Lecture Notes in Mathematics
Editors:
A. Dold, Heidelberg
B. Eckmann, Ztirich
F. Takens, Groningen

1534


Cornelius Greither

Cy

clic Galols° Extensions
"

of Commutative Rings

Springer-Verlag
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London Paris Tokyo
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Author
Cornelius Greither
Mathematisches Institut
der Universit~it Miinchen

Theresienstr. 39
W-8000 Mtinchen 2, Germany

Mathematics Subject Classification (1991): l lRt8, 11R23, 11R33, 11S15, 13B05,
13B15, 14E20

ISBN 3-540-56350-4 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-56350-4 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of
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Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1992
Printed in Germany
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46/3140-543210 - Printed on acid-free paper

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CONTENTS

vii

Introduction

Ch,,,~r O: Galots theory of commutative rinsa

§1

§2
§3
§4
§s
§6
§7
§8

Definitions and basic p r o p e r t i e s
The main t h e o r e m o f Gatois t h e o r y

6

Functoriality and the Harrison p r o d u c t

8

Ramification
Kummer t h e o r y and Artin-Schreier t h e o r y

17
19

Normal bases and Galois module s t r u c t u r e

25

Galois d e s c e n t


28

ZP -extensions

30

Chapter I: Clmlotamic deacent
§l
Cyclotomic extensions

§2
§3

1

Descent o f normal bases
C y c l o t o m i c descent: the main t h e o r e m s

32
38
45

H: C m ' e s t r i c ~ o n and "I-Iflbert's T h e o r e m 90"

§1
§2
§3
§4


Corestriction

SS

Lemmas on g r o u p c o h o m o l o g y

60

"Hilbert 90": the kernel and image o f the c o r e s t r i c t i o n

62

Lifting t h e o r e m s

64

HI: Odcttlatlona w i t h units

§1
§2
§3

Results on t w i s t e d Galois modules

67

Finite fields and g - a d i c fields

70


Number fields

73

IV: Cyclic p-extenalona and Z~-extensions of number fields

§1
§2
§3
§4
§5

C p ~ - e x t e n s i o n s and ramification
Z -extensions
Calculation of qr: examples
Torsion points on abelian varieties with
complex multiplication

88

§6

Further results: a s h o r t survey

95

P

The a s y m p t o t i c order o f


P(R, Cp~)

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77
79
83

91


vi

V: Geometric theory: Cyclic extensions o f finitely generated fields

§1
§2
§3

Geometric prerequisites
97
Zp-extensions of absolutely finitely generated fields 101
A finiteness result
106

~ w t e r Vh Cyclic C ~ o i s theory without ~!~_ condition -p-i  R"
Đ1 Witt rings and Artln-Schreier theory for rings
§2
§3
§4

§S
§6

of characteristic p
Patching results
Kummer theory without the condition ,,p-I ~ R"
The main result and Artin-Hasse exponentials
Proofs and examples
Application: Generic Galois extensions

109
113
116
120
126
135

References

140

Index

144

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Fiir Liane und f~r Margarete.


INTRODtI~I'ION

The subject o f t h e s e notes is a part o f c o m m u t a t i v e algebra, and is also
c l o s e l y r e l a t e d t o certain topics in algebraic number t h e o r y and algebraic geometry.
The basic p r o b l e m s in Galois t h e o r y o f c o m m u t a t i v e rings are the following: W h a t
is the c o r r e c t definition o f a Galois extension? W h a t are their general p r o p e r t i e s
(in particular, in c o m p a r i s o n with the field case}? And the m o s t fruitful q u e s t i o n
in our opinion: Given a c o m m u t a t i v e ring R and a finite abelian g r o u p G, is there
any possibility o f describing a// Galois extensions of R with group G?
These questions will be dealt with in considerable generality. In later chapters,
we shall t h e n apply the r e s u l t s in n u m b e r - t h e o r e t i c a l and geometrical situations,
which means t h a t we consider more special c o m m u t a t i v e rings: rings o f integers
and rings o f functions. Now algebraic number t h e o r y as well as algebraic g e o m e t r y
have their own refined m e t h o d s to deal with Galois e x t e n s i o n s : in number t h e o r y
one should name class field t h e o r y for instance. Thus, the m e t h o d s o f the general
t h e o r y for Galois extensions o f rings are always in c o m p e t i t i o n with the more
special m e t h o d s o f the discipline where they are applied. It is hoped the reader
will get a feeling t h a t the general m e t h o d s s o m e t i m e s also lead to new r e s u l t s
and provide an interesting approach to old ones.
Let us briefly review the d e v e l o p m e n t o f the subject. Hasse {|949} seems t o
have been the first to consider the t o t a l i t y o f G-Galois extensions L o f a given
number field K. He realized t h a t for finite abetian G this set admits a natural
abelian g r o u p s t r u c t u r e , i f one also admits certain "degenerate" e x t e n s i o n s L / K
which are n o t fields. For example, the neutral e l e m e n t o f this g r o u p is the direct
p r o d u c t o f copies o f K, with index set G. This c o n s t i t u t e s the first f u n d a m e n t a l
idea. The s e c o n d idea, initiated by Auslander and Goldman {1960} and t h e n b r o u g h t
t o p e r f e c t i o n by Chase, Harrison, and Rosenberg (1965L is t o admit base rings R
instead o f fields. It is n o t so obvious what the definition o f a G-Galois e x t e n s i o n
S / R o f c o m m u t a t i v e rings should be, b u t once one has a good definition (by the
way, all g o o d definitions t u r n o u t t o be equivalent}, then one also obtains nice

