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the Galois group of the separable closure. One would like a similar description of separable algebras over an arbitrary base. Since a Galois theory is now.
transactions of the american mathematical

society

Volume 170, August 1972

THE SEPARABLE CLOSURE OF SOMECOMMUTATIVERINGS BY

ANDY R. MAGID(1) ABSTRACT. The separable closure of a commutative ring with an arbitrary number of idempotents is defined and its Galois theory studied. Projective separable algebras over the ring are shown to be determined by the Galois groupoid' of the closure. The existence of the closure is demonstrated for certain rings.

The finite,

separable

the connected

strongly

ring are described closure

group

of separable available

to contain strongly

certain

algebra

cases

are

locally

algebras,

and their

Galois

theory

of Boolean

rings

Galois

as a special

is shown

maximal

that the Galois algebras

over the base.

separable

closure.

In general,

Received by the editors September AMS 1970 subject classifications. Key words oid,

profinite

and phrases.

requirement

separable.

sheaf

spectrum,

Science

Foundation

below

of quasi-separable

closure

we consider closure

Galois

We call

studies

theory,

the

closure algebras,

determines

not exist

is deand it

the strongly

the existence

does

an

over the Boolean

the separable

24, 1971. Primary 13B05; Secondary

Boolean

for and in

As an illustration,

In §3,

in the set

Finally,

of every as is shown

above.

§2

is developed.

the separable

image

separable,

closure.

of the separable

separable

case

strongly

is not possible,

the first

strongly

is now for a separable

is to find a definition

is discussed.

element

to be locally

of

description

theory

a homomorphic

paper

subgroups

to asking

of the corresponding

theory

theory

a Galois

however,

from weakening

to open

from the connected

closure

and to contain

if the stalks

base

Since

commutative to the separable

like a similar

is equivalent

of the separable

comes

of the

correspond

generalization

of the present

a construction

quasi-separable

spectrum

fined

Direct

this

over the base)

The purpose

or, more generally,

connected

One would

base.

[Ml],

the separable

no new idempotents,

Our definition algebra

case.

is, requiring

separable

below (3.1).

in question

over an arbitrary

in the general

[J> s I] (that

over a given

closure.

base

field

theorem of Galois theory applied

the algebras

algebras

of a given

algebras

of the separable

over an arbitrary

closure

such

separable

fields

by the fundamental

of the base:

the Galois

extension

of the (see

(4.5)

13B10, 13B20.

separable

algebra,

group-

space.

( )Supported

by National

grant GP-21341A1. Copyright © 1972, American Mathematical Society

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110

A.R.MAGID

below).

We do demonstrate

is extremally conditions

its existence

disconnected

examples

§1

are collected

begins

In particular,

all rings

generic

ring.

base

with some

and notational

are commutative

X(S) is the associated

same

image

A normal,

locally

weakly functions

Galois

We use "regular" identify

triples

means

a groupoid

homomorphism

groupoid

in the category category,

homomorphism. inverse

limits

R

/" l: X{R) —> ß-algebra

x in X(R)) is weakly

limit

has the

under

every

Galois,

and

C{X, K) denotes

is the R-algebra

is a

is a direct

is one which

of such.

is

e of R,

the con-

automorphisms

of S.

of von Neumann. nature

a groupoid

A subgroupoid

is also

[G]).

of Galois

a groupoid.

spaces

Topological

groupoid

of

b. A

is a subcategory

con-

(for the definition

groupoids

finite

We will

a and domain

A topological

A homomorphism

and every

groupoids.

G is a collection

in G with range

of topological

closed

groupoid

of a groupoid

of topological

groupoids

form a category, (with

is a

the discrete

which

is is

topology)

to the category.

Let R, R1 be rings, suppose

limit

is a functor.

see

R-algebra

algebra

AutR(5)

the

x in X(R),

separable

(for each

the topological

which

a continuous

belongs

the direct

g is a morphism

of groupoids

under

of R

throughout.

For an indepotent

R-algebra

with its set of morphisms:

all the identities

in an arbitrary

separable

separable

K, and

examines

(a, b, g), where

taining

closure

in the sense

1. The section

and

R denotes

For

at x.

A strongly

A normal

strongly

from X to

of [Ml]

faithful.

of R.

spectrum

strongly

subalgebras.

in the separable

tinuous

preliminaries,

conventions

function.

and a locally

separable

homomorphism.

of the base

and sufficient

R -* S is a homomorphism,

continuous

R-module

topological

spectrum

over the Boolean

NR{e) = \x e X{R): 1 - e e x\. Iff:

of strongly

spectrum

necessary

and algebras

X(R) is the Boolean

of the sheaf

projective

and give

in §4.

We will use the definitions

the stalk

when the Boolean

or countable,

for existence.

[August

there

S, S'

locally weakly Galois

are ring homomorphisms

b:

R —» R

R- and R -algebras,

and

k:

S —►S

such

and that

5 --S'

R ->R' commutes.

Let F:

S ®„ S —»S ' ®R , S ' be given by F(x ® y) = k{x) ® h(y).

Proposition

1.1. Let R, R', S, S ', b, k, F be as above. Then F~ : If z' = (a,' , b' , g1) is in X(S' ®R, S'), F"1U') = {a, b, g) where a = k~l{a'), b= k~l(b') and g

X{S' ®R , S ' ) —* X{S ®„ S) is a groupoid homomorphism. is the unique

isomorphism

of S,

to Sa making

the following

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diagram

commute:

111

THE SEPARABLE CLOSURE OF SOME COMMUTATIVE RINGS

1972]

(*)

•V Proof. implies

We begin

that

F~l(z')

by showing

that

it is a homomorphism.

the uniqueness Let

= z = (a, b, g) and F~l(W

commutes,

and hence

uniqueness.

