algebra. A is R-separable

if and only if A is R/a -separab19-

Property l.l3: [7, Proposition l.ll, p. 46] If A is a separable

 

R-algebra and I is an ideal of A, then A/I is a separable R-algebra.

Property l.l4: [7, Corollary l.7, p. 44] If A is a separable

 

R-algebra and S is any commutative R-algebra, then A<8R S is a

separable S-algebra.

Property l.lS: (Transitivity of Separability) [7, Proposition l.l2,

 

p. 46] Let S be a commutative, separable R-algebra, and let A be a
separable S-algebra. Then A is a separable R-algebra. Conversely, if
A is a separable R-algebra and S is any R-subalgebra of the center of

A, then A is separable over S.

Properoy l.l6: [7, Proposition l.l3, p. 47] Let A1 be an

 

R1-algebra and A2 an RZ-algebra. Then ATfl-qEA2 is a separable
[LrSERz-algebra if and only if A1 and A2 are separable over R1

and R2 respectively.

Property l.l7: [l5, Theorem 5, p. 5] If A is a finitely generated

 

R-algebra, and if S is a separable subalgebra of A, then S is a

finitely generated R-algebra.

§3. Wedderburn Factors and Inertial Subalgebras.

 

In l907 Wedderburn proved the famous Nedderburn Principal Theorem

 

in the case where A is a finite-dimensional algebra over a field F

of characteristic zero. We state its more general form.

Theorem l.l8: [l7] Let A be a finite-dimensional algebra over

 

a field F with A/J(A) separable. Then there exists a (separable)
subalgebra S of A such that $513J(A)=A.

S is called a Wedderburn factor of A.

 

In l951 Azumaya generalized Wedderburn's result to finitely generated
algebras over Hensel local rings. Recall that a local ring R with

maximal ideal m is called a Hensel local ring if it satisfies Hensel's

 

Lemma; that is, if f(x) e R[X] is a monic polynomial such that
f(x) = g(x)h(x) in R/m[x1, where g(x) and h(x) are monic and
relatively prime, then there are monic polynomials G(x) and H(x) in

R[X] with f(x) = G(x)H(x), G(x) = g(x), and H(x) = h(x).

Theorem l.l9: (Azumaya's Theorem) [4, Theorem 33, p. l45] Let A

 

be a finitely generated algebra over a Hensel local ring R with maximal
ideal m, and suppose A/J(A) is separable over R/m. Then there exists

a separable subalgebra S of A such that S + J(A) = A.

Azumaya called S an inertial subalgebra. In l965 Ingraham extended
this definition to algebras over arbitrary commutative rings. According
to [l2, Definition 2.l, p. 79] if A is a finitely generated algebra over
a commutative ring R, then a subalgebra S of A is called an inertial

subalgebra if S is a separable R-algebra such that S + J(A) = A. We

 

list two properties of inertial subalgebras.

Property l.20: [l2, Lemma 2.5, p. 80] If' S' g S are two inertial

 

subalgebras of a finitely generated R-algebra A, then S' = S.

Property l.2l: [l2, Proposition 2.6, p. 80] If A is a commutative,

 

finitely generated R-algebra, then A contains at most one inertial

subalgebra.

In [5] Bogart characterized certain subalgebras of a finite-
dimensional algebra over a field k which "become” Hedderburn factors
upon tensoring up with an appropriate field extension of k. She called
these subalgebras Wedderburn specters. He will examine these in Chapter III,
where we will generalize some of her results to algebras over commutative
rings.

In [l2, p. 85] Ingraham defined a commutative ring R to be an

inertial coefficient ring if every finitely generated R-algebra A for

 

which A/J(A) is separable contains an inertial subalgebra. By Theorem l.l9
we have that Hensel local rings are inertial coefficient rings. Finite
direct sums and homomorphic images of inertial coefficient rings are inertial
coefficient rings [l2, Proposition 3.2, p. 85, and Corollary 3.4, p. 86].

Also Noetherian Hilbert rings are inertial coefficient rings [T3, Corollary 2,

p. 553]. (Recall that a commutative ring R is a Hilbert ring if every

 

prime ideal of R is an intersection of maximal ideals of R.)

If A is a finitely generated R-algebra and I is an ideal of A,
then we say that we can ”lift idempotents from A/I to A” if every
idempotent in A/I is the image of an idempotent in A under the natural
map from A to A/I. Ingraham has conjectured that if R has the property
that idempotents can be lifted from A/J(A) to A for every finitely
generated R-algebra A, then R is an inertial coefficient ring. In [l4]

Kirkman proved the converse of this conjecture.

Theorem l.22: [l4, Theorem 4, p. 22l] Let R be an inertial

 

coefficient ring and A a finitely generated R-algebra. Then idempotents

can be lifted from A/J(A) to A.

We give one last property of idempotent lifting, due to Greco.

Property l.23: [l0, Corollary l.3, p. 46] Let R be a commutative

 

ring and A a finitely generated R-algebra with the property that
idempotents can be lifted from A/J(A) to A. Let I be an ideal of A

with I g J(A). Then idempotents can be lifted from A/I to A.

CHAPTER II
A THEOREM ON THE LATTICE OF SUBALGEBRAS
OF AN ALGEBRA
Let R be a commutative ring and A a finitely generated
R-algebra. We are interested in finding conditions under which a
maximal separable subalgebra T of A is inertial. It is clear that
if S is an inertial subalgebra of A, and if a is a unit of A

such that aTa"1

g S, then T is inertial. Thus, we are led to look
for conditions under which we can conjugate T into S.

We prove a more general result. If T is any R-separable subalgebra
of A, and S is a subalgebra of A with the property that S + J(A) = A,
then under the condition that R is an inertial coefficient ring, T can
be conjugated into S. Ford has given an example in [8, Theorem 2.3,
p. 43] of a certain class of rings R which are not inertial coefficient
rings and for which a finitely generated algebra A over R exists
having nonisomorphic inertial subalgebras. In view of this result, we
cannot expect to be able to conjugate T into S in general.

The theorem will be proven in six steps. The first two steps reduce
to the case where. A/J(A) is separable and S is an inertial subalgebra
of A. Then we prove the theorem where R is successively a field, a

local ring with nilpotent radical, a Noetherian local ring, and finally,

a Noetherian ring.

lO

ll

Theorem 2.l: Let R be a Noetherian inertial coefficient ring

 

and A be a finitely generated R-algebra. Let T be a separable

subalgebra of A, and let S be a subalgebra of A with the property

1

that S + J(A) = A. Then there is a unit a of A such that aTa' g S.

Proof; Stop_l; We first reduce to the case where A/J(A) is
separable over R. Let A1 = T + J(A). By Proposition l.2,
J(A) g J(Al)’ so that A1/J(A]) is a homomorphic image of T and
hence is separable by Pr0perty l.l3. Setting S1 = S FlA], Awe clearly
have S1 + J(A) g A]. To show equality, we write an arbitrary element
a1 of A1 = T + J(A) as t+-n where t e T, n e J(A). Since a1 also
lies in A = S + J(A), we have t + n = s + n1 for s c S, n1 8 J(A).
It follows that s = t + n - n

is in S Fl(T+J(A)) = S so that

l l’
a1 8 51 + J(A), and we have S1 + J(A) = A]. If the theorem is true for A
l

“I,
then there is a unit a e A1 5 A such that aTa' g S1 5 S. Thus, it
suffices to prove the theorem in the case that A/J(A) is separable over

R.

Stop_2; We will now reduce to the case where S is separable,
hence inertia]. By Proposition 1.2, 5 mm) 5 J(S), but S/S fl J(A)
z A/J(A) is semisimple, so S FlJ(A) = J(S). Since R is an inertial
coefficient ring, S contains a separable subalgebra S1 such that
S1 + J(S) = S, and it follows that S1 + J(A) = A. Clearly, if we can
conjugate T into S], we can conjugate it into S. Therefore, we

assume S is an inertial subalgebra.

l2

The remainder of the proof involves the following setting. Let

K = A/J(A), f = T/(TflJ(A)), and R = R/(RflJ(A)). Let

f:S®R T° —-—+7ie-§T° and ng®R T° ——>A_®ET° be the natural

maps, and let e be a separability idempotent for T with E'= g(e).

