KUMMER EXTENSIONS OF RINGS
A. Z. Borevich
UDC 519.48
The theory of Kummer extensions of commutative rings is constructed, generalizing the theory of Kummer extension fields. group of degree ~ the ring element
~
A Galois extension
~/~ with Abelian Galois
is called Kummerian if in the group of invertible elements of
there is a cyclic subgroup
~-~ is invertible in
~ .
~
of order ~
such that for any @ ~ , @ # ~
Any cyclic Kummerian extension can be obtained
by means of a suitable factorization of the tensor algebra over a finitely generated projective
~ -module of rank 1 (an arbitrary Kummerian extension is a tensor product
of cyclic ones).
The group of equivalence classes of Kummerian extensions with
fixed Galois group is studied.
i.
Definition of Kummer Extensions of Rings In field theory one means by a Kummer extension a finite Galois extension
Abelian group a primitive
~
such that if 9
is the degree of the group
~ -th root of unity.
striction on the ground field
K/K
~, then the field
K
with contains
The definition of a Kummer extension thus imposes a re-
K .
In [i] the concept of Galois extension of a field was carried was carried over to ring extensions. Let
We give the definition of a Galois extension of a ring.
~
be a commutative ring with unit 1 and let
unit of the ring
~
is
I ).
Further, let
~
~.
The extension
i) the subring of ~-invariant elements of 2) in
~
be a unital subring of it (the
be a finite group which acts on
faithful group of operators, and such that for any ring automorphism of the ring
~
6"6~ the map ~ - ~ - ~ S/~
~
=~(~),
~
as a
~e~, is a
is called a Galois extension if
coincides with
~ , i.e., ~ @ = ~
there exists a finite system of elements ~,...~0G~ ~ , . . . ~
;
such that
~=~ where 0~0~ is the Kronecker delta (~,~ = I and ~ , ~ = 0 for 0 ~ The system of elements ~/R
, and the group Definition.
Galois group
~.
Let
~
~,...,m~; ~ , ....~
; ~,~E G) .
is called a Galois system of the extension
is called its Galois group. ~/~
and ~ R
They are called
which leaves the elements of ~
be two Galois extensions of the ring
~
with the same
~-equivalent if there exists a ring isomorphism
~.~_,~r,
fixed, and such that
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 8-30, 1976. 514
0096-4104/79/1104-0514507.50
9 1979 Plenum Publishing Corporation
m~(=) = ~ C ~ ) for all
and all
ac6S~.
In Galois ring theory it is sometimes expedient to consider Galois algebras instead of Galois extensions
(e.g., in the study of tensor products).
over a commutative ring
~
A faithful commutative
is called a Galois algebra with Galois group
~-algebra
~ , if {/~'~
is a Galois extension. The goal of the present paper is to construct a theory of Kummer extensions of rings, generalizing
the well-known
theory of Kummer extensions of fields.
Firstly, we must introduce a reasonable restriction on the ground ring (cf. [i0, p. 57]. Definition. ~-Kummerian group ~
Let
~
be a natural number.
if in its groups i(~l
A commutative ring ~
with unit is called
of invertible elements there exists a finite cyclic sub-
of order r~ such that
for all elements
G~(D
except i.
It is easy to see that for fixed subgroup
~
r ] one has in the polynomial ring ~LX_ the
factorization
X~-~ ~ F] Further,
in the ring
~
Cx-~).
one has the equations:
O< In particular, r~= t~.~
it follows from the second equation that in an
~ -Kummerian ring the element
is invertible.
Definition.
Let
g'
be a commutative ring with unit and
,~ be a unital subring.
The
extension
,~/~ is called Kummerian if it is an Abelian Galois extension with Galois group
of degree
~
2.
and
~
is an
n-Kummerian
ring.
Invertible Modules Connected with One-dimensional In this section
Galois group
Cocycles
~/~ is an arbitrary finite Galois extension of commutative rings with
,~ , '!.i."~S, ~i is the group of invertible elements of ring
{ .
For our purpose we present here certain facts connected with the monomorphism
,~(~
[L(q_!)--~c'~(R) , which is at the head of the seven-termed exact sequence of Galois ring theory (cf. [i, p. 31]; of ring i, cf. In ring
~(~
is the group of classes of finitely generated projective
,~-modules
[4]). f~ there exists an element
LL~ such that
qS.!R "
515
(cf. [i, Lemma 1.6]). Fixing the element ~*, with each one-dimensional ~,~(~)) we associate an element ~ ~ R ~ ,
cocycle ] from
by setting
(cf. [3, p. 403]). The
Proposition i. sequently,
~e~R~
is a projector, i.e. ~o~
~ { = ~ r ~ is a finitely generated projective
Proposition 2. any ~e~
~-endomorphism
An element
~ of ~
and con-
-module.
belongs to the
~ -module
~
if and only if for
we have
Proposition 3. In particular, if
If the cocycle # is a coboundary, i.e., ~(~)-4 for all o'e~, then @i=R.
~_--~--~ %~e~(~), then ~f = ~ .
If the cocycles $ and ~ are cohomolog-
ous , i.e. , ~(~)=~C~)---~--~ for some %~[i(~) then~ = I ~ . The proofs of these propositions are obvious. We note that the
~
~ -module
does not depend on the choice of the element
97 with
trace equal to i. Along with ~ we consider the
Proposition 4. Proof.
~-homomorphism ~e E ~ R
The image of the A
It is clear that ~ # =
Module
Proposition 5.
Q~
~, defined by the equation
~-homomorphism ~: ~-~-~ coincides with
~r~.
~#.
The opposite inclusion follows from the fact that
generates the unit ideal in
~ , i.e., ~ ~ = ~.
Namely, if
is a Galois system of the extension ~/~, then
~=I The proof is obvious. COROLLARY.