F~

We retain

,.

c',

h'),

8

i>'

g'

(z ' w O = F~

the above

of F~

g1 ), w ' =(b',

) = w = (b, c, h). Then

*

Sli

in the description

z ' = {a ', b',

(0 ' , c ', g'h'

notation

for the rest

) = {a, c, gh) = zw, by of the argument.

The dia-

gram

S--5'

{s'®R>s')z,

(S ®R S\ commutes,

where

the vertical

maps

are

and the lower horizontal

map, which

injective

since

potent

on idempotents

k~ (a ' ). Similarly

/, comes

and domain

1 - k{e) is in a'.

b = k~ {b').

on the first

from

F.

factor

Clearly

are connected.

commutes,

so that

/ is

For an idem-

tive on idempotents.

in 2,

e

=1

, g ' (k(t.)b

reversible,

b = k~

Let

K

0 = 1, so

so a =

Sa

z

s'

For uniqueness,

(b ' ).

e = £s.

so ¿(s.)

so 2 = F~

is reversible

The diagram

(*) commutes.

a = k~ l(a ' ) and

The argument

^s-%,s-)z,

Y

^jk(s.)a

we will call

its range

S.b -.(S«Lî)-

is

by inclusion

e of S, if 1 - e is in a, (e ® 1) =1 and f{(e ® 1) ) = 1. Thus

(k{e) ® 1) , = 1 and

where

induced

g((t\),)

suppose

Both vertical

® r. be an idempotent

= 1. Applying

F{e)z , = 1 and

1 - F(e)

that

maps

(*) commutes,

in (*) are again

of S ®R S.

k and

using

injec-

If 1 - e is

(*),

is in 2 ' . The argument

(2 ' ).

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is

112

A.R. MAGID

Lemma 1.2. Let R = dirlimR. for some'

i there

Proof.

S. such

with

that every

S

that

5 = 5. ®^

of G comes

T = S.

and

(S.,

R.

There

R over

HomR(S,

S) and, since

R -projective,

Theorem 1.3.

Proof.

[VZ2,

T is strongly

of finite

First

suppose each

We can assume

Galois

groupoids

that

R. has

that the index

(i.e.

5 is a weakly finitely

weakly Galois 1

oids.

Each

finite

topological

S.) is a homeomorphism

i

X(S. ®R 1

i

S.) is finite l

groupoid.

S = dir lim 5., where each topology,

is compact

Since

T. is

Then X(S ®R S)

groupoid).

R-algebra. (e.g.,

element

Write

R =

R- is Noetherian).

0 and that,

by (1.2),

SQ. Let S. = SQ ®# by (1.1),

and discrete,

R..

unknown and totally

limit

weakly

S =

S. is

groupoids

a topological

disconnected

of group-

limit is a pro-

Galois

R-algebra,

Then X(S ®R S) —>

and a groupoid

of profinite

whether

an isomorphism

so the inverse

S. is weakly Galois.

for later

use some

of a groupoid

Proposition

isomorphism. is again

groupoid

is a profinite

1.4.

Let

K be a closed

onto homomorphism

facts

subgroupoid

By ele-

profinite.

whose

groupoid,

underlying although

this

Then

G —> G'

subgroupoid

such

groupoids.

A weak

a subgroupoid.

that

H is the intersection

of all

H. of G.

z in G, 2 not in H, there /:

profinite

is also

groupoid. of G.

containing weak

about

which

G be a profinite

subgroupoids

(a) Given

simple

is a subcategory

H be a closed

open and closed

Proof,

S) =

for groups.

We record

Let

R ..

Now if S is any locally

an inverse

It is apparently

Let

R, T = R.

Since

tensor-

HomT(S,

over

Galois

and,

'

proj lim X(S. ®R S.) is a homeomorphism

subgroupoid

after

over R. and S = dir lim S.. It follows that X{S ®R S)—>

proj lim X{S. ®r

is the case

T.-algebra.

over

many idempotents

set has a least

exten-

separable

Thus

a profinite

R for some weakly Galois RQ-algebra

(b)

of S. by scalar

3. 1, p. 90], we have,

separable

5. is weakly

G of

i, if necessary)

T. is a strongly

5) is surjective.

Then

R.

separable

subgroup

(by increasing

Let S be a locally weakly Galois R-algebra.

limit

dir lim R., where

(a)

S = S. ®r

i and a strongly

is a finite

assume

Then

R ., S[G] —'Homr(5,

T. = R. and

is an inverse

mentary

that

from an R -automorphism

T = T. ®r . R.

5.) is surjective

ing with

space

S . such

[M2, Theorem 0.9, p. 709] and S . is a weakly Galois

S¿[G] —» Hornf

SQ ®r

R .-algebra

= R and we can further

element

Let

/^.-algebra

and let S be a weakly Galois R-algebra.

Galois

As in [EW, 1. 2, p. 235] we can find an

R¿-algebra

AutR(S)

sion.

is a weakly

[August

f(z)

Then

is a finite

K is profinite.

groupoid

is not in f(H),

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since

G

and an

G is pro-

1972]

THE SEPARABLE CLOSURE OF SOME COMMUTATIVE RINGS

finite.

Then

/

(f{H))

is an open and closed

subgroupoid

113

of G containing

H

but not 2.