Then ker f = i[(SflJ(A)) eR T°+S ®R(TflJ(A))°] g J(S‘PRTC’)

[4, Theorem lO, p. l27]. By Theorem l.22 idempotents can be lifted from
(S®RT°)/J(S®RT°) to S®R T°, since R is an inertial coefficient ring.
Thus, Property l.23 implies that we can lift idempotents from A’S’ET"

to S<8R T°, so let e1 be an idempotent in S’ER T0 such that

f(e1) = 51 The picture looks like this.

S’9R T
f
el
\
\
\ \ A®—:['-°
\ R
9V5
’Ifi
/
o /
/
e “’T

If u :A<®R A° ——+ A is the multiplication map, then we will show that

u(e1) is the conjugating element we seek. In other words, providing e1

is an idempotent preimage of a; we will show that [J(e1)Tp(e1)'1 g S.

 

Step 3: Let R be a field. The proof of [5, Lemma 2.7, p. l27]
gives the existence of a unit a in A of the form a = li—n for n

1as. Ifwedefine o:T——>S by

in J(A) such that aTo'
¢(t) = oToT], then the map ¢ Bhl :leR T0 -+ SteR To makes the diagram

above commute. Furthermore, ker f = i [(Sf7J(A))’gRT°-+S‘9R(T(TJ(A))°] = O,

13

since S and T are separable over the field R. Therefore, if we
let e1 = Opal)(e), then e1 is the unique preimage of 5' in
SISR T°, and e1 is also an idempotent.

Because e is a separability idempotent for T, we have
(l®t-t®l) - e = O for every t in T. Applying d) 8 l, this becomes
(h8t-¢(t)§fl) ~e] = 0. Next apply p, recall that ¢(t) = ata-A, and
notice that u(e])t - ato"]u(e]) = 0. It follows that
u(e1)tu(e1)-] = ata‘A is in S, provided u(e]) is invertible. But
u(E) = l, so u(e]) = l + n for some n in J(A); consequently,

u(e]) is invertible.

Stop_4; Suppose (R,m) is a Noetherian local ring with mn==O
for some positive integer n. We proceed by induction on n. If n==l,
then R is a field, and the result follows from Step 3.

Assume the statement is true for n s k, and consider the case

where n = k + l.

k

Let A = A/(mkA), ii = R/m , i = T/(mkAflT), and §= S/(mkAflS).

~

Since mkA g J(A) by Lemma l.l, then J(A) = J(A)/mkA. Letting e]
and E be the images of e1 and e and taking f and E to be the

induced maps from f and 9, we have the following situation.

S SE T
E

  

l\'

 

 

l4

Both T and S are separable over R by Properties l.l3 and l.l2,

5 is a separability idempotent for T, and S + J(A) = A.

I

Then the induction hypothesis gives that h(e,)ia(e,)' g S. Pulling
k

this inclusion back to A, we have u(e])Tu(e1)'1 g S + m A.

kA and T' = p(e1)Tu(e])']. S is an inertial

Now let C = S + m
subalgebra of C, TI is a separable subalgebra of C, and C is a
finitely generated R-algebra, because R is Noetherian. Write
e = ET,- 8 61., where Y,- e T, 6_i e To, and let
e'==Z[u(e1)yiu(e1)'1<8 u(e])6iu(e])']]. One easily sees that e' is
a separability idempotent for T'. Write e1 = Zojls Bj where “j e S
and Bj e T°, let e; = 2oj<s u(e1)8ju(e1)'1, and notice that ei is
an idempotent. It is not hard to see that J(C) = Cl? J(A), C/J(C) = A,
and T'/(T'flJ(C)) = T: Thus, we have natural maps

f':S® T'°-—->A®-T° and g':T'e T'°-——>A®—T° with

R R R R
f'(e&) = 5‘: g'(e'). We can now use the same argument here for C that
we used previously for A to conclude that u(e,')T'pl(e]')"1 g S + ka =
S + mk(Si-mkA) = S. Equivalently, h(ei )u(e1)Tu(e])'1u(e1')'1 g S. But

H(e1)u(e]) = (Zo.u(e])8ju(e1)-1) ' u(e]) = Zo.u(e])8- = [ZaJ-®Bj]-l1(e])

J J J
= e]- u(e]) = u(e]~e]) = u(e]). Thus we have shown that
“(9])TU(e])-] 5 5°

Step 5: Let (R,m) be a Noetherian local ring. Let k be a
positive integer, and pass to the factor algebra A = A/mkA over

= R/mk. Letting l = T/(mkAflT) and E = S/(mkAflS), we have that

xn

T is R-separable, and ~S is an R-inertial subalgebra of A. Taking

N

e, E], f, and 3 to be defined as they were in Step 4, we can again

refer to the diagram on page l3. R is a local ring with maximal ideal

m = m/mk, and mk = 0, so we can apply the result of Step 4 to get

l5

[l('§])fl'u('€il)'1 g S. Pulling back to A we have p(e])Tu(e])'] g S + mkA.

This containment holds for every positive integer k, so we can write

[J(e1)Tu(e])'1 g,r% (Si-mkA). But R is a Zariski ring [l8, p. 263, 264],
k=l

00

so by [l8, Theorem 9, p. 262] we have (A (Si-mkA) = S, and again we
k=l

-1 c S.

have shown that u(e])Tp(e1)

, . . ._ -l
Step 6. Let R be a Noetherian ring and T - u(e])Tp(e1) .

We will show that T' g S by showing that Z = (T"+S)/S is the zero
module. Z=O if and only if Zm = Z 8R Rm = O for every maximal ideal
m of R, by Proposition l.7. By tensoring everything in the diagram
on page l2 with Rm over R, we again place ourselves in the setting

of Step 5, where we have T$ 5 Sm, or equivalently, Zm==0. We conclude

that 2 =0, and it follows that u(e])Tu(e1)"] g s.

Z/4Z[x]

(xi-2)

By Example l.lO, S is separable over R. It is not hard to see that

Example: Let R = 2/42, A = M2( ), and s = M2(Z/4Z) g A.

A) = A(3 2) + 2A, so that s + J(A) = A, and s is an inertial

subalgebra of A. T1 = {(8 g>|a,bs:R} is a separable subalgebra of A

with separability idempotent e = ((1) 8) ® ((1) 8) + (g (1)) 9 <8 (1) . Then
_ l x 3a+2b (a+3b)_'
the subalgebra T - <§.1T%> {( 3a+b) )x 2a+3b X) a,be:R}.

 

separable subalgebra of A with separability idempotent

e=<3Xx 2 ‘ g(3x 2>+< (2 3X) 2 (x3_> It is interesting to illustrate
the method of proof of Theorem 2.l with this example, so consider the

following diagram.

S®RT
f
e]\
\
‘T‘T ‘-c Itéfiizf°
3 i=<$8 an +(8?)®<8?)
T®RT / /
/
e: (§z§)®(§:§>+<§?3 WM???)

We will consider various candidates for e].

l. Let e1 = O 8)l® (ii-g) + (8 $>48 §-§;>. This element is

an idempotent, and u(e1) = §'§> , so that u(e])Tp(e1)"1 = T1 E S.

It is not surprising that (§'§> arises as p(e1), since it is the

inverse of <% f), the element used to conjugate T1 to T in the
first place, although we have not shown that all conjugating elements

arise as u(e]) for some idempotent e].

. _ “l 2 3 x O 2 2 3x . .
2. Let e1 - (O 0 9(3x 2) + (O l ®<Y3 . This element 15 also

an idempotent, but here u(81)= (i §+—>, and
-l _ ‘a 2a+2b
H(e])TU(e‘I) "' {<0 I) >

that as e] varies, both u(e]) and u(e])Tp(e])‘1 may vary. The

a,be:Z/4i} g S. We see from this example

 

theorem, however, guarantees that as long as e1 is an idempotent,

h(epime >" s s.
_ (O O . 2 32' . .
3. Let e — (g 8>® \3x 2) 3)<8 <§.3 > . This element 15 not

an idempotent, although it is a <preimage of 5' in S'SR T°. However,

this element is simply a scalar multiple of the first candidate for e],

 

l7

so that u(e])Tu(e])'] g S. Thus, some nonidempotent preimages of '5

may satisfy u(e])Tp(e])-] g S.