QS is a faithful
Propositio n 6. Proof.
Let
If ~, ~ ,
R-module (in particular,
~, ~Z1(~, 1~(~)). Then
Proposition 7.
516
Q~Q~ = Q~.
then for ~=~(~) and ~=~C~) one has the equation: ~=~'(~I
~=0C~(~} 9 Consequently, Q # ~ c ~ . the form z = ~ $ ~ 6 ,
Q~(0)).
where O~e~
~e~,
Conversely, if c=~(~), ~ , we get
With the same notation one has the equation
where
then representing
~ in
where |
--
i,='I
Proof.
Since the systems of elements ~4,.,"~ , ~ and ~p...,V~
, one has that
~(~41, ...~ ~ ( ~ )
and ~(~4) ....,~(~r~) generate
Hence it suffices to prove that ~---~-~(~c~)|
for ~ f ~ .
n%
n~
K-i
K=4
generate the
~
and
~ -module
~-4, respectively.
Since~Q~-, =~g, one has
A
~# The module
~,
"
as the tensor product of faithful projective modules, is faithful, and hence
free. Thus,
~
is an invertible
of rank i), while
~-~
Proposition 8. The
is isomorphic with the
Let
~
be an arbitrary
~ -homomorphism ~:~ O ~ - - ~ Proof.
~ -module (i.e., a finitely generated projective module
~,
~-module
~=~O~%R(~)
~ -submoduie of
for which ~ 9
(cf. [4])~
~, ~ Z 4 ( ~ ( ~ ) ) ~6M)is
This assertion is obvious if the module
~
is free.
and
~
~
an isomorphism.
In the general case the
proof is carried out by the method of localization. THEOREM.
Let $
and ~ be one-dimensional coeycles of
Z4(~v ~(~)).
Then the ~ -homo-
morphism
for which g|
(~E~ s , 6 ~ )
is an isomorphism.
The assertion of the Theorem follows from Propositions 8 and 6. 3.
Construction of Kummer Extensions Let {/~ be a Kummer extension with Galois group ~
cyclic subgroup of order since
~ is Kummerian.
character of the group
of order ~
and degree ~
~ of the group ~(~) of invertible elements of ring We call any homomorphism & .
All the characters
~ of the group ~ ~ form the group
,~be
~ , which exists
into the group ~ ~
a
a
&=~O~%(&~)
with
respect to the usual multiplication of characters. Since each character
~6 C~g~ & is obviously a one-dimensional cocycle of Z4(@, ~(g)),
according to See. 2 there are defined the
of
~ -homomorphism
~ and the
~ -submodule ~ =
~
g. THEOREM i.
If
~/~
is a Kummer extension with Galois group
~, then for the
~ -module
we have the direct sum decomposition
where
~ runs through all characters of the group
~ .
517
Proof.
We note first that for characters ~ 6 C ~ g ~
~[6~) --_ fFt%~ if ~
one has the formulas:
~ - is the identity character
0 ~ otherwise
~e~/ca~Along with the degree of the group the element ~
and elements ~66
~
L 0, if its order
(cf. Sec. 2) we can take ~
4
~
.
~% is invertible in
(~%~
g .
Consequently, as
To prove the theorem it suffices to
4).
see that /%
(~
is the identity endomorphism of the ring ^
~ ) and
A
for
For 0[~ ~ we have
Further,
^
^
4 ~
~r
) =0,
The theorem is proved, THEOREM 2.
In the notation of Theorem i, the
for w h i c h ~ @ ~ - - ~
(~e~
-homomorphism
, ~eQ~), is an isomorphism.
This is a special case of the theorem of See. 2. COROLLARY I. group ~(~I
the
The map ~--'-Q~ determines a homomorphism of the group 6~g~ ~ into the
of finitely generated projective
COROLLARY 2.
Let
R -module
=~|174
Proof.
~
R -modules of rank i.
~ be the order of the character
~ from C~a~ 0, i.e.,
( ~ factors) is isomorphic with
~=~.
Then
~ .
By virtue of Theorem 2 and Proposition 3 of Sec. 2 we have
Theorems 1 and 2 show us a way to construct K u ~ e r extensions with fixed Galois group (over a fixed Kummerian ring 4.
~ ).
Ring ~[Q] In this section
~
is an arbitrary commutative ring with unit I.
In algebra the ring of polynomials is an important instrument for the construction of new algebraic structures by factorization.
518
Starting from an arbitrary finitely generated
projective invertib!e
~-module
@, we construct a ring
~[@], which can be considered
as a generalization of the ring of polynomials on one variable. In preparation we prove some auxiliary assertions. Let im be an arbitrary finitely generated projective module. a so-called coordinate system.
Then in
By this is meant a system of elements
system of linear maps ~r ,., ~n~ from
~0r~E(P~)=P'such
~
there exists
~4~'''~ ~n~ of
i3
and a
that for any ~cei~ one has the
equation r~ $=~ It is easy to prove that the element
is independent of the choice of coordinate system. -module ~
It is called the rank element of the
(cf. [5, pp. 113-114]).
By the method of localization one easily proves the following. Proposition.
Let
~
be a finitely generated projective invertible
E -module.
Then
its rank element is equal to i. THEOREM i. invertible
Proof.
Let
E
R-module.
be a commutative ring with unit i, and If g
Let ~4~." ~r
and ~
and
Q As coordinate system for and ~.[,~e(Q@O) ~" (4~< f,, ~,~/'t,l,),
are elements of
~4~... ,~r~e
the
R-module
one has the equation:
~-module
@ | ~ one can take the elements e~@~]e ~ @
where
( ~ ' Q @ E , ~ - - ~ ~ , defined the formula Since ~ ( 6 6 )
~, then in ~ |
be a coordinate system for the
To prove the theorem it suffices to verify that
(cf. [4]).