(b) If G = proj lim G . and /. : G —»G . is the canonical

map, K =

proj lim/.(K). By (1.4a) [Ml,

1.10,

and (1.3), the subgroupoids

p. 96] to locally

separable

of a Galois algebras

groupoid

which

are precisely

correspond

the closed

sub-

groupoids. The next Lemma

lemma 1.5.

will

Let

be used

in the proof

E be a set

of (2.4)

of idempotents

below.

of R

closed

under

the ideal of R generated by E and V(E) = f\ {Ml - e): e £ E\.

unions,

I

Then I =

\f in R\fx =0 for all x in V{E)\. Proof. the

right

Clearly

/ is contained

and let

U = \x in X{R) \f

compactness,

there

is contained

are

in U.

e.,■•■,

Then

2. In this

section

Definition

2.1.

x in X(R),

able

cover

cover

we define

if S is a separable

Clearly

locally

separable quasi-Galois all these

strongly

algebras covers

quasi-separable

a flat,

Proposition separable

Proof. T

- e )

in U and

algebras

A separable

for each

x.

definition

weakly

algebras

5 is a separ-

S is a Galois

in perspective.

strongly

below

are

In general,

There

for each

shows.

strongly

algebras

covers.

shows.

separable (4.2)

covers,

Galois

are Galois

(4.1)

of R, as example

separable

covers. R) if for

5 is a quasi-Galois

are quasi-separable

locally

Galois

S

(of

R -algebra.

the above

covers,

S of R with

cover

cover.

place

separable

faithful,

V, so, by

O • • • O Ml

of quasi-separable

cover.

R -algebra

as example

2.2.

the class

quasi-Galois to help

cover

to

is a

x in X(R)

A separable

cover

algebra.

subalgebra

of a quasi-separable

coyer

is a

cover.

Let

is a separable

and hence,

Galois

and weakly

cover

- e.)

- e) is contained

separable

are proper,

but S not a separable is clearly

strongly

are separable

inclusions

Ml

tj " " " U e > Ml

/ belongs

and contains

that

quasi-separable

weakly

We make some remarks

Suppose

S is a quasi-separable

is a locally

is a locally

fi is open

in E such

and study

An R-algebra

S

= Oj.

side.

= 0 for all x so / = je.

if S is a separable

if S

cover

e

if e = e,

e is in E. Then (/(l - e))

each

in the right-hand

S be the cover subalgebra

of R, T the subalgebra. of the locally

by [Ml, 1.2, p. 90], strongly

Proposition

2.3a.

Let

strongly

For each separable

separable.

S be an R-algebra,

T a subalgebra.

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x in X(R),

R^-algebra

Sx

114

A. R. MAGBD (i)

// T is a separable

a separable (ii)

cover

S is a separable

Proof.

there

cover

that

Then

f4.2)

over

R and

over

T, although

in a natural

2.3-

by av

subalgebras

separable

even

of T, then

S is

generated

over

strongly

oj R then

[VZ1,

of S and hence

cover

strongly

T is in fact

S is strongly

S need

of T.

For p. 731],

follows.

of (2.3ii),

separable,

of locally

rings

T and the result

if, in the notation

algebra

[B, 2.10,

not be strongly

Thus

separable

separable

covers

algebras.

cover

is a separable

cover

which

is separable

at each

x in X(R),

idempotent

as an

p. 94], so a finitely

For the converse,

For each

S

cover

part (i) is standard.

over connected

it is a separable

cover.

for S.

generate

cover

if and only if

as an R-module.

is separable

with separability

(a )

separable

A quasi-separable

A finitely

is a separable

Then

theory

T is locally

generated

idempotent

R is connected.

way in the study

Proposition

of the base

T a quasi-separable

shows,

separable

cover

of R and

by Galois

many

separable

Proof.

cover

S is strongly

As example

it is finitely

S a separable

that

since,

are only finitely

arise

of R and

oj T.

We may assume

separable.

cover

of R.

// S is a separable

(ii), we observe

[August

S

let

generated

localization

quasi-separable

e = V S a surjection

with

R-aglebra,

kernel

I.

S a quasi-

Then

1 is generated

by idempotents.

Proof.

With T = dirlimT.

/ D T . It suffices we can assume each

x in X(R),

idempotent

potents

to show

that

and by [j,

that

2.5-

Let

E

and

I an ideal

/

is generated

strongly

of S generated

(quasi-Galois)

for x in X(R),

cover

(S//)

x, so

separable

by idempotents

{quasi-Galois)

cover

=

and hence

of T

—> S

by an idempotent.

in /, and thus

for each

let /.

by idempotents

is the kernel

of quasi-separable

S be a locally

is a quasi-separable

Since

/

separable,

is generated

1.6, p. 464] /

generates

Then S/I

Proof.

/

separable.

the structure

R-algebra

quasi-separable

T. strongly

to an idempotent

We can now describe

Theorem

that each

T is strongly

can be lifted

of / such

with each

there

is a set

E generates

for This

£ of idem-

/.

covers.

(locally such

weakly

that

of R; conversely,

Galois)

l O R = 0. every

is of this form.

= 5 //

and

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/

is generated

by idempo-

1972]

THE SEPARABLE CLOSURE OF SOME COMMUTATIVE RINGS

tents,

it suffices

Clearly

S/l

to prove

is locally

in the sense

the first

separable

of [Ml, p. 92]);

part

we show

bedding it in a locally

strongly

Let W be a separable

closure

in the case

where

(and if S is locally

Galois,

strongly

R-algebra

of R and

R is connected.

weakly

it to be locally

separable

also

separable

and applying

T - S ®RW.

115

normal by em-

[Ml, 1.2, p. 90].

T is a locally

strongly

separable 5-algebra. T is faithfully 5-flat [Ml, 1.1, p. 90], and hence TI O S = /, so S/1—*T/Tl separably finite

is injective.

closed,

space

T is a locally

by [Ml,

X.

Let

3.7, p. 99]

V(E)

is a closed

subset

jection

C(X, W) -» C(V(E), W) has kernel

C(V(E),

W) which

is locally

Now suppose

strongly

strongly

separable

a in T

(weakly that

TE.

Galois)

over all pairs

as above.

S is locally

and we have

constructed

a surjection

Corollary

2.6.

by idempotents

Let

of S generated

separable

(Galois)

and

S be a strongly

cover

T/Tl

W is pro-

TE = Tl.

Let

the sur-

is isomorphic cover

the techniques

to

of R.

of [M3, §l],

For find a

S(a, x) and a homomorphism S = &S(a,

strongly

/:

x), where

separable

S —* T.