functoriality properties, stability under base change for instance, and the t h e o r y
runs a l m o s t as s m o o t h l y as for fields. Harrison {1965} p u t the t w o ideas t o g e t h e r
and defined, for G finite abelian, the g r o u p of all G-Galois e x t e n s i o n s o f a given
c o m m u t a t i v e ring R modulo G - i s o m o r p h i s m . This g r o u p is now called the Harrison
group, and we denote it by H(R,G), Building on the general theory o f Chase,

Harrison, and Rosenberg, and developing some new tools, we calculate in these
n o t e s the group H(R,G) in a fairly general setting.

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viii
The principal link b e t w e e n this t h e o r y and number t h e o r y is the s t u d y o f
ramification. Suppose L is a G-Galois extension o f the number field K, E a set o f
finite places o f K, and R = Or,~: the ring of ) - i n t e g e r s in K. Then the integral
c l o s u r e S o f R in L is with the given G - a c t i o n a G-Galois e x t e n s i o n o f R if and
only if

L/K

is at m o s t ramified in places which belong t o E. In m o s t applications,

will be the set o f places over p. The r e a s o n for this choice will b e c o m e apparent
when we discuss Z - e x t e n s i o n s below.
We now discuss the c o n t e n t s o f these n o t e s in a little more detail.
After a s u m m a r y o f Galois t h e o r y o f rings in Chap. 0, which also explains
the c o n n e c t i o n with number theory, and Z p - e x t e n s i o n s , we develop in Chap. I a
structure theory for Galois extensions with cyclic g r o u p G = Cpn of order pn, under
the hypothesis t h a t p - I e R and p is an odd prime number. For technical reasons,

we also s u p p o s e t h a t R has no nontrivial idempotents. Since the Harrison group
H(R, G) is functorial in b o t h a r g u m e n t s , and preserves p r o d u c t s in the right a r g u ment, this also gives a s t r u c t u r e t h e o r y for the case G finite abelian, IGI- t e R.
The basic idea is simple. If R contains a primitive p n - t h r o o t of unity ~ {this
notion has to be defined, o f course}, and p - t e R, then Kummer t h e o r y is available
for Cp~-extensions o f R. The s t a t e m e n t s of Kummer t h e o r y are, however, more
c o m p l i c a t e d than in the field case: it is no longer true t h a t every C p n - e x t e n s i o n
S/R can be g o t t e n by "extracting the p n - t h r o o t o f a unit of R", b u t the o b s t r u c tion is under control. The procedure is now to adjoin ~, t o R s o m e h o w (it is a lot
of work to make this precise}, use Kummer theory for the ring S

obtained in this

way, and descend again. Here a very i m p o r t a n t c o n c e p t makes its appearance.
A G-Galois extension S/R is defined t o have normal basis, if S has an R - b a s i s of
the f o r m {y(x) [ y e G} for some x e S. Fo G = Cp~, the e x t e n s i o n s with normal
basis make up a subgroup NB(R, Cpn) o f H(R, Cpn). In Chap. I we prove r a t h e r
precise r e s u l t s on the s t r u c t u r e o f NB(R,

Cpn),

and of H(R,

Cpn)/NB(R, Cpn).

In the

field case, the latter g r o u p is trivial, b u t n o t in general. Kersten and Michali~ek
{1988} were the first to prove r e s u l t s for NB(R, Cpn). Our r e s u l t
is "almost" isomorphic t o an explicitly given s u b g r o u p o f

NB(R, Cpn)


powers}, and Hi R, Cpn )/NB(R,
o f the Picard g r o u p o f S .

Cpn )

says t h a t

S,*/(pn-th

is isomorphic to an explicitly given s u b g r o u p

The description o f

NB{R, Cp~}

is basic for the calcula-

tions in Chap. III and V.
In Chap. II we t r e a t c o r e s t r i c t i o n and a r e s u l t o f type "Hilbert 90". This
a m o u n t s t o the following: We get a n o t h e r description o f NB(R,Cp~), this time as

a factor

S~/(p~-th powers}. This is
lifting theorems which conclude

group of

nessed by the


s o m e t i m e s more practical, as witChap. II: If I is an ideal o f R, c o n -

tained in the J a c o b s o n radical o f R, then every C p n - e x t e n s i o n S of
basis is o f the f o r m

S - T/IT, T e NB(R, Cp~).