@ be a finitely projective
~(&|
e__.,~|
= e#| where
ge~
~,~). , h e ~ ~ is an
The map ~. -isomorphism
= 4, for the element
b we have
Further, since
(~(~)=~
and ~(e6|
Using this, we represent the element
= ~Le{), one has
gf,|174162
of ~ | 1 7 4
in the form
519
To this equation, we apply the map ~ | ~*| ~ - - ~ ,
for which ~ @ ~ |
We get F~
Now for ~ ,
~r~
we have
iv-,( Theorem 1 is proved. COROLLARY.
Let
jective invertible
~
be a commutative ring with unit,
R -module.
If ~ , . . . , g ~ ,
~
be a finitely generated pro-
then in ~@R... |
~ ( % factors) one has the
equation ~4| whe re
|
= ~4|
.., @ ~
,
(~,..., 6~) is any permutation of the numbers ~ . . . , ~. We denote by
~
the tensor product ~ @ R ... |
(%
factors) and by E [Q] the following
-module OO
R[Q] = Z~ e Q~
(QO= R).
We introduce an operation of multiplication in R[Q] by setting for natural numbers ~ and }
C~|
| ~)(~+~ |
|
|
and extending this law to all elements of ~[~] by distributivity. turned into an THEOREM 2.
In this way E[~] is
~ -algebra (cf. [2]). Let
jective invertible
~
be a commutative ring with unit,
~ -module.
Then the
~-algebra
Q
be a finitely generated pro-
R[Q] is commutative.
The assertion of the theorem follows from the Corollary of Theorem i. Remark i.
If Q = ~ X
is a free ~-module of rank i, then Q ~ = ~ X m ,
(~ factors) and the algebra
where
X m = X @..'@x
~[~] is isomorphic with the polynomial algebra on one variable
REX] Remark 2.
Analogously one can transform the
R[QI,...,Q ~] = ZI
~
~)O,...,~n)O
( Q~
are finitely generated projective invertible
set
where
520
~
Q~ ,
~
(~ ~ >~o).
m -module
Q~ |
|
R -modules) into an algebra.
For this we
The algebra ~[@4,'"~ Q~] is commutative and if all it is isomorphic with the 5.
Q~
are free modules of rank I, then
algebra of polynomials in ~ variables.
Factorization by a Homomorphism Let
~
be a commutative ring with unit and
-module of rank i. :~
~=~
~ .
We define an ~ of
Suppose for the natural Using this
If ~ r ~
~K, then for ~ = ~ 4 |174
and for all K ~
projective invertible
If 0 ~
n4-{ and
is already defined on elements
~ .
For elements
X
and ~
of ~
we set
~ be a finitely generated
~=~='f@~. The multiplication operation introduced above,
~-homomorphism q: Q"~--~ ~, turns
~
into a commutative algebra.
It suffices to note that the map q:R[~]-'-~ is a ring homomorphism.
Proposition 2.
Under the conditions of Proposition i, the
as a unital subring, while considered as an tive over
into a ring.
~ that for x, ~E~[~] one has the formula
be a commutative ring with unit,
R-module
constructed for a given Proof.
~
the map q
R-homomorphism
(~% we set
We introduce a multiplication operation in
Let
we turn the module ~
~ : ~[Q] --~ ~ in the following way.
It follows from the definition itself of the map
Proposition I.
be a finitely generated projective
number M% there is given an
~-homomorphism,
~-homomorphism
, then ~ ( m ) = ~ .
~
~-algebra
Actually,
~
contains
~-module it is finitely generated and projec-
~.
The proof is obvious. Proposition 3. in ~[~]
where
The kernel of the ring homomorphism ~: ~ [ ~ ] ~
~" coincides with the ideal
generated by the elements
~ runs through all elements of
any system of
Q~
(it suffices, clearly, to take as elements
~-generators in Q ~ ) .
We omit the proof. The ring ~Q,~
~, which depends on the choice of
~-homomorphism
~, will be denoted by
, and the process of its construction will be called factorization by
(~ .
521
6.
Loca! Separability Criteria Galois ring extensions are always separable.
Hence in constructing extensions it is
important to have criteria which allow one to establish their separability.
We give here
the local separability criteria which we need in what follows. In this section
~
is an arbitrary commutative ring with unit,
(not necessarily commutative) Let
C
R-algebra
coincides with the center
~-algebra
A
C
A.
According to one of the definitions of
is called separable if its Noetherian different
of the algebra A
that for the separability of the algebra A~over
is an arbitrary
~-algebra.
be the center of the
separability, the
A
A,
(cf. [3, Proposition i.i, p. 369]).
~A/~ We show
the separability of all its localizations
R ~ with respect to all maximal ideals
~
of the ring
~
is necessary and suf-
ficient (with respect to the notation, cf. [4]). Proposition i.
Let ~ = ~ A / ~
be the Noetherian different of the
be a maximal ideal of the ring Aw
is equal to the localization of
~ . ~
algebra.
Let
f~0 be the opposite
If we denote by
which ~ @ ~ - - ~
~
by
~,
~
f o r which H~ = 0 .
~-algebra for
A
The image
~-algebra
~, i.e.,
~ be the enveloping ~: Ae--~A
(under
t h e c o l l e c t i o n of a l l ~ A e ,
(~(~), i s an i d e a l i n t h e c e n t e r of t h e a l g e b r a
for
A .
C o f the a l g e b r a
A , and i s
By v i r t u e of t h e c a n o n i c a l isomorphism
A~v can be i d e n t i f i e d
w i t h (Ae)~.