4. Let e1 = (3 g ‘6 (37 32?) + (g I) ® (32? 3x . This preimage of e

 

. . _ l 2*?
15 not an idempotent. p(e]) - (2+3R'3 > , and
-l _ a 2a+2b+2a§42b_'
“(9])”(91) ' {(2a+2b+2af+2bx_ b > M E R}?- 3'
Hence, not all preimages of 5' have the property that [J(e])Tp(e])-1 g S.
D

In the proof of Theorem 2.l, once the reductions of Step l and
Step 2 are made, the only place we use the fact that R is an inertial
coefficient ring is when we wish to lift idempotents. Thus, if we
start with the assumption that A/J(A) is separable and S is an

inertial subalgebra of A, we have the following corollary.

Corollary 2.2: Let R be a Noetherian ring with the property

 

that for every finitely generated R-algebra idempotents can be lifted
from the algebra modulo its radical to the algebra. Let A be a finitely
generated R-algebra with A/J(A) separable, and let S be an inertial

l

subalgebra of A. Then there is a unit a in A with aTa' g S.

Conjugates of inertial subalgebras are inertial subalgebras, so
if we are in a setting where inertial subalgebras exist and Theorem 2.l

applies, the following corollary holds.

Corollary 2.3: If R is a Noetherian inertial coefficient ring

 

and A is a finitely generated R—algebra with A/J(A) separable, then
every separable subalgebra is contained in an inertial subalgebra, and

every maximal separable subalgebra is an inertial subalgebra.

 

l8

When A is a commutative, finitely generated algebra over a

commutative ring R, the situation becomes much simpler.

Proposition 2.4: Let A be a commutative, finitely generated

 

algebra over a commutative ring R. Let S be a subalgebra of A
with S + J(A) = A. If T is a separable subalgebra of A, then
T g S.

Proof: If we consider A as an S-algebra, then S is an
S-inertial subalgebra of A. By Pr0perty l.l4, S<8R T is an S-separable
algebra, and S- T is a homomorphic image of S<SR T, so by Property

l.l3, S- T is an S-separable subalgebra of A. Furthermore, since

5 S s. T, s. T is also an S-inertial subalgebra of A. Therefore, by

Property l.2l, S- T = S, and, consequently, T g S.

It would be nice to be able to eliminate some of the restrictions
on R in Theorem 2.l. The following result lifts both the Noetherian
and the inertial coefficient ring conditions on R, but we are forced

to replace the Jacobson radical of A with a nil ideal of A.

Proposition 2.5: Let R be a commutative ring and A be a

 

finitely generated R-algebra. Let I be any nil ideal of A, and let
S be an R-subalgebra of A such that S + I = A. If T is any

separable subalgebra of A, then there exists an element a in A

I

such that aTa' g S.

l9

Proof; We shall use the technique of selecting a suitable Hilbert
subring R1 of R and an Rl-algebra A1 which satisfy the conditions
of Theorem 2.l. We then lift the result back to A.

By Property l.l7, T is a finitely generated R-algebra, so write
T = Rt1 + Rt2 + ... + Rtm. T is R-separable, so there are elements
x1. and y, in T such that 2x1. ®y1. is a separability idempotent
for T in T 8h'T°. Thus, we have, for every j==l,...,m,

(*) (thl-l®tj)(2x1.®y1.)= 0 ln T®RT .

Think of T‘8R T0 as a free abilian group with subgroup a7 of relations
factored out, and notice that there is a finite subset Mj of TlJ R
such that the elements of d’ making (*) zero in T®RTo are
expressible in terms of the elements of Mj'

Let a],...,an generate A as an R-module. Since S + I = A,

there exist 5],...,s in S and u],...,u in I with a. = s. + u-

n n l l l

for 'i=l,...,n.

Now set 8 = {l,a1.aj,s1.,t1.} and C = {l,t1.tj,x1.,y1.}d (sJJMj).
Write each element of the finite set B as an R-linear combination
of a1,...,an, and write each element of the finite set C as an
R-linear combination of t],...,tm. All of this will involve only
finitely many coefficients from R. Let R1 be the Noetherian subring
of R generated by this finite set and the ”prime" subring P of R.
P is a homomorphic image of the Hilbert ring Z, the integers, so
P is a Hilbert ring. R1 is finitely generated as an algebra over P,
so R1 is a Hilbert ring [9, Theorem 2 and Theorem 3, pp. l36-l37].

Therefore, R1 is an inertial coefficient ring [l3, Corollary 2,

p. 553].

20

Define A1 = Rla1 + Rla2 + --' + Rlan' By construction of R1
we have 8 g A], *so A1 is a finitely generated R1-algebra containing
the 51's and the ti's. Consequently, we can take S1 to be the
R1-subalgebra of A1 generated by s,,...,sn. Next let
T1 = R1t1 + th2 + ~-- + thm’ so T1 is a finitely generated R1-algebra

containing the set C. Furthermore, in<8 yi is an element of

T1®R1 T1 satisfying (*) in T13},1 T1. Consequently. XX,- Pr,

is a separability idempotent for T], so T1 is Rl-separable. Finally,

we let I1 = If? A]. Since I is nil, I1 is nil, and it follows that

I1 5 J(A1). Recall that a, - Si = p, is in I, and since a, - Si
is also in A1, then a, - Si = pi is in I1 for i =l,...,n. The
relations a, = s, + pi imply that S1 + I1 = A], and it follows that
31+ J(A) = A].

Now A1 satisfies all the conditions of Theorem 2.l, so there is

a unit a in A1 such that aT1a"1 5 S1. We next extend back up to

g S, and RT = T.

A by multiplying by R to get RA1 = A, RS 1

1
Considering a now as an element of A, we have

-l

aTa = a(RT )a'1 = R(aT]a']) g RS1 g S, and we are done.

I

Remark: If R is a Hilbert ring and A is a finitely generated
R-algebra, then it is not difficult to show that J(A) is nil. This is

one setting in which Proposition 2.5 applies.

CHAPTER III
PURE INSEPARABILITY AND NEDDERBURN SPECTERS

In [l6] Sweedler defines an algebra A over a commutative ring

R to be purely inseparable over R if the multiplication map

 

u : A ®R A° -—-> A gives an A ®R A°-projective cover of A. This is
equivalent to ker u being a small left ideal of AwsR A°; that is,

if M is a left ideal of A 3R A° with M + ker u = A® A°. then

R
M = A<8R A°. Furthermore, A is purely inseparable over R if and
only if ker a g J(A eRA°).

If A 3 S are rings, then Bogart [5] extends Sweedler's definition
by taking A to be purely inseparable over S if the A-A bimodule
map u :Algs A0 -—+ A has small kernel. If C = Z(A)(W S, where Z(A)
is the center of A, one can consider u to be an A mp A°-map, and
if A is an R-algebra and S is a subalgebra of A, then u can be
considered to be an A<8R A°-map. The smallness of ker p is inde-
pendent of which of the three module structures one uses.

In the case where S is a commutative ring and A is an algebra
over S, Bogart's definition reduces to Sweedler's definition. If A
and S are fields, Sweedler has shown that his definition is equivalent

to the usual definition for purely inseparable field extensions

[l6, Theorem l2, p. 35l].

21

 

22

Example: Let R = Z/2Z(o), the field of functions over Z/2Z,

and let A = R[X]/(xz-ta). Then A is a purely inseparable algebra
over R, since A is a purely inseparable field extension of R. As
an illustration of the definition of a purely inseparable algebra, we
will show that ker u g J(A SRA)‘ In Chapter I we saw that ker u is
generated as an ideal of A @R A by elements of the form y QT + l 3 y
for y in A. Moreover, y can be written as ai-bx, where a and

b are in R. Therefore, y @l + l ®y = (a+bx) sol + l 5o (a+bx) =

a ®l +bx ®l +l®a+l®bx=b°(x®l+l®x), so

ker u = (A @RA) ° (x ®l+l @x). Furthermore, (x®l +l ®x)2
on 81 + l so = 0, so that x ®l + l ®x is in J(AQRA), and it

follows that ker p g J(A @RA).