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R/I

with normal


ix
In Chap. Ill we set out t o calculate the order o f NB(R, Cp,~), where now
R ----O r [ p - i ] , K a number field. A l t h o u g h one a l m o s t never knows the g r o u p s S t l*
explicitly, which are closely related to the group o f units in the ring o f integers
o f K(~ n), one can nevertheless do the calculation one wants, by dint o f some tricks
involving a little c o h o m o l o g y o f groups. All this is p r e s e n t e d in a quite e l e m e n t a r y
way. We d e m o n s t r a t e the s t r e n g t h o f the m e t h o d by deducing the Galois t h e o r y
o f finite fields, and a piece of local class field theory. The main r e s u l t for number
fields K is t h a t with R as above, and n not "too small", the order o f NB(R, Cpn)
equals c o n s t . p (1 +r2)n, where r 2 is half the number o f nonreal embeddings K -- C
as usual.
The goal o f Chap. IV is t o g e t an understanding, h o w far the s u b g r o u p

NB(R, Cp,~) differs
Cpn. Here H(R, Zp)


from

H(R, Cp,~),

and a similar q u e s t i o n for Zp in the place o f

is the group o f Z p - e x t e n s i o n s of R. A Z p - e x t e n s i o n is basically

a t o w e r o f C p , - e x t e n s i o n s , n -~ co. It is known t h a t all Z p - e x t e n s i o n s of K are
unramified outside p, and hence already a Z p - e x t e n s i o n s o f R, which justifies the
choice o f the ring R.
We prove in IV §2: NB(R,Zp) - Zp1+r 2 . This was previously proved in a speclal case by Kersten and Michali~ek (1989). The r e s u l t is what one expects from
the f o r m u l a for INB( R, Cp~)t, but the passage to the limit p r e s e n t s some subtleties.
The index

qn = [H(R, Cpn):NB(R,Cp~)]

is studied in some detail, and we s h o w t h a t

q~ either goes to infinity or is eventually c o n s t a n t for n -* ~. The first case c o n jecturally never happens: we prove t h a t this case obtains if and only the f a m o u s
Leopoldt c o n j e c t u r e fails for K and p. Another way o f saying this is as follows:
NB(R, Zp) has finite index in H(R, 7p) if and only if the L e o p o l d t conjecture is true
for K and p. We give r e s u l t s a b o u t the actual value of t h a t index; in particular, it
can be different f r o m 1.
Apart f r o m adjoining r o o t s of unity, there is so far only other explicit way
o f generating large abelian extensions o f a number field K, namely, adjoining t o r s i o n
points on abelian varieties with c o m p l e x multiplication. We show in IV §S t h a t
lip-extensions obtained in that way tend to have normal bases over R = O r [ p - i ] ,
and a weak converse t o this s t a t e m e n t . These r e s u l t s are in tune with the much

more explicit r e s u l t s o f C a s s o u - N o g u ~ s and Taylor (1985) for elliptic curves.
There is a change o f scenario in Chap. V. There we consider function fields
o f varieties over number fields. Such function fields are also called absolutely fini-

tely generated fields over

Q. A f t e r some prerequisites f r o m algebraic geometry, we

show a relative finiteness r e s u l t on Cpn-Galois coverings o f such varieties, which
is similar t o r e s u l t s of Katz and Lang (1981), and we prove t h a t

all

Zp-extensions

of an a b s o l u t e l y finitely generated field K already come f r o m the g r e a t e s t number
field k contained in K. In other words: for number fields k one does not know how

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many independent Zp - e x t e n s i o n s k has, unless L e o p o l d t ' s conjecture is known to
be true for K and p, but in a g e o m e t r i c situation, no new Z p - e x t e n s i o n s arise.
The last chapter {Chap. VI) p r o p o s e s a s t r u c t u r e t h e o r y for Galois e x t e n s i o n s
with g r o u p Cpn, in case the g r o u n d ring R contains a primitive p ~ - t h r o o t o f unity
~n but n o t necessarily p - I e R. It is assumed, however, t h a t p does not divide zero
in R.

Even t h o u g h Kummer t h e o r y fails for R, we may still associate t o many


Cpn-extensions S/R a class ~0n(S) = [u] in R* mod p~ - t h powers. If R is normal,
S will be the integral closure o f R in R[p-l,P~Z-u]. The main question is: Which
units u • R* may occur here? In §2 we essentially p e r f o r m a reduction to the case
R p - a d i c a l l y complete. Taking up a paper o f Hesse {1936), we then answer our
question by using s o - c a l l e d A r t i n - H a s s e exponentials. It t u r n s out t h a t the admissible values u are precisely the values o f certain universal polynomials, with parameters running over R. Reduction mod p also plays an essential role, and for this
r e a s o n we have t o review Gatois t h e o r y in characteristic p in § I. In the final § 6
the d e s c e n t technique o f Chap. ! c o m e s back into play. In §4-5 a "generic"

Cpn-

extension o f a certain universal p - c o m p l e t e ring containing ~ {but not p - l ) was
c o n s t r u c t e d , and we are now able to see in detail how this extension descends
down to a similar g r o u n d ring w i t h o u t ~,,, to wit: the p - a d i c c o m p l e t i o n of Z[X].
This e x t e n s i o n is, roughly speaking, a p r o t o t y p e o f C p n - e x t e n s i o n s o f p - a d i c a l l y
c o m p l e t e rings. All this is in principle calculable.
Most c h a p t e r s begin with a s h o r t overview o f their c o n t e n t s . C r o s s references
are indicated in the usual style: the chapters are numbered O, I, II . . . . . VI, and a
reference number not containing O or a Roman numeral means a reference within
the same chapter.