Since
~
is a flat
[4, Theorem 1 ] ) , we have the e x a c t sequence
e ~j~A 0 - - ~ H ~ ---'~ A~, We assert that the right annihilator with the localization clusion
Ae=A|
and
~H--~Ae ~-%-A----0.
o f t h e submodule
c a l l e d the N o e t h e r i a n d i f f e r e n t module ( o f .
= "ff~ 9
), then we have the exact sequence
be t h e r i g h t a n n i h i l a t o r
the enveloping
Let
i.e.,
the kernel of the augmentation epimorphism
0 Let
A .
Then the Noetherian different of the Rv,-algebra
~^~IR~ Proof.
~-algebra
~
A~/ of the kernel ~
of the right annihilator
A ~ c A ~ i s obvious.
Let ~
__~0
~
(g~A~ER\Y@).
in the algebra
of the kernel
~
e
A~ in
coincides
Ae .
The in-
Then for any @ell and a n y ~ X ~
we have
OK. ~ which means o~(0~)=0 for some
0~\~
~
=0,
But then ~ e A
-~-= #~ e A~.
and hence
A4 =A~, . Now we have r~A~/R ~
which is what had to be proved. 522
= q~(A 0 = q~(A~)=(~(A))~, =~
,
Thus,
Proposition 2. A ~
Let
A
and
~
be two
~ -submodules of
~ .
In order that the modules
and B should coincide, it is necessary and sufficient that their localizations should coincide for all
9; as
~-submodules
of
A~
and
Am .
We omit the proof. THEOREM.
Let
~
In order that the maximal ideal Proof.
be a commutative ring with unit and A be an arbitrary
~-algebra
%W of the ring Since
A~ ~ A|
~ -algebra~
A be separable, it is necessary and sufficient that for any ~
the localization
R~,
A~
be a separable
~-algebra.
the necessity of the condition follows from Corollary 1.6
of[3]. To prove the sufficiency, we consider the Noetherian different ~ = ~ A / ~ A
According to Proposition I, the Noetherian different of the
with the localization of the
~-algebra
~%W. A~w .
Let
~=C(w~)
~W, one has (Proposition 2)
Remark. algebra
A
be the center of the algebra
In view of the inclusion
it follows from the condition ideals
C
that ~ = C ~
~=C,
A~
be finitely generated.
proved under the assumption that
coincides
A and C(~) be the center
.
i.e., the
Since this is true for all maximal ~ -algebra
A
is separable.
The assertion of the theorem with certain restrictions on the ring is given in [3, 6, 7].
A
~-algebra
C~w~ O~%WI and the inclusion ~ c C ~ ,
~
and the
Thus, in [3, p. 378, Corollary 4.5] in the conditions
for local separability criteria it is required that the ring module
of the algebra
~
be Noetherian and the
~ -
In [6, p. 468, Proposition 2.3] the same theorem is A
is a commutative
~ -algebra, which as an ~
-module is
finitely generated and projective, and in [7, p. 94~ Corollary 2.10] it is required that the algebra 7.
A
be a finitely representable
A e -module.
Radical Ring Extensions Again let
~
Definition.
be an arbitrary commutative ring with unit. Let
~
be a finitely generated projective invertible
suppose given for the natural number
~
~ -module and
a module homomorphism q : ~ - - ~ ~.
Then the
~-
~-~
algebra
S'--~Q,q=~o@ ~, obtained by factorization by the homomorphism
extension of the ring THEOREM.
Let
~.
=~.o~
be a radical extension of the ring
finitely generated projective invertible that the
~ -algebra
I) the element
~ ~ is called a radical
~ -module
~
~ , constructed from a given
and homomorphism q : ~ - - ~ ,
in order
~ be separable, it is necessary and sufficient that ~.f be invertible in
~ ;
2) the map q be bijective. For the proof we need the following auxiliary assertions~ Proposition i.
In order that the element ~
and sufficient that for any maximal ideal in the ring
of ring
~
be invertible, it is necessary
~W of the ring
~
the element
~
be invertible
~.
523
The proof is obvious. Proposition 2.
Let ~(X)=~-~ be a polynomial in the ring R[X].
R[X]/(~(X}) is separable if and only if the elements ~.I and ~
The quotient algebra
are invertible in
~ .
Proof [6, Corollary 2.4, p. 468]. The proof given in [6] does not require the hypothesis about the absence of nontrivial idempotents.) Proof of the Theorem. -module
Q
Let ~
be an arbitrary maximal ideal of the ring
is invertible, there exists in ~
g~= R~(~R~,~ e . . . where ~,~'=9(~...O9 (~, factors), 4~ ~-'~-I. the ring
an element ~
Since the
such that Q~ = ~ .
Then
9 p,~m-~,
The ring S'~ is, obviously, the extension of
R~, obtained from the ring R~[~]=
p.,~
~
~'
"
~,
induced by the homomorphism
~,-o Q~=,-0 'p" e Rw,V~
by factorization by the homomorphism ~v,: ~
other words (Proposition 3 of Sec. 5), the ring ~ ring ~ [ ~ ]
over ~
are invertible in ~
~.
In
is the quotient ring of the polynomial
by the ideal generated by the polynomial~-~(~$~l.
2, the algebra ~
~
~.
By virtue of Proposition
is separable if and only if the elements ~
and ~v,(~~)
. The second of these conditions is equivalent with the fact that
is an isomorphism ~
onto ~
. Now applying the Theorem of Sec. 6, Proposition 1
of the present section, and Theorem 1 of [4]~ we get the assertion of the theorem. 8.
The Construction of Cyclic Kummer Extensions We assume here that
~
is an ~-Kummerian ring and ~
is some fixed cyclic subgroup of
order ~ of the group ~=~(~) of invertible elements of the ring theses of Sec. i.
Further, let
We denote by ~--~(R)
~
be a cyclic group of order
~, satisfying the hypo~.
the subgroup of elements of period ~ of the group ~(RI of classes
of finitely generated projective invertible there exist isomorphisms of
@~
onto
~-modules.