While Bogart and Sweedler present results in the general setting
of these definitions of pure inseparability, most of their work deals
with algebras over fields. This chapter extends certain of their results
to algebras over commutative rings. We begin with a proposition giving
some basic properties of pure inseparability which will be used through-

out this chapter.

Proposition 3.l: Let A be a finitely generated algebra over a

 

commutative ring R.

a) Let S be a subalgebra of A such that A is purely inseparable
over S, and let I be an ideal of A. Then A/I is purely
inseparable over S/(STTI).

b) Let A be commutative and S be a subalgebra of A. Then A is
purely inseparable over S if and only if Am is purely insep-

arable over Sm for every maximal ideal m of R.

 

23

c) Let A be commutative, S be a subalgebra of A, and P be a
finitely generated, commutative, faithfully flat R-algebra. Then
A is purely inseparable over S if and only if A<8R P is

purely inseparable over S<8R P.

.Ppoof; (a) The proof of this is essentially the same as Bogart's
proof of [5, Proposition 2.lO, p. 128]. We include it here for com-
pleteness.

Let A'= A/I, n :A ——+ A. be the canonical map, and

S-= S/S FlI. Consider the commutative diagram with exact rows given

 

below.
0 >L >A®SA°————u-——->A——————+O
TT®TT°IL ln®n° [TI
O-—-——+K—-——-—>A® A° E A.——————>0

Here ker u = L and ker fi'= K. Since the diagram commutes,
(Tr®ii°)(L) 5 K. To show equality, we pick E c K, and let

a c (non°)“(‘e') ngS A°. If b = a - (u(8)®l), then u(b) = 0,
so b e L. Furthermore, (n®n°)(b) = B, since TT(IJ(B)) = IKE) = O,

—0

so that (ii®ir°)(L) = K. If M is an A-A submodule of A®§A
with M+ K = NEE-IA”, then for M = (n®n°)‘](M), we have
M + L = Ass A°.- Since L is small, M = A®S A°. This implies that

(ii®ir°)(M) =M=A®§A°, so K is small.

24

(b) Let m be any maximal ideal of R. Consider the following

commutative diagram.

A®SA __1.J__+A

ml ($2

u
m
Amls A -—————+ A

lll m
Sm

Here p and “m are the multiplication maps, o2 is the natural map
from A to Am, and to] TS ¢2®o2. Since Am=A®RR and

m
Am :35 Am z (A ®RR (A CORR ) = (A® we can rewrite

A)®R
m R

m m’

) e
m S®RRm S

the preceding diagram.

u®l
Alfis AlgR Rm -—-——-—#-A<8R Rm
Since Rm is flat over R, then ker(u®l) = (kerp) ®R Rm,

and furthermore, ker(u®l) z ker pm.

(<=) Suppose M is an A-A submodule of A®S A such that
M + ker u = A ®S A. Tensoring up with Rm, this becomes
M ®R Rm + (kerp) ®R Rm = A83 A®R Rm. Then M ®R Rm + ker(u®l) =

A69 A®R Rm, and, since ker(u ®l) is small, we have

S
M 8R Rm = A ®S A 58R Rm. This equality holds for every maximal ideal m
of R, so by Proposition l.7, M = A®S A. Therefore, ker p is

small, and A is purely inseparable over S.

25

(=)) Since A is purely inseparable over S, then

ker u g J(A @SA). Tensoring up with Rm, we have

(keru) 3R R _c_ J(A @SA) ®R Rm. By Proposition l.4,

lTI

J(A @SA) ®R Rm 5 J(A @SA ®RRm), and we have ker(u®l) g J(A®SA ®R m).

Consequently, ker pm 9 J(Am @SmAm), and Am is purely inseparable

over Sm.

(c) Consider the following commutative diagram.

 

A ®s A . U > A
.1] 1.,
u
(A @RP) ®S®RP (A ®RP) —-———+ A oR P

Here u and D are multiplication maps, 62 is the natural inclusion,
and o] is 422 ® 62. It is not hard to see that
(A oRP) ®soRp (A eRP) = (A ®SA) oR P via (a1® [31) e (azepz) ——->

(a1®a2) ® pIpZ. Therefore, we can rewrite the preceding diagram.

 

A®SA “ >A
(At P2
(A®SA) eR [DJ-LL AeR P

Since P is flat over R, then ker(u®l) = (kerp)® P, and also

R

ker(p®l) 2* ker [1.

(<=) Let M be an A-A submodule of A® A such that

S
M + ker u = A®S A. Then M®R P + (keru) ®R P = (A®SA) ®R P.
Since (keru) ®R P z kerfi is small, then M®R P = (A @SA) ®R P,
and it follows that M = A62)S A because P is faithfully flat over

R. Thus, ker p is small, and A is purely inseparable over S.

26

(==)) Let a(R) denote the set of maximal ideals of R.
Alas A is a finitely generated R-algebra, so by Lemma l.l there is
SAM”: F‘s m-(A®SA).
meO(R)

Pg[ O m- (A®SA)] e
m€Q(R)

J(A 83A @RP), where the last inclusion follows

a positive integer n such that [J(A®

[J(A e A)]n o

P1" 3

Then [J(A @SA) ®

P:

R R R

| n

[l m-(A® A® P)
ch(R) S R

from Lemma l.l. Consequently, J(A ®SA) ®

since ker u S. J(A ®SA), then (keru) ®

P _C_ J(A ®SA ®RP), and

A)®RPE

R

RPEJ(A®

J(A GSA @RP). Therefore, ker u _c_ J((A ®RP) ®

5

A ®R P is purely inseparable over S ®R P.

Example: The conclusion of Proposition 3.l(c) need not be true if
P is not finitely generated over R. If R = Z(q) (the integers
localized at a prime ideal q), A = Z(q)[y]/(y2-q), and P = Z(q)[x],
then P is faithfully flat over R because it is free, but P is not
finitely generated over R. Let m be the ideal of A<8R A generated
as a module over R by the set {q cal, l®y, y®l, y®y}. m is
maximal because (A ShA)/m is a field, and in fact m = J(A<8RA)
because the square of each generator of m lies in J(AlsRA).
Consequently, A 8R A is local, so ker p g_m = J(A<8RA), and A is
purely inseparable over R.

= . 2 ° f ..__L .f f
A®RP A®Z(q)2(q)[X] A[X] Vla a® (x) a (x) or

a c A, f(x) 8 Z(q)[X]. Therefore, (A ®RP) ®P (A ® P) 2 A[x1 3 AU] =

R P
(A eRA)[xi via ax" e bxm —+ (a®b)xn+m for a,b c A. We have the

following commutative diagram.

27

 

p
P
(A eRp) op (A ®RP) ————-+ A @R P
(A ®RA) [X] [J > A[X]

The map u acts as multiplication on Al8R A and leaves x fixed,

and the element y ®l - l ® y is in ker ii. For the polynomial ring

(A ShA)IXl, the Jacobson radical is equal to the nilradical. If n

is any even integer, then

new" = [<8)+(2)+~'+<2>lt“/2®l)-[(i)+(3)+-"+(n9i)lqfl%<y®y>

750. Therefore, y ®l - l ®y Pf J[(A ®RA)[X]], so that

ker u g J[(AtsRA)[X]], and consequently, A<8RP is not purely

inseparable over R<®R P 2 P.

The next result shows the relationship between pure inseparability

and maximal separable subalgebras.

Proposition 3.2: Let R be a Noetherian ring and A be a finitely

 

generated R—algebra. Let S be a separable subalgebra of A with A
purely inseparable over S. If

a) R is local with maximal ideal m, or

b) A is commutative,

then S is a maximal separable subalgebra of A.

28

Proof: (a) Step l: Assume mk = O for some positive integer k.

