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).