(locally

By (2.4),

(a, x) weakly

the kernel

/ of

/ O R = 0.

separable

by idempotents

since

R.

of / . Let

Galois)

I an ideal

Thus over

R-algebra

a is in the image

map is generated

We have

(quasi-Galois)

using

ranges

this

in Tl.

separable

we can,

W-algebra;

to C(X, W) for some

of X and by (1.5) again

T is any quasi-separable

x in X(R) and

S(a, x)—> T such

Galois

E be the set of idempotents

V(E) be as in (1.5).

each

weakly

T is isomorphic

such

of R; conversely,

(weakly that

every

Galois)

I C\ R - 0.

separable

R-algebra

Then

(Galois)

S/l

and

is a

cover

is of

this form. Corollary

2.7.

Every

of suba Ige bras which

Proof. algebra

separable

are separable

By (2.5) there

S mapping

(weakly

quasi-separable (Galois)

is a locally

onto the cover

Galois).

(quasi-Galois)

is a direct

limit

covers.

strongly

T.

cover

separable

(locally

S = dir lim S. where

If T. is the image

weakly

each

Galois)

S. is strongly

of S. in T, T. is a cover

of the

proper type and T = dir lim T.. We recall

5

that

is connected

and T, AlgR(S, Lemma

2.7.

as an R-module

an R-algebra

(equivalently,

T) is the set Let

S is locally

every idempotent

of R-algebra

S be a locally

by n elements

if for each

of S is in R).

homomorphisms

connected

and let

connected

separable

T be a locally

For

x in X(R),

R-algebras

from S to cover

connected

S

T.

of R generated R-algebra.

Then

Card(AlgR(S, T)) < n ■Card (X(R)). Proof.

If / and g are in AlgR(S, T) and fx = gx fot all x in X(R) then

/= g. Thus AlgR(5, T) — nxAlgRx(Sx, Tx) is injective.

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By [J, 1.3, p. 462],

A. R. MAGID

116 Card(AlgR

(S , T )) is at most

Corollary algebras

2.9.

[August

n, and the result

The cardinalities

follows.

of quasi-separable

locally

connected

R-

are bounded.

Proof. locally

By (2.7),

connected

only a bounded

number

We will use

conclude

this

any such

separable

of ways

(2.9)

Since

proofs.

Let

each

Let

/:

of subalgebras

of the Galois identical

R-algebra.

which

and,

are

by (2.8),

in the algebra.

of the separable

are almost

X(C®R

limit

is only a set of such

can be embedded

with a statement

the arguments 5 be a quasi-Galois

as follows:

is a direct there

in the construction

section

covers.

algebra

covers;

closure

theory

with

We make

We

of quasi-Galois

[Ml,

^l],

X(5®R

S) —►X(R) be the canonical

below.

we shall

omit

S) into a groupoid

map, and let

x be in

X(R). Then /" Hx) = X(S x ®R x Sx ) and this latter is a groupoid by [Ml, 1.8, p. 931.

Thus

tween

X(S ®„ S) becomes

members

Galois

of the same

R-algebra

X(T ®r

T) is also

fibre-wise,

clude

by (1.4b)

Theorem

2.10.

covers

Let

Proof.

(2.10)

2.11.

Let

field.

Then

of finite separable.

limit

of finite

Corollary

that

weakly on

over points

X(S ®R S) is iso--

of X(T ®R T).

We con-

Now by repeating

the

2.12.

S

that

field

Conversely, of finite

Let

X(S ®R S) is a

between

the quasi-

of X(S ®R S).

Separable

of Boolean ring such

reduced

rings-

that for every

extension

x in X(R),

R

S of R is a regular

ring

and conversely.

x in X(R),

products

Then

subgroupoids.

an integral,

separable

of R.

correspondence

subgroupoids

in the case

cover,

For all

strongly

groupoid.

cover

is a one-one

R be a regular

R , so we may assume

products

structure

to the fibres

it follows

S be a quasi-Galois

to open-and-closed

and a quasi-separable

field

applied

weak subgroupoid

of R in S and closed

We will illustrate Lemma

(1.1),

only be-

find a locally

The groupoid

g is injective,

to a closed

and there

correspond

is a perfect

(2.5),

allowed

we have

groupoid

separable

Using

X(S ®R S) is a profinite

of [Ml],

topological

covers

that

X(r). T —> S.

and hence

since

and homeomorphic

arguments

over

with compositions

map g: X(S ®R S) —»X(T ®RT) shows that g is a

homomorphism;

morphic

fibre

T and a surjection

of X(R) of the induced groupoid

a groupoid

is a reduced

integral

R = R . But then

extensions

of R, and hence

a quasi-separable separable

S be a Boolean

extension

S is a direct

field

ring,

cover extensions,

R a subring.

Galois cover of R.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

of the perfect limit

regular

of a field

of finite

and locally is a direct

and the result

Then

S is a

follows.

1972]

THE SEPARABLE CLOSURE OF SOME COMMUTATIVE RINGS

Proof.

By (2.11),

any image

of 5

in a separable

We continue

and

X and

Again

S = C(X,

by the Stone

theorem,

to closed

tion whose

quotient

contained

defined

We now show from that

strongly

of [Ml],

separable.

Lemma

2.13-

is a separable

Let

work,

rela-

equivalence

rela-

are the closed

by

X xy X = X(S ®R S), and the groupoid

of Boolean ring extensions

however,

cannot

need

be deduced

not be locally

in more generality.

S be an R-algebra,

e an idempotent

of S.

Then

R + Re

R-algebra.