Let 'R = R/m, AI= A/mA, and S'= S/(SflmA). By Proposition 3.l(a),

A is purely inseparable over S: A is a finite-dimensional algebra
over the field P‘, so by [5, Proposition 2.], p. l24] s is a
maximal separable subalgebra of A:

Suppose there is a separable subalgebra S' of A with S 5 5'.
Setting '§‘ = s'/(S'rlmA), we have S,l = S, since S- is maximal
separable, and it follows that S' + mA = S + mA. Consider the R-algebra
B = S' + mA = S + mA. Clearly mA g J(B), so both 5' and S are
inertial subalgebras of B. Since R is Noetherian, B is finitely
generated over R, and Property l.20 yields S = 5'. Therefore, S is
a maximal separable subalgebra of A.

Stop_2; Let (R,m) be a Noetherian local ring. Let k be a
positive integer, and pass to the factor algebra A'= A/mkA over
R'= R/mk with separable subalgebra S-= S/(mkATlS). By Proposition 3.l(a),
A' is purely inseparable over S; so by Step l, S- is a maximal

separable subalgebra of A' for each k.

Suppose there is a separable subalgebra S' of A with S g S'.

Setting 3 = s'/(mkArls'), we have ‘s‘ = 5', and it follows that
k k

S' + m A = S + m A for every positive integer k. In particular,

8' g S + mkA for each k, or SI 5,]; (Si-mk ). Then, since R is
k=l

a Zariski ring, '3' c r] (S+mkA) = 3 [l8, Theorem 9, p. 262], and

‘ k=l
therefore, S is a maximal separable subalgebra of A.
(b) Let SI be a separable subalgebra of A with S g S', and
let m be any maximal ideal of R. By Proposition 3.l(b), Am is

purely inseparable over Sm. By Property l.l4, Sm and 8% are

separable over Rm’ so by (a), Sm = 8%. This equality holds for every

29

maximal ideal m of R. Applying Proposition l.7 we have S = 8', so

that S is a maximal separable subalgebra of A.

The next three results, Lemma 3.3, Proposition 3.4, and Proposition
3.5, are the unpublished work of Edward C. Ingraham. I am grateful to
him for allowing me to include them here. They provide us with a link

between pure inseparability and inertial subalgebras.

Lemma 3.3: Let A 3 S be rings, and let C = Z(A)(7 S, where
Z(A) is the center of A. Then A is purely inseparable over S if
and only if for every x in A e A° with u(x) = l,

S

(A®CA)-x=A®SA.

Proof: (-—->) Let ch®S A° with p(x) =l. Then

ker p + (A ®CA°) - x = A35 A°, whence (A ®CA°) - x = A <83 A° by the

smallness of ker p.

(<=) Conversely, suppose ker u + M = A®S A° for some
A ®C A°-submodule M of A88 A°. (We can write l ®l r: A®S A° as
k + x, for some k e ker u, x e M. Then p(x) = l, and by the

assumption, A®S A° = (A ®CA°) ° x g M, so ker p is small.

Remark: If L is a finitely generated R-module, then J(R)- L
is small in L, since J(R)- L + M = L implies M==L by Corollary
l.6.

30

Proposition 3.4: If A is a ring, S a subring, and

 

C = Z(A)TW S such that
. l. A = S + N for some ideal N in J(A) such that
2. i(A®CN ) +i(N®CA)_<;J(A®CA),

then A is purely inseparable over S.

Proof: A®S A° = i(S e 5°) + i(N ®SA°) + i(A® N°). Choose

S S
x e A®S A° with u(x) = l. Then x = s + n, where s e i(S®SS°) z s
and n e i(N ®SA°) + i(A®SN°). Now l = p(x) = (1(5) + u(n) = s + u(n),

and u(n) e N g J(A), so 5 is a unit in A. Thus,

A®SA =(A®CA)°s, and (A®CA):s=(A®CA)(x-n)=

(A ®CA°) - x + (A ®CA°) . n. Since n e i(A ®SN°) + i(N eSA°),

(A epA°): n has elements of the form Zoi<8 Bi’ where a1 or

Bi 5 N, which implies that (A QEAO) on g
[i(A ®CN°)+i(N ®CA°)] - (Ao A°) 3J(A ®CA°) - (A® A°). Thus, by the

S S
preceding remark, (A ®CA°) - n is small, since A $5 A° is generated
by l ®l over A ®c A°. Therefore, A 85 A° = (A ®CA°) ° x. Then

Lemma 3.3 gives that A is purely inseparable over S.

Proposition 3.5: Let A be a finitely generated R-algebra over

 

the commutative ring R, and let S be an inertial subalgebra of A.

Then A is purely inseparable over S.

Proof: Let C = Z(A)f7 S, and let N = J(A). By [4, Theorem lO,
p. 1271, i(A ®CN°) + i(N®CA°) g J(A®CA°), since A is finitely

generated over C. Apply Proposition 3.4.

3l

The next lemma is a generalization of a standard field theory
result and also of a result of Sweedler [l6] for algebras over fields.
Recall that in the category of rings a homomorphism f is called an
epimorphism if for any two homomorphisms g and h the equality of
9° f and h° f implies the equality of g and h.

Lemma 3.6: Let R be a commutative ring and A be an R-algebra.
If A is both separable and purely inseparable over R, then R—-+ A
is an epimorphism. If in addition A is finitely generated over R,

then A = R.

Epoof; A purely inseparable over R means that for
p :A ®R A0 -—+ A, ker p is small. A separable over R means that
ker p is a direct summand of A ®R A°. Consequently, ker p = 0,
and by [l5, Theorem l, p. 2] the map R~—+ A is an epimorphism. If
A is finitely generated over R, then [l5, Corollary 4.2, p. 4]
yields A= R.

Example: If A is not finitely generated over R in Lemma 3.6,
we do not necessarily have A= R. Q (the rational numbers) is both
separable and purely inseparable over Z (the integers), since

oezoea. but an.

Now we are able to combine several of the preceding results to get
the following proposition, which provides a link between pure insep-
arability, inertial subalgebras, and maximal separable subalgebras of

A when A/J(A) is separable.

 

32

Proposition 3.7: Let R be a commutative ring and A a finitely

 

generated, commutative R-algebra. If A/J(A) is separable, then the
following are equivalent:

a) S is an inertial subalgebra of A.

b) A is purely inseparable over S, and S is separable over R.
If R is a Noetherian inertial coefficient ring, then (a) and (b)
are equivalent to

c) S is a maximal separable subalgebra of A.

M:
(a) =5) (b) is Proposition 3.5.
(b) =fi> (a) A/J(A) is separable over R, so A/J(A) is separable
over S/[SflJ(A)] by Property l.l5. Since A is purely inseparable
over S, then A/J(A) is purely inseparable over S/[SflJ(A)] by
[l6, Proposition 6(e), p. 345]. Lemma 3.6 then gives
A/J(A) = S/[SFlJ(A)], which implies A = S + N. Since S is separable,
then S is an inertial subalgebra of A.

Assume now that R is a Noetherian inertial coefficient ring. Then
(b) ==> (c) is Proposition 3.2.
(c) =e> (a) Since R is an inertial coefficient ring, A contains an
inertial subalgebra T. By Proposition 2.4, S g T. Since S is

maximal separable, S==T, so S is an inertial subalgebra of A.

Let A be a finite-dimensional algebra over a field K. In [5]

Bogart defines a separable subalgebra B of A to be a Wedderburn

 

specter if for any field extension K of k for which (A @kK)/J(A ®kK)

is a separable K-algebra it follows that B®k K is a Wedderburn

33

factor for Al®k K over K. An algebraic closure k of k is always
an extension of k for which (A ®kk)/J(A ®kk) is R—separable, so we
can always check to see whether B is a specter for A over k by
checking to see whether B @k k- is a Wedderburn factor of A®k k.

In addition, Bogart proved that B is a specter for A over k if
and only if A is purely inseparable over B, which provides us with
an internal characterization of specters.