If C 6 ( Q ) ~ ,
then consequently
~ . For each isomorphism ~: Q~--~ ~ we can, according
to Sec. 7, construct the radical extension ~@,@. We choose and fix some isomorphism ~ of the group element in the group of characters C~g~--~o~i~,~). ~Q,~ , operators from
for any ~ E ~
~
onto
~, i.e., some generating
Further, we introduce on the ring
~ , by setting
It is easy to verify that
and any ~ Q s
$, by virtue of the
definitions introduced, is a faithful group of operators for ~Q,~ . We note that the action of the operators from
~ on
~Q,q depends on the choice of
this dependence explicitly, the ring THEOREM I.
524
~,~
~ .
Wanting to indicate
will also be denoted by ~Q,@,%.
With the notation introduced, the radical extension
is a Galois extension of the ring degree
with Galois group
~
(a cyclic Kummer extension of
~ ).
Proof.
Firstly we prove that the subring of
incides with ing element
R . Let~=~§
[~Q6)
6" of the group
~, then
of the elements ~-~(o') ( ~ - ~ ) ,
=~
~
~-invariant elements of the ring
be an element of
(~->~(~))~=0,
~.
~
co-
If O'~=~ for some generat.
whence, by virtue of the invertibilit~
it follows that ~4 . . . . . ~.~ = 0, which means that
R. The separability of the extension
~/R
follows from the results of the previous section
It remains to verify that all automorphisms from (cf. [I, Theorem 1.3, p. 18]). suppose for some idempotent e
from
invertibility of %(0")-~(~}, ~ e = 0 . ~e=0 for all ~ ,
are pairwise strongly distinct
Let 6" and r be two different automorphisms from ~
In particular, for all ~ from
so we have
~
one has the equation ~
we have
~(~)e=~(~}e
~(~)~e=~(~)~,
But elements from
~
~, and
for all y from
whence, by virtue of the
obviously generate the ring
~,
which means that ~=0o
Theorem I is proved. THEOREM 2.
Let
~
with Galois group
~Q,q,~/~ for
be a cyclic Kummer extension of degree ~ .
some module
an arbitrary isomorphism Proof.
Let
(cf. Sec. 2).
~
If
q
q
Q
such that
~(~)e~,
of the Kurmmerian ring
~ -equivalent with the extension
and some
~ -isomorphism
is the isomorphism of ~
onto
~h~r ~ .
We set ~ = ~ m ~
[ , defined by the formula
we denote the radical extension of the ring
and on which the action of the operators
Now we define a map
~:~a-~-[ (for
~:~--~).
be an arbitrary generating element of the group
(~ie @ ), then by ~ q , ~ and
Then the extensionT/~ is
~
~ from
~
~ , constructed from
is defined by the chosen
~:~@,q,~--~7' by setting
Q) (for ~=0 the map ~ is the identity of ~ -isomorphism, establishing the 9.
onto ~ ). It is easy toverify that ~is a ring
~ -equivalence of the extensions
S~,%~/Rand T/~.
Equivalence of Cyclic Extensions We preserve the notation of Sec. 8.
In Theorems i and 2 the isomorphism
~:~-~- ~
is
assumed to be fixed. THEOREM 1.
In order that the Kummer extensions ~ = ~Q,q
(with cyclic Galois group that there should exist an
~
of order
~ ) be
I-isomorphism
and
~ z ~(~I
of the ring
~ -equivalent, it is necessary and sufficient
f:~__~Qr such that
525
Proof. for which
Suppose for the extensions q=q~ ~.
We extend
is the identity map of onto
~I.
~
and
~! there exists an
~ to an
R-isomorphism
R
R ).
onto
If
~: ~--*-~, by setting:
We verify that ~ is a ring isomorphism of
It suffices to see that for ~e~ ~ and ~
tion ~(~)=~(~J ~ ) .
R -isomorphism y:~--~r,
(0~
~-4
) one has the equa-
K+{-= ~, then ~(~z)--9(~@ Z)=~(~)@~(~)= 9(~)~(~).
let ~=0~4@...@~ ~ ~ ~ = ~ § 1 7 4
(here g ~ @
~ = @(~ |
,4 ~
K+~).
| ~) R~ |
Let ~ §
and
Since
@ a~§
one has
=
f
)(q
|
9 9 |
=
It is easy to see that the ring isomorphism g:~--~grsatisfies for any
m
from
~
and ~ from
&.
Conversely, let the map~: ~__,~I be a ring the elements of ~ unchanged. with the condition
For elements
@! i Here, if ~ @
~%...~)=~(~)...~(~) Thus,
~=~
~,
THEOREM 2. which ~ ( @ } e ~ e ;
~ the condition r
~ -isomorphism of the
~I, which leaves
~(r
is equivalent
R -module
~
onto the
R -
@ g~) = ~... g~ (product in ~ ) =
(i=4~...~ ~), then q(~4| (product in ~ )
=~(~)...~(~)=~(~}|174
=C~f~)(~|
and the theorem is proved. Let
~
be a fixed finitely generated projective invertible
further, let
the Kummer extensions ) be
~
~ -isomorphism of ~ onto
~q(~} =~(~)%x), and since ~= ~m(~: g--~ g)and ~'= Ira(9: g[--~ ~) (cf.
Sec. 2), the restriction ~I@ = ~ is an
module
the condition
~,~
~
and
and ~ , ~
~
be twoisomorphisms of ~
of the ring
R
onto
~ -module for
~ .
(with cyclic Galois group
In order that ~
of order
~ -equivalent, it is necessary and sufficient that in the group ~ of invertible
elements of the ring
Proof.
~
there should exist an element
Since ~0~R(Q,Q)~--R, any
invertible element E ~ .