Proof. there

We will

theory

Boolean

the equivalence

of a groupoid

rings

is, that

covered

subgroupoids

Galois

this

to spaces

the structure

here.

that

X —» Y.

of X, the spaces

than

is the same as that given

that

R.

zero-dimensional

S correspond

the closed

hand,

over

theorem,

a surjection

are the closed

X Xy X can be given

On the other

by (2.10)

these

x in X(R),

S is normal

are quotients

on X finer

terms,

so

compact,

R and

relations

For any

representation

S induces

spaces

c) = (a, c), and then

relations.

structure

these

In other

in X Xy X.

(a, b)(b,

equivalence

Y.

Z/2Z,

for suitable

between

since

equivalence

is

is

of R.

By the Stone

of R into

rings

V, and,

correspond

defining

Z/2Z)

cover

of R

of (2.12).

Y and the inclusion

by X and covering

tions

closure

the notation

R = C(Y, Z/2Z)

spaces

5 is a quasi-separable

117

Let T = R[x\l(x2 - x), and let f = x + (x2 - x). Then f2 = / and

are homomorphisms

of

T into

R sending

/ to 0 and

1 respectively.

T =

Tf x T(l - /) and Tf = Rf and T(l - /) = R(l - /). If rf = 0 for r in R, then sending

/ to

fashion,

R(l - /)

separable.

1 shows

that

r = 0, and hence

is isomorphic

to R.

R + Re is a homomorphic

Proposition

2.14.

Let

Rf is isomorphic

Thus image

S be a strongly

to R.

T is isomorphic of T and so also

separable

In a similar

to R x R and hence is separable.

R-algebra.

Then

the in-

duced map p: X(S) —►X(R) is open. Proof. algebra

Let

e be an idempotent

(2.13) and hence

of T and hence summand

have

also

contained

in x implies

that

and assume

that

for the hypotheses particular,

e

empty

X(R)

and

strongly

a projective

of R, say equal

Ns(e)

original

also

of S.

to

separable.

R-module,

R(l - /)

in p~l(NR(f)),

Re is a faithful

is not zero), is a single

R, we have p(Ns(e)\=

summand

is a direct

/ is an idempotent

of R . We first

e = ef, so we can replace

We then from

and we may assume so

re = 0|

e(l - /) = 0, so for x in x(S),

we pass

point,

Te - Re is a direct \r in R:

R-

since

R-module.

when

so

T = R + Re is a separable

where

1 - / is in x f~l R. Also

remain

Then

claim

that

R = Rx-

p is surjective.

But then Thus,

NR(f).

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

R by Rf

p(Ns(e))

R to R% for each

e not

= X(R),

x in X(R) (in Ns(e)

returning

is nonto our

118

A. R. MAGID The converse

will establish Lemma set

of (2.14)

a partial 2.15.

Let

of idempotents.

is,

in general,

converse,

false,

so example

(4.3)

illustrates.

We

however.

S be a separable

Then

[August

cover

S is strongly

of R generated

separable

over

R by a finite

over

R if and only

if the

in-

e be an idempotent

of S. Since

NAe)

is

duced map p: X(S) —» X(R) is open. Proof.

compact

Suppose

and open,

p is open.

Let

so is p(Ns(e)),

so

p(Ns(e))

= NR(f)

foe some idempotent

/ of

R. Also, p(Ns(e)) = {x in X(R):

there is y in X(5) with e = 1 and y n R = x\ =

\x in X(R):

argument

X(R):

Rj,

0\.

A similar

open and closed

V = X(R) - U. By the above,

(2.13),

T is a separable

X(R), V.

ex 4

ex j¿ I] is also

[T

over

R.

The converse

follows

DeMeyer

from compact, Boolean

and thus,

O-dimensional

Proposition

2.16.

that

\x in

U = \x in X(R):

e

and hence

is in

Let T = R + Re.

By

For

if x is in U, the second

constant

T is strongly

x in

if x is in separable

from (2.14).

[D], we will call

ring is, of course,

1 - e shows

by (2.2),quasi-separable.

happening

the rank of T is locally

Following

to

Let

U is open and closed.

R-algebra

: Rx] = 1 or 2, the first

Thus

applied

in X(R).

spaces

a finite

to connected

weakly

product

rings

of rings

of functions

a weakly

uniform

ring.

A

uniform.

A strongly

separable

extension

S of a weakly

uniform

ring

X compact

and

R is weakly uniform. Proof.

We can suppose

O-dimensional.

nected, The

Suppose

R = C(X, K), where

x is in X (= X(R))

strongly separable

K-algebra.

R-algebras

T and

S are both

and isomorphic

at x.

Thus,

hood of x, and on this by such

neighborhoods,

Proposition

Then

S is strongly

=Fx---xF,F.a

con-

generated

and projective

as in [M3], T and S are isomorphic

neighborhood

Let

K is connected, S

Let T = C(X, F, x • • • x F ) = IIC(X, F.).

finitely

S is weakly

S is weakly

2.17.

and

on a neighbor-

uniform.

Since

X is covered

ring,

S a separable

uniform.

R be a weakly

separable

as R-modules,

uniform

if and only if the induced

map p:

cover of R.

X(S) —> X(R)

is

open.

Proof.

We can suppose

x be in X, and define to

T

lifts

surjective

toa

of some

hoods.

Thus

of the

there

that for every

F..

K), K connected,

T as in the proof

homomorphism

on a neighborhood

product

such

F.,

R = C(X,

T—> 5.

of x.

For each

X is covered

is a connected,

x in X, S

of (2.16).

Since

X O-dimensional.

Let

The isomorphism

of Sx

5 is finitely

y in this

number

generated

Galois

is a product

of subalgebras

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

(2.3),

neighborhood,

by a finite

finitely

generated

S

of such

is a

neighbor-

extension

of L.

T is

Let

L of K

W=

1972]

THE SEPARABLE CLOSURE OF SOME COMMUTATIVE RINGS

S ®K L.

hence able

W is strongly

X(W) —►X(R) then

X, then,

separable

is open.

over

W is a separable

so is the subalgebra S

is a product

S so that

S.

of copies

R-algebra;

Thus

we may assume

of L

and

V is between

separable

over

over

strongly

separable

S is generated

over

(2.14)

if it is strongly

separ-

For every

y in X(S),

and

x in

S . = L,

and

R and S, and V is strongly

V by idempotents

V and therefore

is open

W = S.

so for every

S = C(X(S), L). Let V = C(X, L). R.