All rings are supposed commutative {except, occasionally, an

e n d o m o r p h i s m ring), and with unity. Other conventions are s t a t e d where needed.
Earlier versions o f certain p a r t s o f these notes are contained in the journal
articles Greither {1989), {1991).
It is my pleasurable duty t o thank my colleagues who have helped to improve
the c o n t e n t s o f these notes. Ina Kersten has influenced the p r e s e n t a t i o n o f earlier
versions in many ways and provided valuable information. Also, the helpful and
detailed r e m a r k s o f several referees are appreciated; I like to think t h a t their
s u g g e s t i o n s have r e s u l t e d in a b e t t e r organization o f the notes. Finally, I am

grateful for w r i t t e n and oral c o m m u n i c a t i o n s t o S. Hllom, G. Malle, G. Janelidze,
and T. Nguyen Quang Do.

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CHAPTER 0

Galois

theory

of

colnlnutative

rings

§I Deflaltlo~ and b~lc
The s t u d y o f G a l o i s e x t e n s i o n s o f c o m m u t a t i v e r i n g s w a s i n i t i a t e d by A u s l a n d e r
and G o l d m a n (1960) and d e v e l o p e d by Chase, H a r r i s o n , and R o s e n b e r g (1965). In
t h i s s e c t i o n we s h a l l t r y t o p r e s e n t t h e b a s i c s of t h i s t h e o r y . O c c a s i o n a l l y we
r e f e r to t h e p a p e r of Chase, H a r r i s o n , and R o s e n b e r g for a p r o o f . A l m o s t e v e r y thing we say in t h i s s e c t i o n is can be f o u n d t h e r e , or in t h e c o m p a n i o n p a p e r
H a r r i s o n (1965), s o m e t i m e s w i t h p r o o f s which d i f f e r f r o m ours.
Let G be a finite g r o u p , K c L a field e x t e n s i o n . Then, as e v e r y b o d y a g r e e s ,

L / K is a G a l o i s e x t e n s i o n w i t h g r o u p G if and only if:
G is a s u b g r o u p of A u t ( L / K ) , t h e g r o u p of a u t o m o r p h i s m s of L
which fix all e l e m e n t s o f K; and
K = L c, t h e field of all e l e m e n t s of L which are fixed by every

a u t o m o r p h i s m in G.
A l i t e r a l t r a n s l a t i o n o f t h i s d e f i n i t i o n w o u l d r e s u l t in a t o o weak d e f i n i t i o n in t h e
f r a m e w o r k o f c o m m u t a t i v e rings, f o r many r e a s o n s . Let us n o t p u r s u e this, b u t
r a t h e r p o i n t o u t t w o a l t e r n a t i v e d e f i n i t i o n s of "Galois e x t e n s i o n " in t h e field c a s e
which t u r n o u t t o g e n e r a l i z e well, and which indeed give e q u i v a l e n t g e n e r a l i z a t i o n s .
Thus, we will have f o u n d t h e " c o r r e c t " n o t i o n o f a G a l o i s e x t e n s i o n of c o m m u t a tive rings. S u p p o s e t h a t G is a finite g r o u p which a c t s on L by a u t o m o r p h i s m s
which fix all e l e m e n t s of K. We t h u s have a g r o u p h o m o m o r p h i s m G -~ A u t ( L / K ) .
D e f i n i t i o n 1.I, The K - a l g e b r a L a G is t h e L - v e c t o r s p a c e (~oe¢LuG ( t h e u o are j u s t
f o r m a l s y m b o l s ) , with m u l t i p l i c a t i o n given by (Xuo)(itu~) ---- X.c(it)-uc~ (X,iteL).
The map j: L a G -) E n d r ( L ) is given by

jC~u o) = (it ~

X'o(~)) ~ E n d r ( L ) .

~ 9_ j is a well-defined K-algebra homomorphism, which is bijective i f f
G is embedded in A u t ( L / K ) and L / K is a G-Galois extension.

~poslUon

Proof. The f i r s t s t a t e m e n t is e a s y to check. A s s u m e G c A u t ( L / K )

and L / K

is

G - G a l o i s . Then by D e d e k i n d ' s L e m m a t h e e l e m e n t s a of G are L - l e f t l i n e a r l y i n d e p e n d e n t in E n d x ( L ) , hence j is a m o n o m o r p h i s m . Since d i m r ( L ~ G ) --- [L:K] 2 ---d i m r E n d x ( L ) , j is bijective.