R-automorphism
Now if # = g'i~ (~Q
$, such that
~:Q--~
is multiplication by an
is the identity endomorphism of the module
~ ),
then ~ = g~.i~, and the assertion of the theorem follows from Theorem i. i0.
Group of Equivalence Classes of Abelian Extensions Let
~
be a commutative ring with unit and
Abelian group ~ R ) , (with the group
&
whose elements are classes of as Galois group) denoted by
is defined in the following way. 526
~
Let ~ / ~
be a finite Abelian group.
In [9] the
& -equivalent Galois extensions ~/R
~(~) is considered.
Multiplication in ~(&~)
and ~ / ~ be two Galois extensions with Galois
group
6.
Then the tensor product
with Galois group & x 6
~g~z
is a Ca!ois extension of the ring R ~ R | ~ = q |
(cf. [3, p. 402]).
In the direct product f f ~ we denote by D
subgroup consisting of elements of the form (~,~-~), where obviously, is isomorphic with the group
&.
~e &,
ring
~4|
~ =[~@R ~z)b
is a Galois extension of
dicated natural identification
R
in which the symbols
consisting of
~-invariant elements of the ~
(by virtue of the in-
) is the Galois group.
The identity element of the group ~(e,s structed in the following way.
According to the Galois theory
for which the group
~(ff~&)/~
The quotient group (ff~&)/D~
We identify this quotient group with the group
by means of the natural isomorphism ~--~(~4)D = (4,~)D. of commutative rings, the ring
the
We set
is the class #6{S) containing the ring
We choose symbols ~ [ ~ e & )
and we consider the free
S
con-
~ -module
8~ are free generators:
6"~ 6-
The
~-module
E
is turned into an
-algebra if for the basis elements
e~
the multi-
plication is defined by the rules:
e$ The operators ~ & ~
act in
~
f=
according to the rule
It is easy to verify that E/~ is a Galois extension with Galois group g~CE)
is the identity element of the group ~(&~R). We also indicate the construction of the
Galois extension
~/~ we denote by
morphic with the ring ~66
6, and also that
~
~
inverse element in the group ~(&,~).
the Galois extension in which the ring
under the isomorphism X--~-m~ ( ~ 6 ~
act according to the formula
~xo=(~)o.
~o)
One can verify that C~(~-~(~~
direct product ~ x . . . x & ~
Namely, let the finite Abe!ian group
of cyclic subgroups~4,...~ ~ "
from
~
~
C{(~}.
reduces to the
be represented as the
Further, let ~{/~...~ ~%/~ be
Galois extensions with Galois groups &i~...~%, respectively. ~@R"" |
~0 is iso-
and on which the operators
The study of the group ~(ff~R)for an arbitrary finite Abelian group case of cyclic Galois group.
For a
We consider the tensor product
and on its elements we define the action of operators
~=
~...~ ~e
ff~)
~ : ~(~
|
~ ~,1 = ~
~. 9 9 ~ ,
(~
~'~) .
Then the map
induces an isomorphism
527
.
.
.
The inverse for it will be the map ~--~(~,...,~), where
~=~H~, 11.
~ =~...~&~...~.
Group o f C y c l i c Kummer E x t e n s i o n s Now we c o n s i d e r t h e group o p e r a t i o n i n ~ { ~ }
the notation of Sets. THEOREM 1.
Let
Further, let ~ f f i ~ , ~ Galois group
~.
8, 9; f i r s t l y , ~
f o r c y c l i c Kummer e x t e n s i o n s .
t h e isomorphism ~: ~ - - ~
be a c y c l i c group o f o r d e r
and ~ = ~ , ~
~
and
We p r e s e r v e
i s assumed to be f i x e d . ~
be an ~-Kummerian r i n g .
be two n -Kummerian e x t e n s i o n s of t h e r i n g
~
with
Then
where Q=~ r
(we denote by ~ the natural
R|162
and
~ _--(~4| ~)~ : ~
~-isomorphism of Qm onto ~7@ ~ ~; we identify
with
~| For the tensor product ~4| ~r we have the direct sum decomposition
Proof.
=
If ~ :
~
and
~Q~
9
, then for any ~" from
(~,~~)( ~ ~ ~} = ~6-) ~ ~(~-)~ = ~-J(~) ~ ~. of elements of
Whence it follows that the subring respect to
~|
~ ' which are invariant with
(~6"-~) has the form
Further, for ~& @: and ~&
we have ~(~|174174
This proves that for some
R-isomorphism
=~C~)~. ~:~--~ R there is a
-isomorphism
~= ~Q,~ 9 It remains to find
where ~ = ~ @ . . . @ x ~ ,
528
~.
Let Z~eQ~, ~ Q ~
~=~@...@ ~.
(~=~,...,~).
Then in ~Q,~ we have
On the other hand, in the ring
i.e.,
q=(q4~qZ)o#,
and Theorem i is proved.
For a class s
we set
~(c~(~1) = c ~ where ~ = ~ ( ~ :
~--~).
e~(R),
According to Sec. 8 and Theorem 1 of the present section,
epimorphism of ~(@,~) onto
# is an
~n(R).
Further, for EE~ we set (cf. [8]):
~(~ modU~)
=
Ce(~'R(~,~),
q(a ~)
In this way we define a monomorphism A : ~ / U " - - ~ ( & , R )
8.
=
(Theorem i and Theorem 2 of Sec. 9).
It is easy to verify that ~ m ~ = ~g%Q, so that we have the following exact sequence
4 ---~-u/u ~ ~ ~(G~,R)~-!-~(~) -)-- 4.. THEOREM 2. Proof.
The exact sequence
(I) splits, i.e.,
any isomorphism ~ : @ ? ~ - - - ~ ( ~ = 0 ~ d ~ { ( @ 0 ~ ~ .