If we wish to extend this definition of specter to a finitely
generated algebra A over a commutative ring R, we need to find an
extension P of R for which (A ®RP)/J(A ®RP) is P-separable and
then look for a subalgebra B of A for which 8 8R P is an inertial

subalgebra of A 8RP' We know by Proposition 3.7 that A<8 P must be

R
purely inseparable over BlsR P, so if we hope to characterize a specter
B of A as a separable subalgebra for which A is purely inseparable
over B, then it would be useful to find a P which is finitely
generated and faithfully flat over R so that Proposition 3.l(c) holds.
The next set of results will show that if R is a commutative, semi-
local ring, there is always a finitely generated, commutative, free
R-algebra P such that (A @RP)/J(A 8 P) is P-separable. We begin

R
with two lemmas.

Lemma 3.8: Let R be a commutative ring, A a finitely generated
R—algebra, and S a finitely generated commutative R-algebra. If
A/J(A) is separable over R, then (A<8RS)/J(A<®RS) is separable over
S.

34

Proof: By [4, Theorem lO, p. l27] i(J(A) ®RS) g J(A ®RS), so
(A ®RS)/J(A ®RS) is a homomorphic image of (A ®RS)/i(J(A) 8R3)

2 [A/J(A)] ® S. By Property l.l4, [A/J(A)] ®R S is S—separable, and

R

it follows from Property l.l3 that (A ®RS)/J(A 8R5) is S-separable.

Lemma 3.9: Let R be a commutative ring, S a commutative,
finitely generated R-algebra, and A a finitely generated, separable

S-algebra. If S/J(S) is R-separable, then A/J(A) is R—separable.

Proof; By Lemma l.l J(S) -A.g J(A), so A/J(A) is a homomorphic
image of A/J(S)- A. But A/J(S)° A is a separable S/J(S)-algebra,
so by Property l.l5 A/J(S) -A is a separable R-algebra, and it follows
that A/J(A) is R-separable.

The next two results, Propositions 3.l0 and 3.ll, show that if A
is a finitely generated algebra over a commutative, semilocal ring R,
then we can construct a finitely generated, free, commutative R-algebra
P with (A<®RP)/J(A ShP) separable over P. Proposition 3.lO is the
special case where R is a field k, and here P is actually a finite
field extension of k. Since this result is used to prove Proposition
3.ll, the finiteness of P is important, which explains why we cannot
use 'k, the algebraic closure of k, for P. In Proposition 3.lO P
is shown to be a purely inseparable extension of k, but it is not
known whether the P constructed in Proposition 3.ll is purely insep-
arable over R.

The proof of Proposition 3.lO is in five steps. A is successively

a finite field extension of k, a finite-dimensional division ring

 

35

over k, a matrix ring over a division ring which is finite-dimensional
over k, a finite-dimensional, semisimple k-algebra, and, finally, an

arbitrary, finite-dimensional k-algebra.

Proposition 3.l0: Let A be a finite-dimensional algebra over a

 

field k. Then there exists a finite, purely inseparable, TTGId 9X-

tension P of k such that (A ®kP)/J(A®kP) is separable over P.

Proof: Step l: Let A be a finite field extension of k. Then

there are elements a],a2,...,an such that A = k(a1,a2,...,an). Let

fi(x) denote the minimal polynomial in k[X] for ai, and let F
be a splitting field for the family {fi(x)}1.:1 over k. Then F is
a finite normal extension of k containing A, and it follows that
there is a purely inseparable field extension P of k contained in
F with F separable over P.

If p :Plsk P -+ P is the multiplication map, then
ker u g J(P<sk ), since P is purely inseparable over k. Therefore,
(P GkP)/J(P ®kP) is a homomorphic image of (P ®kP)/ker p 2 P, which
is P—separable, and consequently, (Plng)/J(P<8kP) is P-separable.
Furthermore, since F is separable over P, then F ®P(P®kP) .- F®kP
is separable over P 8k P by Property l.l4. Thus if we take
F®k P to be A, P®k P to be S and P to be R in Lemma 3.9,
we conclude that (F ®kP)/J(F ®kP) is P-separable.

Now A ®kP g F ®k P, and by Proposition l.2
J(F ®kP) 0 (A @kP) _c_:_ J(A ®kP). Therefore, (A ®kP)/J(A ®kP) is a

homomorphic image of (A ®kP)/[J(F ®kP) H(A ®kP)] , which is isomorphic

to a subalgebra of (F ®kP)/J(F ®kP). Since homomorphisms preserve
separability, if we can show that all subalgebras of (F ®kP)/J(F @kP)

 

36

are P—separable, then it will follow that (A ®kP)/J(A ®kP) is
P-separable, and we will be done.

Let 8 denote the separable P-algebra (F ®kP)/J(F®kP). B is
commutative, semisimple, and satisfies the descending chain condition
on ideals, so by the Wedderburn structure theorems it is a direct sum
of fields, say 8 = 81638263 «~GEBm where each B1. is a field
extension of P. In fact, Theorem l.8 yields that each Bi is a
separable field extension of P. Let C be any P-subalgebra of B,
and let G be any field extension of P. Then C<®P G is a subalgebra
of B op G : (B1®PG)EE(B2 ope) 6E 63 (am ope), and for each i,

B1. ®P G has no nonzero nilpotent elements, since 81. ®P G is
G-separable. Hence, B<8P G has no nonzero nilpotent elements, and so
J(ClapG) = 0. Therefore, C is separable over P, and we are done.

Stop_g; Let A be a finite-dimensional division algebra over k,
and let C be the center of A. By Step l there is a finite, purely
inseparable, field extension P of k such that (C @kP)/J(C®kP) is
P-separable. A is a central simple C-algebra, so by [ll, Lemma 4.l.l,
p. 90] A is a separable C-algebra, and it follows that
A ®C(C @kP) 2 A 8: P is separable over C ®k P. Then Lemma 3.9 yields

k

that (A ®kP)/J(A ®kP) is separable over P.

Stop_§; Let A be a matrix ring over a division ring D, say
A = Mn(D), where. D is finite-dimensional over k. By Step 2 there
exists a finite, purely inseparable, field extension P of k such
that (D @kP)/J(D®kP) is separable over P. Since
Mn(D) ®k P 2 Mn(D ®kP) and J(Mn(D ®kP)) 2 Mn(J(D ®kP)), we have
[Mn(D) ®kP]/J[Mn(D) ®kP1 2 Mn(D ®kP)/Mn(J(D ®kP)) 2 Mn((D @kP)/J(D ®kP)).

If K is any field extension of P, then [(D ®kP)/J(D®kP)] ®P K is

 

37

semisimple, because (D ®kP)/J(D ®kP) is P-separable. It follows that
Mn[((D ®kP)/J(D @kP)) ®pKl 2 Mn[(D ®kP)/J(D ®kP)] ®p K is semisimple,
so Mn[(D ®kP)/J(D ®kP)] 2 (Mn(D) ®kP)/J(Mn(D) ®kP) is P-separable.

Step 4: Let A be a finite-dimensional semisimple k-algebra. Then
A is a direct sum of matrix rings over division rings, say

2 ...CD . . . _
A Mn](D])G}Mn2(DZ)EEa \L Mnm(Dm), where each Di is finite

dimensional over k by the Wedderburn structure theorems. By Step 3 we
have for each i a finite, purely inseparable, field extension Pi of

k such that [Mni(Di) ®kpi]/J[Mni(Di) ®kPi1 ls Pi-separable. Then

), where each a.. is purely inseparable over

P. = k(afl,a1.2,...,aimi 1J

1
k.
Let f1j(x) (denote the minimal polynomial for aij in k[X],
and let K be a splitting field for the family {f1j(x)} of poly-
nomials. Then K is a finite normal extension of k, so that K
contains a maximal purely inseparable field extension P of k.
Furthermore, P contains an isomorphic copy of each Pi'

Consider A ®k P 2 (Mn](D]) ®kP) Qw- @(Mnmwm) ®kP). By

Proposition l.3, the radical of a direct sum of algebras over a field

is the direct sum of the radicals. Therefore, (A ®kP)/J(A ®kP) 2

(Mn1(D,) ekP)/J(Mn1(o,) ekP) GE $(Mnmwm) okPl/amnmwm) okP),

and, moreover, tensoring up by any field extension of P preserves

the semisimplicity of the sum if and only if it preserves the semi—
simplicity of the factors. Consequently, (A ®kP)/J(A ®kP) will be
P-separable if and only if (Mni(Di)<8kP)/J(Mni(Di)<8kP) is P-separable

for each i. But the latter follows by taking R = Pi’ S = P, and

A = M (D.) ®k P- in Lemma 3.8.