X(W)—*X(S)

119

and hence,

over

R.

The converse

ring,

R a subring.

by (2.13),

implication

is

(2.14). Corollary

separable X(R)

2.18.

over

If S is locally separable

is an open

covering

Example

(4.1)

over

shows

bras

alone

limit of spaces

strongly

mapping

to

X(S) = proj

The converse

that

lim X(S.).

of (2.18)

(which,

covers;

S.

X(S .)

(2.17). satisfied.

as shown

above,

theory

of separ-

in the Galois

to go to separable

where

each

using

is not always

rings

theorem)

lim S.,

By (2.14),

similarly

of Boolean

representation

R, S = dir

follows

the condition

theory

over

strongly

separable

alge-

are insufficient.

(4.4),

separable.

the base

R, and

it is necessary

3. In this Example

separable

the Galois

only on the Stone

algebras,

strongly

of X(R).

to encompass

able

is an inverse

S is locally

open sujections.

is strongly

depends

S be a Boolean

R if and only if X(S)

by compatible

Proof.

Thus

Let

section

we define

we shall

However,

we would

Definition

3.1.

A separable

closure

closures.

closure

our definition

idempotents.

cover of R which receives

separable

the separable

like

ring has no nontrivial

separable

and construct

not require

leads

with that

of [j] when

to

of R is a locally

a homomorphism

of

strongly

to agree

This

Because

to be locally

connected

from every

quasi-

strongly

separ-

able R-algebra. Proposition

3.2.

Suppose

R is connected.

R if and only if S is a connected no proper,

connected

strongly

locally

separable

Proof.

Suppose

S is a separable

is a locally

strongly

separable

strongly

able.

separable

extensions.

Then

strongly

S is a separable

separable

closure

R-algebra

which

of has

extensions.

closure

R-algebra

of R.

SQ which

As in [j, contains

SQ = dir lim S. where

each

1.4, p. 463] there S and has

no proper

5¿ is strongly

separ-

Thus AlgR (SQ, S) = proj lim AlgR (S{, S) and the latter is nonempty, being

an inverse

limit of nonempty

Composed

with the inclusion,

465] an automorphism. (3.2)

shows

that

finite

Thus (3.1)

sets.

Choose

a homomorphism

it gives

an endomorphism

5 = SQ.

The converse

reduces

to [j,

Definition

from

of SQ, hence is found

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

5Q to S. 1.7, p.

in [j, p. 464].

3, p. 463] when

connected.

[J,

the base

is

Proposition X(R),

S

3.3.

X(R), and

T.

X

R-algebra

Again

S

X

T a strongly

separable.

as in [M3], these

5 is a separable

that

is a separable

X

x in

T

there

There

If each

S

is, for each

/ 's can be pieced

x,

closure,

x in

we can find [M3, 1.6, p. 116] a

= Ta.

closure.

R-algebra

if and only if for each

If S is a separable

R -algebra,

T such

separable

of R if and only if for every

and quasi-separable

strongly

is a strongly-separable

separable U

closure

of R .

connected

and locally

h : T. —» S , and

and

closure

S is locally

is connected

strongly

5 is a separable

is a separable

Proof. S

LAugust

A. R. J4AGID

120

together

is an b:

T —» S, so

is a separable

X

x, some

to give

closure

/ : T

an

—>S .

/ : T —> S.

Thus

closure.

Corollary

3.4.

A separable

closure

is a quasi-Galois

cover.

Proof. This follows from (3.3), (3.2), and [j, Remark 1, p. 466]. Using

(3.3),

we can give

ring with separable every

closure

O-dimensional

examples

space

separable

theory

R-algebras.

S X ) = Aut R y (S X ).



of a separable

If 5 is a separable

X(S ®R S) of (2.10) has the property X(S X ®R

that

Definition is a set

a category 3.5.

M with

b in homr

Let

maps

(x, x) and

(i) k(l, a) = a

If R is a connected

closure

of C(X, R) for

closure

of R determines

Thus

every

morphism

if every

G be a skeletal M —>X and

a in p~

of R, the groupoid

in X(S ® t\„ S) is an automor-

of X(S ®R S).

is skeletal

p:

closure

that the fibre over the point x of X(R) is

phism, and X(R) is the set of objects Recall

closures.

X.

We now show that the Galois the strongly

of separable

S, C(X, S) is the separable

isomorphism

groupoid k:

G x„

is an automorphism.

with set of objects M —»M such

that

X.

A G-set

for x in X, g,

(x) we have

(1 = identity

on x),

(ii) k(g, k(h, a)) = k(gb, a). We will write set

in the usual

Let

ga sense

for k(g, a). of group

S be a separable

closure

We make X(S ®R T) an p:

X(S ®

and then

a in the former

and

R-algebra

Proof.

3.6.

is determined

Let

T and

p~

(x) is a homr

(x, x)-

of R and

T a strongly

the inclusion

R-algebra.

R in S ®

R

we define

by its corresponding

T

k(g, a) = ga.

corresponding

R have a separable

be strongly

For this

T induces

For

set

k componentwise.

separable

of

we can assume

g in the latter

Let

fibre

X(S ®R T) = AlgR (T, S) and X(S ®R S) = AutR (S).

X(S ®R T) the Galois

Theorem

each

on a set.

X(S ®R S)-set:

T) —>X(R); we define

is connected, we call

Note that action

Under these definitions,

to T.

closure

S.

Galois

separable

Then a strongly

set.

