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
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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 ' ).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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),
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
/
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.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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 ).
jç
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].
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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.
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and Y. Watanabe,
On separable
Osaka J. Math. 4 (1967), 233-242. R. Pierce,
70(1967).
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MR 34 #5880.
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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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