38

Step 5: Let A be a finite-dimensional k-algebra. Then A/J(A)
is semisimple, and by Step 4 there is a finite, purely inseparable,
field extension P of k with [(A/J(A)) ®kP]/J[(A/J(A)) ®kPl

separable over P. Consider the following commutative diagram.

 

B -
(Wk P f (A ®kP)/J(A @kP)
d1 16
(ll/J(A)) ck P -—Y—-+ i(A/J(A)) ®kPl/J[(A/J(A)) ®kPI
J(A) is nilpotent, so J(A) ®k P 3J(A ®k ), and

ker o: _c_ ker B = J(A ®kP). Therefore, or maps the radical of A ®k P
to the radical of (A/J(A)) @k P, or we can write a(ker B) = ker Y:
so 6 is an isomorphism, and it follows that (A ®kP)/J(A ®kP) is
P-separable.

It is now possible to use Proposition 3.lO to construct P with
the desired properties when A is a finitely generated algebra over a

commutative semilocal ring R.

Proposition 3.ll: Let A be a finitely generated algebra over a

 

commutative semilocal ring R. Then there exists a finitely generated,

free, commutative R-algebra P such that (A<®RP)/J(A<3RP) is

separable over P;

Proof: If P: R/J(R), then P: F1$F2$~-EEF where the

n)
F, are fields, since R is semilocal. If Ah: A/(J(R)- A), then
A = F17: {E 03an. For each i FiA. is a finite-dimensional algebra

over the field F1, so by Proposition 3.lO there is a finite, purely

39

inseparable, field extension P1 of F1 such that

(F. A 31:1 1.1)/J(F A $1: P.) is P1-separable.
i

We now construct an algebra P over R for which (A ®RP)/J(A®RP)
is separable over P. Since each P1 is a finite field extension of
F1, write P1 = F1(a11,a12,...,a1ki). Let F11(x11) be the minimal
polynomial for a11 in F1[x11]. There is a natural map of polynomial
rings ¢11: R[x11]-——+ F1[x11], so we can choose f11(x11) to be a

. -l . ~ _ N
monic element of ¢11(f11(x11)). Define P11 - R[x11]/(f11(x11)),
and notice that ¢11 induces a natural map from P11 to F1(a11).

Now let f12(x12) be the minimal polynomial for a12 in
F1(a11)[x12]. Let ¢12: P11[x12]-——+ F1(a11)[x12] be the natural map,

and let f 12(x12) be a monic element of ¢1%(f12(x12)). Define

P12= P11[x12]/(f12(x12)), and notice that ¢12 induces a map from
P12 to F1(a11,a12). Continue this procedure for j =3,4,...,k1. At
the final stage we have a map from Elk to P1 = F1(a11,a12,...,a1k ).

l

Next let 621zP1k1[x21] -—+-F2[x21] be defined by letting

¢21(x1j) = O for j==l,...,k1, ¢21(x21) = x21, and ¢21 :R ——+ F2 be

l

the natural map. Let f21(x21) be the minimal polynomial for a21
in F2[x21], and choose f21(x21) to be a monic element of

-](f

21 Define P21 = P1k [x21]/(f21(x21)), and we have an

I
induced map from’ P21 to F2(a21). Continue on in this way, taking

‘1’ 2l<x2l))'

~

¢.. :P..[x1j]-——+ F1(a11,...,a

11 13 )[x. 1] where 2 s j s k.

ij- -l l

Ol“

¢11z Pi-l,k1_1[xij]'-_+ F1[x11] where j =1.

Then let P = Pnkn'

 

40

P is a commutative, finitely generated R-algebra which is free

over R. Thus, we only need to show that (A ®RP)/J(A ®RP) is separable

over P.

Let A = P/J(R) . P. Then P = F1AEBFZA$~~ GEFn'P, and for

each i, F1A is a finite-dimensional F1-algebra containing P1.

Therefore, by Lemma 3.8 (F1A <81: F1P)/J(F1A ®F FF) is F1A-separable.
i i

We canwrite A®§P —:-=(F1A€3 EEFnA) )1W EEFn) (F1P€}-- .FnP)

e (F1A eF F1P) $- EH (FnA anFnP). Since F1A ®F1F1P is an ideal of

 

 

1 F11 R i F1
J(F1A®1_. F P Q:- GEJ FnA®F F P) c J(A ofiA) However,
n
F1A ®F1 F1P F A®Fn F P
_. _EE'“EB _ __ is a direct sum of semisimple
J(FA® FP) J(FA® FP)
l F1l n Fn

Artinian rings and is itself semisimple. Thus,

 

 

— " N T ‘— 21:... — '—
J(A ®RP) _ J(F1A 8F1F1P) o, 69J(FnA anFnP), and

A 8R3 F1A ®F1F1P F A®Fn FnA
-—_—:—‘_—_— '—‘-’ _ _ EDI-“mg; _ _ . By Property l.l6
J(A ®AP) J(F1A®F1F1P) J(FnA ®FnFnA)

(A ®EP)/J(A @715) is separable over P.

Consider the following commutative diagram.

P -————-—-> (A ® RFD/J(A® P)

®R R

Since A ®RP is finitely generated over R, J(R) - (A ®RP) g J(A ®RP)

41

by Lemma l.l. However, J(R) - (A® P) = ker TT so J(A 8+1: ) =

R I
n1(J(A<8RP)): and it follows that n2 is an isomorphism. Since

(A ®-R-P)/J(A ®RP) is P-separable, (A ®RP)/J(A ®RP) is P-separable.

We can now extend Bogart‘s definition of Wedderburn specter.

Definition 3.l2: Let A be a finitely generated algebra over a

 

commutative semilocal ring R with separable subalgebra B. B is

said to be a Wedderburn specter (or specter) for A over R if B 8R P

 

is an inertial subalgebra of A<8R P for every finitely generated,
faithfully flat, commutative R-algebra such that (A ®RP)/J(A ®RP) is

P-separable. (Such a P exists by Proposition 3.ll.)

Now we establish a connection between pure inseparability and

Wedderburn specters in the case where A is commutative.

Proposition 3.l3: Let A be a finitely generated commutative

 

algebra over the commutative semilocal ring R, and let 8 be a
separable subalgebra of A. Then 8 is a specter for A over R

if and only if A is purely inseparable over 8.

Proof; By Proposition 3.ll, there is a finitely generated, free,
commutative R-algebra P such that (A<8RP)/J(A<8RP) is separable
over P. B 8R P (is separable over P by Property l.l4. A is purely
inseparable over 8 if and only if Al®R P is purely inseparable over
B<8R P by Proposition 3.l. Furthermore, Proposition 3.7 yields that
A ®R P is purely inseparable over 8 8R P if and only if B ®R P is
an inertial subalgebra of A 8R P. Thus, we have shown that A is
purely inseparable over 8 if and only if B is a specter of A.

42

Remark: Bogart has shown that not all maximal separable subalgebras
of A are specters of A, where A is a finite-dimensional algebra

over a field.

The remaining results give some properties of Specters.

Proposition 3.l4: Let R be a commutative, semilocal, inertial

 

coefficient ring. Let A be a commutative, finitely generated algebra

over R and A'= A/J(A). Then A has a specter if and only if A' does.
Proof: Let n :A-—-+'A' be the canonical map.