Then each element
morphism q ~ Q z ~ : ~ - - ~ element
~I%,~=Ke~,J)is
a direct factor in ~(~)R).
By a well-known theorem of PrSfer, the group ~%(~) can be decomposed into the
direct product of cyclic groups ~(~)%~I{Cg(@~)I.
~--~
(1)
where
~-equivalent
is defined a map ~.@~(~)---~(~,~), fl is a homomorphism, Remark.
~6d{=~).
~el we fix the
We denote by
~-module
q~ the isomorphism
0~
and
q~=~#s
C~[@) of the group ~ [ R I defines a fixed module ~ %~I ~[~ and iso-
(0~K~<~,
~(~) the class of
For each
K~= 0
for almost all
6 ).
extensions containing
We associate to the
~O,q/~.
In this way there
It is easy to verify that
for which, obviously, 7P=~=l.
and Theorem 2 is proved.
The maps
~ and
~ in (i) depend on the choice of generating character
~
in
the group C ~ % 6. !2.
Group of Kummer Extensions We proceed to the consideration of the Kummer extensions of an
an arbitrary finite Abelian Galois group
~
of period
~
(~=~
~-Kummerian ring
for all
~6~r ).
~ for
Our goal is
to carry over Theorem 2 of Sec. Ii to noncyclic Kummer extensions. We consider the group
of extensions of the group ~ = ~(R) of invertible elements in Ch&~& = H o m ( 6 ~ )
R
by the group of characters
and we define a map of the group (I) into~(&)~).
Let (2)
be some exact sequence of Abelian groups.
We choose in
~
elements
g~ , for which ~ ( ~ ) = ~ ,
where for the identity character the chosen preimage is the identity of the group assume that
~cX
).
(we
Then 529
where g(~,~)eg.
Now we construct the free
E -module
2]@Re~ taking as free
R -generators, symbols
r
~), among which e~=~e~.
troduce a multiplication and action of operators
eTev
(4)
,
In
~
we in-
~ e 6 by means of the formulas
=u(7,,)eTV,
(5)
~e} = ~(~) e t .
(6)
Equations (5) and (6) turn the ~ -module ~ into a commutative, associative ring, on which 6 acts as a group of operators, while ~& = ~. Proposition i. group
~ , and up to
Proof. ~(~)
The extension ~/~ just constructed is a Galois extension with Galois ~ -equivalence it is uniquely determined by the element of the group
Varying in (3) the choice of representatives
G~,
we change the two-cocycle
to atwo-cocycle cohomologous to it, and this-leads us to a ~ -equivalent extension. Thus, up to
-equivalence,
the extension
apply the concept of
~/~ is uniquely determined by the sequence (2).
~ -equivalence to an arbitrary extension of the ring
~
(Here we
with operators
from ~). We decompose the group C ~
~ into a direct product of cyclic subgroups and in each of
these cyclic factors we choose a generating character. their orders by
~,..., ~ ,
(2) in the following way. ~,...~,
respectively. Let ~,..., ~
We denote
We choose the system of representatives
be any representatives
i.e., let ~(~i)=~i (~=~,...,~).
= ~7s
Let these be ~t...~%.
from
~e~
in
X for the characters
If
~:"
(0~ K# < ~ i ) ,
(7)
then we set
a~= ct:~. .. a,,K~ Since ~ i ~ e ~ , law in
while
530
one has
~i=
E~EU.
Now if ~ = 6 . ~ ,
~, for the system of representatives
(8) then by virtue of the multiplication
(8) one has the equation
Whence it follows that the ~=~[e~] ~ e? ~= 8~.
~ -algebra
~
decomposes into the tensor product of subalgebras
According to Theorem 1 of Sec. 8, the extension ~ / R is a Galois exten-
sion with respect to the group of operators
~/~r
But then
~/R as a tensor product of
Galois extensions is also a Galois extension with respect to the group of operators
$, and
Proposition 1 is proved. Thus, we have a single-valued map
X : Ext (Cha~ if, ~)--~-~(~, R). Proposition
Proof.
The map (9) i s
a monomorphism.
Omitting the verification that the map
Galois extension E/~
2.
~ is a homomorphism, we assume that the
~/~, defined by (4), (5), and (6), is
(cf. Sec~ i0).
(9)
This means that
~
~ -equivalent with the extension
can be represented in the form
From (6) it follows easily that
where
V~E ~
9
Further, from (5) we get that =
and this proves that COROLLARY.
~
v
is a monomorphism.
The identity element of the group ~(&,~l is determined by the
(4), for which in (5), N(~?)=i
for all
~-algebra
~ and ? from ~k~%&.
According to Corollary 1 of Theorem 2 of Sec. 3, for each class
~(~I~ ~ ( ~ ) ,
there is
uniquely defined a homomorphism
Consequently, we have a map :
Proposition 3. Proof.
The map
~
~ (e,R.)--*/-/o~K@..'~6,ff(,q)).
is an epimorphism.
We omit the verification that
homomorphism of the group Cha~& into ~(~).
~ is a homomorphism.
Let ~
be an arbitrary
Using the notation from the proof of Proposition
i, for each ~ = ~, ..~% we choose a projective invertible and a homomorphism q~ :@~_-~ ~.
(io)
g-module
@6, so that ~(~)=C{~
Now if, following the notation of Sec. 8, we set
= ~'Q~,q,,y~ oR 9
~)~ ff~.,,q~.~,y.~ ,
531
then we will have f(~(~))= ~. Proposition 4.
Proof.
For the maps
The inclusion ~
~c
% and
Ke~
~ one has the equation
9 is obvious.
Let C{(~)e K ~ V.