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2

chap. 0
If G -* A u t ( L / K )

is not injective, then there exist ~ ~= z in G with ](o) = j(z).

i.e. j c a n n o t be monic. If G embeds into A u t ( L / K )

but L / K fails to be G-Galois,

then there exists x e L \ K fixed under G. A s h o r t calculation s h o w s then t h a t l =
(left multiplication by x) c o m m u t e s with Im(j) c Endr(L}. If j were surjective, we
w o u l d have l
~ o n

in the c e n t e r o f E n d r ( L ) , i.e. x e K, contradiction.

L3. The K - a l g e b r a L(a) is defined to be the set o f all maps G -* L, e n d o -

wed with the obvious addition and multiplication. (Note t h a t L ~, w i t h o u t brackets,
d e n o t e s a fixed field.) Let h: L ®K L --~ L(c~ be defined by h ( x ® y ) = ( x . o ( y ) ) o e G.
~ o n
L4. The map h is a L - a l g e b r a h o m o m o r p h i s m (here L operates on the left
f a c t o r o f L ®r L), and h is bijective i f f G e m b e d s into E n d r ( L ) and L / K is G-Galois.
Proof. The first s t a t e m e n t is obvious. Pick a K - b a s i s Yi . . . . . Yn o f L. Then I® Yr
. . . . l®y~ is an L - b a s i s o f L ®r L. Thus we see t h a t h is bijective iff the matrix
(o(yt))oeG, lcondition says t h a t the images o f all 0eG are L - l e f t linearly independent in

E n d r ( L ) , or (what is the same) t h a t the map j of 1.1 is injective. Hence 1.4 f o l l o w s
f r o m 1.2.
Motivated by these descriptions o f Galois field extensions, we define for any
finite g r o u p G:
l~tlm'Llon 1~. An e x t e n s i o n R c S o f c o m m u t a t i v e rings is a G-Galois extension, if
G is a s u b g r o u p o f A u t ( S / R )

= {~o: S -~ S]~ R - a l g e b r a automorphism}, s u c h t h a t

R = S c (fixed ring under G), and the map h: S®t~S ~

S (c), h ( x ® y ) = ( x o ( y ) ) o e G

exactly as in 1.3, is bijective, or (what is the same) an S - a l g e b r a isomorphism.
lSAtamplml: a) Galois extensions o f fields are obviously a special case.
b) For any c o m m u t a t i v e ring R we have the trivial G - e x t e n s i o n S = R (~) which
is defined as follows: The algebra R (c) is again just Map(G,R) with the canonical
R - a l g e b r a s t r u c t u r e , and the action o f G is given by index shift:
=

for

o

G,

It is an easy exercise to prove t h a t in this case indeed S a = R and h is bijective.
We shall see more examples below.
There exist plenty o f o t h e r definitions, or r a t h e r characterizations, o f G-Galois
e x t e n s i o n s o f c o m m u t a t i v e rings. Some o f t h e m are listed in the next theorem:

~ L 6 .

[ C h a s e - H a r r i s o n - R o s e n b e r g (1965), Thm. 1.3]:

tative rings, G c A u t ( S / R )
conditions are equivalent:

Let R c S be c o m m u -

a f i n i t e s u b g r o u p such that S ~ = R. Then the following

(i) S / R is G-Galois (i.e. p e r def.: h: S ®e S ~
(ii) h: S @1~S----* S (~) is surjective;

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S (~) is bijective);


§1

3
(iii) S is a f i n i t e l y g e n e r a t e d p r o j e c t i v e R - m o d u l e , and the m a p j: S : : G ~ E n d a ( S )
( d e f i n e d as in 1.1 ) is bijective;
(iv) F o r any o e G \ { e ~ } and any m a x i m a l ideal M c S, there exists y ~ S

with

o(y) - y not in M.

Proof. Let us first reformulate condition (ii).O n e sees easily that h is compatible
with the G-action, where G acts naturally on the second factor of S ®~ S, and by
index shift on S ~}, exactly as in example b) above. Therefore h is surjective iff
the element (I, 0 ..... O) is in Ira(h) (the I is at position ec). Letting ~xt®y ~ be a
preimage of (I, 0 ..... O) under h, w e get the following reformulation of (ii):

(ii') T h e r e e x i s t n e ~

and x I ..... x , Yl ..... Yn ¢ S s u c h t h a t ~ l x ~ o ( y t )

is 1 or 0,

according to whether o = e G or c 4: e c. ( W e m a y write ~=ixto(yt) = ~c,e.)