(==9) Let B be a specter for A, and let 8': h(B). B' is
separable by Property l.l3, and A' is purely inseparable over B. by
Proposition 3.1. Then Proposition 3.l3 gives that B' is a specter for
AI

(«@=) Sipp_l; Assume R is local with maximal ideal m, and let
8 be a specter for A: Then 8 is finitely generated over R by
Property l.l7, so we can write B = R31 + R32 + --- + RS". Let a1 be
a preimage of 31 in A for each i. Each a1 is integral over R
[2, Proposition 5.l, p. 59], so we can construct a finitely generated
R-subalgebra B' of A with the property that h(B') = B in the
following way. Let f1(x1) be a monic polynomial for a1 over R,
and take 81 = R[x1]/(f(x1)). Continue inductively, so that
81 = B1_1[x1]/(f1(x1)), where f1(x1) is a monic polynomial for a1

over R. Then map Bn into A by taking x1 to a1. Let B' be

the image of Bn in A under this map. Then B' is finitely generated

over R because Bn is, and h(B') = B.

43

By Proposition l.2, J(A)lA B' g_J(B'), and since

5: B'/(J(A)flB'), then J(B) = h(dm'n = man/(ammo). A is
a finitely generated B-algebra, so by Lemma l.l, J(B)- A g_J(A) = 0.

~

Consequently, J(B)

o, and it follows that J(A) r13' = J(B').

Moreover, B'/J(B') B is R—separable. Since R is an inertial

coefficient ring, 8' has an inertial subalgebra B; that is, B is
separable, and B + J(B') = 8'. Notice that h(B) = B/(B(1J(A)) =
B/(Bnammw = B/(Bmls'n == (B+J(B'))/J(B') = B'ms') = “8’.

Now m: A g_J(A), by Lemma l.l, so if we let A_= A/mA, we can
let h1 :A ——+-A_ and n2 :A_——+-A' be the natural maps, with
n = nzo'n1. Furthermore, if B_= n1(B), then n2(8) = B. We have the
following commutative diagram, where L, H, and K are the appropriate

kernels.

O -—--—-* L -————-+ A<SB A -—-—————+

l (.1. .1

A
o -——————-+ H ——————-—+ Aye g -———————+ 5_————————+ 0
‘A

o ————————+ K-————————+ A'

B is our candidate for a specter of A, so by Pr0position 3.13,
we need to show that L = ker p is small as an A<8R A-submodule of
A®B A. If ker('tTo (iTZGTTZ) ° (n1®ii1)) is small, then L will be
small. It is easy to see that ker(fi° (n2®ii2) ° (TT1®TT1)) is small
if ker II, keY‘(TT2®TT2), and ker(n1®ir1) are all small. kerII is
small by Proposition 3.13, and since A is an R/m-algebra, where R/m
is a field, by [5, Proposition 2.l0, p. l28], ker(n269n2) is small.

Thus, we only need to show that ker(mp81H) is small.

44

It is not hard to see that ker(ii1®ir1) = m: (A 83A). Suppose

there is an A QRA-submodule of A®B

Then M + m . A @B A = A ®B A. If we view these modules as R-modules, it

A such that M + ker(ii1®ii1) = A®B A.

follows from Corollary l.6 that M = A®B A. Therefore, ker(ir1®ir1)
is small, and we are done.

§_t_e_L2_: Let R be semilocal, and let B be a specter for A.
By [6, Proposition 2, p. ll], R = R1$R2€P nogfiRn, where each R1
is local, and, moreover, each R1 is an inertial coefficient ring.

Then A = R1AEi}°-°EERnA, A= RAEE-neaRnA, E = R1BEE-“EERnB,

1
and aniA :R1A -—+~R1A. Furthermore, R1B g_R1A, and R13 is
R1-separable by Property l.l6. Proposition 3.l yields that R1A is
purely inseparable over R1B, so apply Step l to find an Ri-subalgebra
31 of R1A which is a specter for R1A. Let B = B1®B2€E-~QEBn,
and note that B is separable over R. Thus, we only need to show
that A is purely inseparable over B.

We know that R1A is purely inseparable over 81, because 81

is a specter for R1A. If we define p1 :R1A®B R1A —-> R1A to be
i

the multiplication map, then it is clear that u : A®B A —-> A can be
written as p1 @E:-°€~Iun. We want to show that ker(u1'$°°"$pn) is
R A-module, so let M be an A ®R A-submodule of

A with the property that M + ker(u1 {PH-$1.1") = A®B A. Multiply

small as an A 8

by R1 to get R1M + R1[ker(p1QE~-§Eun)] = R1 : (A ®BA), or,

equivalently, R1M + ker p1 = R1A ®S R1A. Since ker [:1 is small, it

.1

n
- = a: =
follows that R1M - R1A @131 R1A. Then A®B A \L 1:] R1A $81 R1A

R1M = M. Thus, ker(p1§9~°€un) is small, and we are done.

n
GEE

i l

 

45

The next result is a slight modification of Proposition 3.l4.

Proposition 3.l5: Let R be an inertial coefficient ring and A

 

be a projective, commutative, finitely generated R-algebra. Then A
has an R-separable subalgebra B with A purely inseparable over 8
if and only if A'= A/J(A) has an R-separable subalgebra B with A-
purely inseparable over B. If n :A ——+'A' is the canonical map,

then h(B) = B.

Proof: (==>) This is the same as the proof of Proposition 3.l4

(=>)9 With E: E.

( ¢=0 This is nearly the same as the proof of Proposition 3.l4 (¢==),
Step l. We need to modify that proof by replacing m by J(R) and
letting A = A/J(R) - A. The arguments that ker E and ker(ii1®ir1) are
small go through as before, so we only need to Show that ker(iT2® 1T2)

is small. keY‘(TT2®TT2) = lT1(J(A)) ® A_+ A88 1T1(J(A)), and since A

B

is projective over R, n1(J(A)) is nilpotent in 5338 A_ by Lemma l.l.
Therefore, ker(n2®n2) is nilpotent, so that kEY‘(1T2®TT2) _c_ J(A ®B_A_),

and it follows that ker(n2®n2) is small.

Proposition 3.l6: Let A be a finitely generated, commutative

 

algebra over the commutative, semilocal ring R. If 81 and B2 are

specters for A over R, then B1 = B2.

Proof: Let P be a finitely generated, free, commutative R-algebra
such that (A ®RP)/J(A ®RP) is P-separable. The existence of P is
proven in Proposition 3.ll. Since B1 and B2 are specters for A

over R, then B1T8R P and 82%R P are inertial subalgebras of

46

A ®R P over P. Proposition 2.4 yields 81 8) P = B P.

R 2®R

Now consider the short exact sequence:
0 ————+:B1 -———+~B1 + B2 ————+-(B1-tBZ)/B1 -———+-O.
Since P is flat over R, the following is exact:

0——->B o P—-—+(B +8

1 R ® P-—-->((B +BZ)/B1)®RP-—-+O.

l 2) R l
This yields [(31+32) ®RP]/(B1 o P) = ((B +BZ)/B1) e1, P, or
P,

R 1
[B1 oRP+B2 ®RP]/(B1®RP) z ((31+32)/B1) eR P. Since B1 311 P = 32 oR
then ((B1-l-BZ)/B1)®R P = O, and because P is faithfully flat over
R, (B1i-BZ)/B1 = 0. Therefore, 82 g_B1, and by symmetry, 81 E-BZ'
[3

Proposition 3.l7: Let R be a commutative, semilocal, inertial

 

coefficient ring and A be a finitely generated, commutative algebra

over R. Then A has a unique specter B.

_P_l_roof_: The technique is to find a specter for A = A/J(A), and
then use Proposition 3.l4. A. is an algebra over R'= R/J(R) =
EEF n’ where the F1 are fields. Write
A$ ~93FnA, and notice that F1A is a commutative,
finite-dimensional Fi-algebra. Bogart has shown in [5, Corollary 2.l3,
p. l30] that such an algebra has a (unique) specter; call 81 the

Specter for F1A1- Let .B'= B155 ---EEiBn, and use the argument given

in the proof of Proposition 3.l4, Step 2, to show that B' is a specter

for A over R. Proposition 3.l4 then gives the existence of a specter

B for A over R, and B is unique by Proposition 3.l6.

BIBLIOGRAPHY

 

l0.

ll.

12.

BIBLIOGRAPHY

Albert, A. A., Structure of Algebras, American Mathematical
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