Then for
~, the de-
composition of Theorem i of Sec. 3 assumes the form (4), where
e~ are invertible elements
in ~
(by virtue of Proposition 5 of Sec. 2).
in R(~), generated by the
group
~
and the elements 6 > ( ~ e C h g ~ ) ,
The subgroup
X
defines an extension (2) for which, as is easy to
see, >,CC~(X))=C(~(s The facts we have proved make it possible to formulate the following theorem.
in
THEOREM I.
Let
R,
be a finite Abelian group of period
and
~
be an
~-Kummerian ring,
~
be the group of invertible elements ~.
Then one has the following exact
sequence
Remark.
In the case of a cyclic group
~
of order
~
(a divisor of
of Theorem I coincides with the sequence of Theorem 2 of Sec. ii. character
~ of
C~a~.
To each element e ~
The map 8-~-~ determines an isomorphism coincides with the map % of Sec. ii. ~:HQ~ICh=~,~(R))--~(~I Now let
~
and
a
and
We fix the generating
we associate the two-cocycle
C:~/Rm-,-F~t(C~a~@,~), and
~o~ coincides with the map ~
of Sec. ii.
be two finite Abelian groups of period , ~(H,R),
~.
We denote by
K
set ~ =(~|
~.
With a homomorphism
defined as follows (cf. [9, p. 3]).
and let C~I~) be the identity element of the group
x ,~ acts on ~|
XoC, as is easy to see,
Further, the map ~-~-k(~) determines an isomorphism
60:~-~-H is connected a homomorphism 00~:~(~R) Let ~ ( ~ I 6 ~ R )
~) the sequence
"~(H,~). The
direct product
F~ according to the rule
the kernel of the epimorphism GxH --~-H, for which (6~r Then
~4/~ will be a Galois extension with Galois group
and we ~
and ~ ( ~ ) =
oo*(c~(s Further, we define a homomorphism
If ~ e Z ~ ( ~ , ~ ) acters from
532
is a two-dimensional symmetric cocycle C~g~,
then the equation
(~{~;,~')=~u(?,'~))and
~,# are char-
from Z~ (Ch~%H,~).
defines a two-cocycle
The map ~-~-~ induces the defined homomorphism
HCC.a, F i n a l l y , i f for L~o~rl,(C~l,o,'tGfJ~(R)) and e~eC~c~ H we set
then in this way we define a homomorphism
THEOREM 2. morphism
Let
co:~--~
~
and
H
be two finite Abelian groups of period
.
For a homo-
there are the following commutative diagrams :
Ez~: (Ck(~'~/4,i~) - A .... ~(H,R),
I c~
1 ~ov
The proof of Theorem 2 reduces to purely technical verification and we omit it, THEOREM 3.
The exact sequence of Theorem i splits, i.e., ~ = ~ e ~
is a direct factor
in Proof. groups. subgroups
According to the Remark after Theorem I, Theorem 3 is valid for cyclic Galois
We represent the finite Abelian group ~4~.., ~ ~ .
Let ~ : ~ - - ~ 6
inclusions (~=4,...,~}. , respectively.
For the group
and ~
~
~:~--~
in the form of a direct product of cyclic be the corresponding projections and
the maps (9) and (i0) will be denoted by
~
and
Since the theorem is already proved for cyclic extensions, for each
there exists a monomorphism
such that #~.o~= I.
We set
=rl i oAo [ Th en
v
~
)% ~v=
and Theorem 3 is proved.
533
COROLLARY. ring 13.
For a finite Abelian group
~
of period
~ and an arbitrary
R, the group ~(~,~) is a periodic group of period
~ -Kummerian
~ .
Normal Basis Let K/~ be a Galois extension with Galois group
extension ~/~ has a normal basis, if can find an element THEOREM.
~
is a free
@, such that the elements
A Kummer extension
~
of order
~-module of rank
(r
One says that the ~
and in
form a basis for the
~/~ with Galois group
basis if and only if for all characters ~ E C h ~
~.
~
(of order
the module ~
~
one
R-module
~.
rR ) has a normal
=2~t~ is a free
R -module
of rank i. Proof. A
Let
(0"~e ~ be a normal basis for the ring ~
cording to P r o p o s i t i o n
4 of Sec. 2,
over
R
and let
~[~C/1,0L%~'.
sin=e
CsFr one has ~tt%~=,R~{F 1 and the
~-module
@~ is free.
Conversely, let us assume that ~ = R e l [~m}[r
is invertible, one has that
for all
(O~-e&
~ . We s e t ~ Y - ~
is a basis for the free
e I. Since the matrix R-module
~.
LITERATURE CITED i. 2. 3. 4. 5. 6. 7. 8. 9. i0.
534
S. U. Chase, D. K. Harrison, and A. Rosenberg, "Galois theory and Galois cohomology of commutative rings," Mem. Am. Math. Sot., No. 52, 15-33 (1965). S. A. Lang, Algebra, Addison-Wesley (1965). M. Auslander and O. Goldman, "The Brauer group of a commutative ring," Trans. Am. Math. Sot., 97, No. 3, 367-409 (1960). N. Bourbaki, Commutative Algebra, Addison-Wesley (1973). A. Hattori, "Rank element of a projective module," Nagoya Math. J., 25, 113-120 (1965). G. J. Janusz, "Separable algebras over commutative rings," Trans. Am. Math. Soc., 122, No. 2, 461-479 (1966). H. Bass, Lectures on Topics in Algebraic K-theory, Bombay (1967). T. Nagahara and A. Nakajima, "On strongly cyclic extensions of commutative rings," Math. J. Okayama Univ., 15, No. i, 91-100 (1971). D. K. Harrison, "Abelian extensions of commutative rings," Mem. Am. Math. Soco, No. 52 1-14 (1965). K. Kishimoto, "On Abelian extensions of rings. II," Math. J. Okayama Univ., 15, No. i, 57-70 (1971).