(i} ~
(ii') ~

(ii): This is trivial.
(iii): W e f i r s t s h o w t h a t RS is finitely g e n e r a t e d projective. Define t h e trace

tr: S --* R by tr(y) = ~,acGo(y). (tr is w e l l - d e f i n e d since S ~ = R, and R - l i n e a r since
all o e G are R - l i n e a r . ) Let ~0t: S -*R be d e f i n e d by ~0t(z) = tr(zy~), z ~ S. T h e n t h e
f o r m u l a o f (ii') implies by d i r e c t c a l c u l a t i o n : z = ~.~t(z).y~ f o r all z e S, i.e. t h e
pairs (xt,~o t) are a dual basis f o r aS, w h i c h is h e n c e finitely g e n e r a t e d projective.
N o w w e may, by l o c a l i z a t i o n , a s s u m e h a t S is even finitely g e n e r a t e d f r e e
over R, w i t h b a s i s x l',...,x n', say. W e m a y t h e n a s s u m e t h a t t h e x~ in c o n d i t i o n (ii')
are j u s t t h e xl', b e c a u s e every e l e m e n t o f S® S can be w r i t t e n in t h e f o r m ~.xt'®yt',
and it d o e s n o t m a t t e r j u s t h o w we w r i t e a p r e i m a g e o f (1,0,...0) u n d e r h. Let us
therefore

o m i t t h e ' again. F r o m t h e c a l c u l a t i o n

just

performed

we g e t x j =

~tq)t(xy).xt, h e n c e by d e f i n i t i o n o f ~ , and since t h e x l are a basis, tr(xyy t) = ~y.
As in t h e field case, bijectivity o f j is e q u i v a l e n t t o invertibility o f t h e m a t r i x A
= ( o ( x t ) ) o , u O n e c a l c u l a t e s as f o l l o w s : Let B = (x(yj))y, T. Then A B = (So,z) =
unit m a t r i x (use (ii'}), and BA = (tr(xyyt))j~---- unit matrix. H e n c e A is invertible,
a n d j is bijective.
(iii) ~

(i): Since S is finitely g e n e r a t e d p r o j e c t i v e over R, we m a y again a s s u m e

t h a t S is free over R, w i t h basis x I ..... x n. As in t h e l a s t p a r a g r a p h , j is bijective iff
t h e m a t r i x A = (o(xt))o, l is invertible. As in t h e field case, this is again e q u i v a l e n t
t o t h e bijectivity o f h.
(ii') ~ (iv): J u s t s u p p o s e o * e a ( = id), a n d o ( y ) - y ~ M f o r all y ¢ S .
~ t x t ( y t - o ( y t ) ) e M, c o n t r a d i c t i o n .
(iv) ~

Then 1 =

(ii'): W e f i r s t c o n s t r u c t a s o l u t i o n o f t h e f o r m u l a in (ii') f o r a single o * e~

By (iv), t h e ideal o f S g e n e r a t e d by all y - o ( y ) is c o n t a i n e d in no m a x i m a l ideal,
h e n c e is equal t o S. O n e finds h e n c e n o e IN and x l ( ° ) ,

x(O) y(O) .... y r ~ ) e S
•--, n O ,
with ~.,x,(o).(y,(O)-o{yl(°)))----I. N o w one lets x 0 = ~,:~x,(O).o(y {°)) and Y0 -- -I.

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W e t h e n g e t ( s u m m a t i o n f r o m 0 t o no): ~ , x t ( ° ) ' y t ( ° ) = 1 a n d ~ x f ( ° ) . c ( y t ( ° ) ) = O.
Now one shuffles together these solutions for individual o to a solution for all o
as f o l l o w s : L e t I b e t h e i n d e x s e t 1-[ceG\e{0 ..... no}; f o r e a c h i E I, l e t x i b e t h e
(0)
p r o d u c t o f a l l x | ( o ) w i t h o # e, a n d Yi s i m i l a r l y . O n e c a n t h e n c h e c k t h a t i n d e e d
f o r a l l c e G: ~ i x l . c ( y

l) is e q u a l t o ~c,e' q.e.d.

In o u r o p i n i o n , it is i n s t r u c t i v e t o u s e t h e t h e o r y o f f a i t h f u l l y f l a t d e s c e n t a l r e a d y a t t h i s e a r l y s t a g e o f G a l o i s t h e o r y o f r i n g s . To t h i s e n d , r e c a l l t h a t a n
R - m o d u l e M is f a i t h f u l l y flat if M is f l a t , a n d M / P M

:6 0 f o r e a c h m a x i m a l i d e a l

P o f R. I t is a n o t h e r c h a r a c t e r i z a t i o n o f f a i t h f u l f l a t n e s s t h a t t h e f u n c t o r M ® a preserves and detects short exact sequences of R-modules. One has the following
easy results:
P r o p o a t t l o n 1 3 . [ K n u s - O j a n g u r e n (1974), B o u r b a k i A l g . c o m m . I §3] Let M be a f a i t h -

f u l l y flat R - m o d u l e , and ~p: A -~ B a h o m o m o r p h i s m o f R - m o d u l e s . Then q~ is an isomorphism i f f M ® Rq~: M ® a A ~ M ® R B is an isomorphism. The statement remains correct, i f the word "isomorphism" is replaced by " m o n o m o r p h i s m " , or by "epimorphism".