R-algebras.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

separable

T is isomorphic

to

1972] T'

121

THE SEPARABLE CLOSURE OF SOME COMMUTATIVE RINGS

if and only if for each

work

componentwise

T = T1x

and assume

■•■ xTn

AutR

(S)-sets.

AlgR

(T.,

x in X(R),

Since

T. C S is determined

AutR

of Galois

theory.

Thus

AlgR

S), and hence

We note that

case,

since

spectrum.

closure

be locally

all data

s) as of S.

of the inclusion

is the subgroup

of

theorem

by the Galois

set

set.

to which

localize

is the main reason

(F.,

any definition

of

from the known theory properly

with respect

for the requirement

that

in the

to the the separable

connected. 3.7.

A separable

closure

has

no proper,

locally

connected

covers.

Proof.

If T is a locally

S, for each

x in X(R),

connected

T

(3.7)

suggests

quasi-separable

a possible locally

connected

Proof.

3.8.

Since

cover

separable

of the separable

connected

S -algebra

closure hence,

T = S.

construction

connected

separable

Proposition

separable

is a strongly

by (3.3) and (3.4), equal to Sx. Thus

locally

result

immediately

involved

is standard:

T. by the fundamental

by its Galois

follows

we can

it is a subalgebra

up to isomorphism

is the basic

lead,

This

Proposition separable

which

Thus

the stabilizer

stabilizer

determines

T is determined

should

Boolean

and thus But this

and which

T'.

Here the argument

we may assume

T. is determined

(3.6),

closure

to

so AlgR (T, S) = UA1gR

(S)-set,

up to conjugation.

T. fixed,

connected

R is connected.

T. is connected,

AutR (5) leaving

separable

is isomorphic

where T. is connected,

S) is a transitive

(T.,

T

cover

of the separable a maximal

cover

closure.

We call

a

if it has no proper

covers.

Maximal

covers

the cardinalities

exist.

of locally

connected

quasi-separable

covers

are bounded (2.9), the idea of the proof of [J, 1.4, p. 463] works here. A maximal images

cover

of strongly

S will be a separable

separable

algebras,

( + ) If T is a strongly

5 ®

T, generated

locally

connected

If (+) separable

obtains,

if it contains

homomorphic

if it satisfies:

such that

there

exists

an ideal

/ of

/ O S = 0 and 5 ®R TA is a

5-algebra.

then since

S ®R T/1

cover of S, it must equal

correspond

i.e.

R-algebra,

by idempotents,

T/I = S is the necessary tents

separable

closure

p. 37], we can translate

subsets

connected

S, and hence the composite

homomorphism.

to closed

is locally

Since ideals of the Boolean

and, by (2.5),

T —>5 ®R T —►5® R

of a ring generated spectrum

by idempo-

of the ring [P, 9.3,

(+) to

( + +) If T is a strongly

X(S ®„ T) —►X(S) (= X(R))

separable

R-algebra

has a continuous

the canonical

section.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

a

surjection

122

A- R- MAGID (In making

the translation,

one goes

from

X(S ®R T/I) —» X(5 ®R T) for the section the ideal

quence

corresponding

to the image

[August

(+) to ( + +) by taking

X(S) =

and from (++) to (+) by taking for /

of the section.)

As an immediate

conse-

of ( ++) we have

Corollary

3.9.

// X(R)

is an extremally

disconnected

space,

R has

a separ-

admits

a sec-

able closure. Proof. tion,

for

By [W, Example Y any compact

Thus

if X(R)

a separable

closure.

This

happens

spaces

3.10.

Y—► X(R)

compactification

if R is a product

of a discrete

space,

of connected

rings,

R has

or if R

field.

of Example

for arbitrary

any surjection

space.

in a number

If X is the space

Proposition

Hausdorff

is the Stone-Cech

is the ring of adeles

section

8, p. 311],

4.1, not every

Y. However,

Suppose

that

surjection

Y —►X admits

a

we do have

X(R)

is countable.

Then

R has a separable

closure.

Proof. weakly

We will show the following:

Galois

R-algebra.

and a surjection algebras

T —►T

inductively

°

locally

connected

suffice

to construct

x while

(e)x

proceed

to find such

be a minimal

+ • . • + gk(ex) p. 724],

e„ of S which

an

neighborhood.

mains

is connected.

T

Gx goes

when

(3.10)

..

(+) implies

idempotent

e.

Let

. Choose

(where

at

x.

e equal

group

that

such

and such

that

will

T

the

be a

it will

¿ 0 for any

= R.

We

Let

e.

that

is possible

since,

by

We can find an idempotent

(g.(eQ)

+ • • • + gk(eQ))

(eQ)

to eQ on the neighborhood only be special exists.

e

in G such

of x . In particular,

closure

T

will work.

that

■ ■ • , g,

n

Clearly,

- e)T = T,

on idempotents.

of course,

the separable

0

g.(e {) = (g¿)x ,(e i)); this

neighborhood

should,

Tn = dir lim T

of T such

g.,

R-algebra

We will construct

the result.

T/(l

G be a finite

transitively

to e,

then

We have

We will

collected

in this

make the following

section

/0

= 1

to 1 off it.

of a general

general

for

in that

and equal cases

What this

C the complexes

(all three

examples

conventions:

{l, 1/2, 1/3, • ■• , 0Î of the unit interval; and

and hence

n+l

Then

If e is an idempotent

= lx acts

We take

and

—>T

n

Galois

result

result

is re-

an open question. 4.

tions.

(T )

that

of (T)x

x in an open-closed

describing

such

of T.

T..

idempotent

[VZl,

(3.9)

is a weakly

is a minimal

gjtej)

ail

n there

so that

'

image

Let X = {x., x2, • ■• \ and let T be a

For each

with

let

referred

let Q denote

the discrete

to in the other

X be the subspace the rationals,

R the reals

topology).

Example 4.1. Let W = X x {0, 1| and let Y be the closed subspace

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sec-

1972]

THE SEPARABLE CLOSURE OF SOME COMMUTATIVE RINGS

\(l/k)\k

123

even! U \(l/k, 0): k oddiu 5(0, 0), (0, L)}. Map W (and hence Y) to X

by projection

on the first factor.

C(W, Z/2Z).

All three

jectively

to E.

Let D = C(X, Z/2Z),

are Boolean

F is a finitely

Y to X is not open is not a strongly

D is a subring

generated

((X x|li)^

separable

rings.

D module,

Y does

D-algebra.

E = (Y, Z/2Z) of E and

hence

so is

however,

F andFmapssurE.

not map to an open set) By (2.12),

and F = The map of

so by (2.18),

E is a Galois

E

cover

of D.

Example 4.2. Let D = C(X, R), E = C(X, C) and F = {/ e E: f(0) e R\. Then X(D) = X(F) = X and for x in X, Dxxx= E R and

able

EQ = C.

E is a weakly

D-algebra

D-algebra,

such

since

F by (2.11)

that

Galois

F

= F

is separable

it is not finitely

but not a strongly

= C for x ¿ 0,' while Dn00 = Fn =

D-algebra.

F is a locally

for each

generated

separable

over

algebra

x but

D.

strongly

F is not a separable

E is a separable

since

separ-

cover

of

its rank is discontinuous.

Example 4.3. Let Y = {(x, x): x € X\ \jX x {0¡. Map Y to X by projection on the first factor. Let D = C(X, Z/2Z), E = C(Y, Z/lZ). For x in X, Ex = (Z/2Z)(2) for x ^ 0, and EQ = Z/2Z. Thus B is not a strongly separable Dalgebra,

but

p:

Example latter

Y —» X is open.

4.4.

Let

(It is a locally

strongly

T be the ring of continuous

with the discrete

topology)

such

that

/(0)

separable

functions is in Q.

algebra.)

/ from X to R (the

Then

X(T) = X and

Tx = R if x ^ 0 and TQ = Q. Let A = T[x]/x2 + 2 and let B = T[x]/x2 + 3. A and

B are locally

connected

weakly

Galois

T-algebras.

Now for x in X,

(A ®T B)x = Cx C if x ¿ 0 while (A ® T B)0 = 2(V- 2, V3). Thus if S is a strongly

separable

isomorphism

hood of 0. locally

T-algebra

and hence

But then

connected

and

(since

/:

AQ^B

/ is

5 is not locally

locally

strongly

—»5

T-split)

a surjection,

/ is an isomorphism

connected.

separable

It follows

T-algebra

which

then

/Q is an

on a neighbor-

that there

is no

contains

homomor-

phic images of both A and B. Example 4.5. Let A be the ring of [AK, p. 477]; A is a subring of the ring B of all (9=1

GF(4)-valued but

functions

on a space

6 has no fundamental

g of B where

g(f)(x)

= f(Q(x))

domain) .

B is a strongly

[CHR, 1.2(f), p. 4]. If A had a separable would

induce

for 6 on X.

a section Thus

X (X has a homeomorphism

invariant

closure

X(C) = X(A) —» X whose

A has no separable

under separable

extension

of A by

C, the homomorphism image

6 with

the ring automorphism

is a fundamental

B —» C domain

closure.

REFERENCES [b].

H. Bass,

tal Research, [Dl].

Lectures

on topics

in algebraic

K-theory,

Tata

Institute

of Fundamen-

Bombay, 1967.

F. DeMeyer

and E. Ingraham,

Separable

ture Notes in Math., no. 181, Springer-Verlag,

algebras

over commutative

New York, 1971.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

rings,

Lec-

124

A. R. MAGID

[GJ. Geometrie

P. Gabriel, Algébrique,

Construction Inst. Hautes

Études Sei., Paris, 1969. LjJ.

G. Janusz,

de préschémas Études Sei.,

quotient. Schémas en groupes, Sem. 1963/64, fase. 2a, Expose 5, Inst. Hautes

MR 41 #1749.

Separable

algebras

over

commutative

rings,

Trans.

Amer.

Math.

Soc.

122 (1966), 461-479. MR 35 #1585. [Ml]. A. Magid, Galois groupoids, J. Algebra 18 (1971), 89-102. [M2]. -, Locally Galois algebras, Pacific J. Math. 33 (1970), 707-724. MR 41 #8405. [M3j.

—-—,

Pierce's

representation

and separable

algebras,

Illinois

J. Math.

15

(1971), 114-121. [VZl].

O. E. Villamayor

and D. Zelinsky,

Galois

idempotents, Nagoya Math. J. 27 (1966), 721-731. [VZ2j.

—•—.—,

(1969), 83-98. [EWj.

Galois

theory

with

[P].

for rings

with finitely

many

many

idempotents,

Nagoya

Math.

J. 35

MR 39 #5555.

S. Endo

and Y. Watanabe,

On separable

Osaka J. Math. 4 (1967), 233-242. R. Pierce,

70(1967).

infinitely

theory

MR 34 #5880.

Modules

over

algebras

over a commutative

ring,

MR 37 #2796. commutative

regular

rings,

Mem. Amer.

Math.

Soc.

No.

MR 36 #151.

[w].

A. Wilansky,

[D].

De Meyer,

Topology for analysis,

Separable

polynomials

Ginn, Waltham, Mass., over

a commutative

ring,

1970. Rocky

Mt. J. Math.

2 (1972), 299-310. [AK].

R. F. Arens

and

I. Kaplansky,

Amer. Math. Soc. 63 (1948), 457-481. [CHR]. cohomology

Topological

representation

of algebras,

Trans.

MR 10, 7.

S. U. Chase, D. K. Harrison and of commutative rings, Mem. Amer.

A. Rosenberg, Galois theory and Galois Math. Soc. No. 52 (1965), 15—33MR

MR 33 #4118. DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YORK, NEW YORK 10027

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