This s i m p l e r e s u l t a l r e a d y h a s a p p l i c a t i o n s . S u p p o s e T is a n R - a l g e b r a w h i c h
is a f a i t h f u l l y f l a t R - m o d u l e , a n d s u p p o s e S is a r i n g e x t e n s i o n o f R s u c h t h a t t h e
f i n i t e g r o u p G a c t s o n S by R - a u t o m o r p h i s m s .
P r o p o a l t J o n L 8 . Under these hypotheses, S / R

One can then state

is a G-Galois extension i f T ® R S is a

G-Galois extension over T.
Proof. W e m a y c o n s i d e r T = T® s R a s a s u b a l g e b r a o f T® R S, s i n c e T is f l a t . W e
use the defining property of "G-Galois" and check that the map

hr: ( T o R S ) ® r ( T o ~ S ) - - ~

(ToRS)C~)

a s s o c i a t e d in Def. 1.5. w i t h t h e e x t e n s i o n T c T ® R S is, u p t o c a n o n i c a l i s o m o r p h i s m , j u s t T ® h ( w h e r e h: S ® a S - - ~

S ~a~ is t h e m a p o f Def. 1.5. f o r t h e e x t e n s i o n

R c S). By 1.7, if T ® h is an i s o m o r p h i s m , t h e n s o is h. W e s t i l l have t o s h o w t h a t

S a = R, i.e, t h e c a n o n i c a l m a p ~: R ~ S a is o n t o . But it f o l l o w s f r o m t h e f l a t n e s s
o f T t h a t (T® n S) a - T® R S a, h e n c e T®t is o n t o . By 1.7, w e a r e d o n e .
The c o n v e r s e o f 1.8 is " m o r e t h a n t r u e " , in t h e s e n s e t h a t b a s e c h a n g e a l w a y s
preserves G-Galois extensions (not only faithfully flat base change). We will see
this a little later.

Lamina 1.9, A n y G-Galois extension S / R is f a i t h f u l l y flat over R.

Proof. F l a t n e s s is c l e a r s i n c e S / R

is p r o j e c t i v e by T h m . 1.6 (iii). Pick a m a x i m a l

ideal P o f R; w e n e e d S / P S ~: O. By N a k a y a m a , it s u f f i c e s t o s e e S p ~ : 0 . B u t R c S,
and localization preserves monomorphisms,

so we are done.

This l e m m a s u g g e s t t o t r y o u t S in t h e r o l e o f T; t h e r e s u l t

is s t r i k i n g l y

simple, but we first need to define morphisms of G-Galois extensions:

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§1

s

Daflm~t~

If S a n d S ' are t w o G - G a l o i s e x t e n s i o n s , t h e n a m o r p h i s m ~p: S -, S' is

a G - e q u i v a r i a n t R - a l g e b r a h o m o m o r p h i s m from S to S'. (G-equivariance means of
c o u r s e : q0(ox) = c g ( x ) for all o e G, x e S.) The G - G a l o i s e x t e n s i o n S / R

is c a l l e d

trivial, if it is i s o m o r p h i c t o t h e t r i v i a l e x t e n s i o n R ( a ) / R .
l ~ m m ' i r . I t is o b v i o u s t h a t w e o b t a i n a c a t e g o r y GAL(R, G) o f G - G a l o i s e x t e n s i o n s
o f a g i v e n r i n g R.
N o w w e c a n see t h a t " b a s e - e x t e n d i n g a n y G a l o i s e x t e n s i o n w i t h i t s e l f gives a
t r i v i a l e x t e n s i o n " . M o r e p r e c i s e l y : Let S / R b e a G - G a l o i s e x t e n s i o n , l e t T = S, a n d
c o n s i d e r t h e r i n g e x t e n s i o n T® a S / T . Since T = S, it is n o w e a s y t o c h e c k t h a t t h e
i s o m o r p h i s m h: S ® a S

~

S (c) gives a n i s o m o r p h i s m o f G - G a l o i s e x t e n s i o n s h:

T o R S-~ T (a). Recall t h a t G o p e r a t e s n a t u r a l l y o n t h e s e c o n d f a c t o r S, a n d by i n d e x

s h i f t o n S (c). W e n o w c a n p r o v e a r e s u l t o n t h e t r a c e :
l ~ l n l l ~ L|O. Let S / R be G-Galois, and tr: S ----* R the trace ( s e e p r o o f (ii') ~

(iii) o f

1.6}. Then:

a} tr: S --> R is s u r j e c t i v e
b} The R - s u b m o d u l e R o f S is a direct s u m m a n d o f S.
P r o o f . a} By t h e p r e v i o u s r e m a r k s , S ® S / S o R

(= S} is i s o m o r p h i c t o t h e t r i v i a l e x -

t e n s i o n o f S. O n e has a c o m m u t a t i v e d i a g r a m
S o R S

"

•

Sotr [
SoaR

S (G)

l trs
"

~

S

w h e r e tr s is t h e t r a c e a s s o c i a t e d t o t h e e x t e n s i o n Sin S ca~, a n d o n e s e e s f r o m t h e way G a c t s o n S d i a g ( x ) for all x e S. H e n c e tr s is o n t o ; by 1.7, tr is o n t o .
b} Pick c e S w i t h tr(c) = 1, a n d let f : S -) R b e d e f i n e d by f ( x ) = t r { c x l . T h e n
f is a n R - l i n e a r s e c t i o n o f t h e i n c l u s i o n R c S, s o R is a d i r e c t s u r n m a n d o f S.
N o w we c a n s h o w :

l ~ m m a 1.1L Let S / R b e G-Galois, and T any R - a l g e b r a . T h e n T o a S / S

is again a

G-Galois e x t e n s i o n .
P r o o f . W r i t e S r f o r T o R S.We w a n t t h r e e t h i n g s : T e m b e d s in S r, Sr a = T, a n d hr:

S r o r S r ---- Sr(a~ is a n i s o m o r p h i s m . The l a s t c o n d i t i o n is t h e e a s i e s t t o see,
s i n c e we k n o w a l r e a d y t h a t h r is {up t o c a n o n i c a l i s o m o r p h i s m ) j u s t Toh, a n d h is
a n i s o m o r p h i s m by h y p o t h e s i s . Since R s p l i t s o f in S, t h e m a p T -* S r a l s o s p l i t s ,
in p a r t i c u l a r T is a s u b r i n g o f S r. To see t h e s e c o n d c o n d i t i o n , we a r g u e as in

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Chase-Harrison-Rosenberg

(1965): Pick c e S w i t h t r ( c ) = 1 ( L e m m a 1.10). a n d l e t

y e S r b e f i x e d u n d e r G.

Then

~o(T®o)((l®c)'y)

=

y

=

(T®tr)((l®c)'y)

(T®tr)(l®c)'y
~

Im(T®tr)

~o(l®o(c))'y
T ® R ---- T, q.e.d.

=
----

=

As another example of this descent technique, we show the following important fact:
~'ol)oBlUon LI2. Let S/R

and S ' / R

b e G - G a l o i s . T h e n e v e r y m o r p h i s m ~p: S ~ S' o f

G - G a l o i s e x t e n s i o n s is an i s o m o r p h i s m .
P r o o f . T h e r e e x i s t s a f a i t h f u l l y f l a t R - a l g e b r a T s u c h t h a t b o t h S r (= T® R S) a n d

S ' r a r e t r i v i a l G - e x t e n s i o n s o f T. ( T r i v i a l G - e x t e n s i o n s a r e o b v i o u s l y p r e s e r v e d b y
arbitrary base change. Hence one can for example take T = S® R S', since base extension with S (resp. S') trivializes S/R

(resp. S'/R).)

I t is o b v i o u s t h a t T®~o is a

m o r p h i s m f r o m S r t o S ' r. W e m a y n o w s u p p o s e , b y v i r t u e o f 1.7, t h a t T = R ( f r e s h
n o t a t i o n ) , S = S ' = R (el. M o r e o v e r it is h a r m l e s s t o s u p p o s e R l o c a l . L e t n o w e a
R c¢) b e t h e e l e m e n t w i t h 1 in p o s i t i o n a a n d 0 e l s e w h e r e (a e G). T h e s e e a, a e G,
a r e a c o m p l e t e s e t o f i r r e d u c i b l e i d e m p o t e n t s o f R (a~, a n d t h e y a r e p e r m u t e d b y G
in an o b v i o u s f a s h i o n . In p a r t i c u l a r , G p e r m u t e s t h e e a t r a n s i t i v e l y . G e t t i n g b a c k
t o o u r m o r p h i s m ~, w e n o w s e e t h a t t h e ~ ( e °) a r e p a i r w i s e o r t h o g o n a l i d e m p o t e n t s
w i t h s u m 1. I f a n y o f t h e m is z e r o , t h e n a l l a r e z e r o s i n c e ~0 is G - e q u i v a r i a n t , s o
no ~ ( e o) is z e r o . T h e r e f o r e ~ m u s t s i m p l y p e r m u t e t h e e a, w h i c h i m p l i e s i m m e d i a t e l y t h a t ~0 is an i s o m o r p h i s m .

§ 2 The main theorem o f Galols theory

W e fix a f i n i t e g r o u p G a n d a G - G a l o i s e x t e n s i o n S / R
Can one find a bijection between subgroups

of (commutative) rings.

H c G and R-subalgebras

U c S?

C e r t a i n l y t h i s p r o b l e m is n o t w e l l p o s e d if w e a d m i t all s u b a l g e b r a s . ( A l r e a d y f o r
R = Z a n d I G I - - 2, t h e t r i v i a l G - e x t e n s i o n S - - Z × Z h a s i n f i n i t e l y m a n y s u b a l g e b r a s . ) T h e c o r r e c t c o n d i t i o n t o i m p o s e o n s u b a t g e b r a s is s e p a r a b i l i t y , an i m p o r t a n t
c o n c e p t in i t s e l f . O n e m a y f o u n d t h e w h o l e t h e o r y o n t h i s c o n c e p t , w h i c h w e
avoided for the sake of simplicity; we shall use separable algebras practically only
in C h a p t e r 0, a n d a s l i t t l e a s p o s s i b l e . Let us j u s t r e c a l l t h e d e f i n i t i o n a n d r e f e r
the interested reader to DeMeyer-Ingraham

(1971). W e r e m i n d t h e r e a d e r t h a t a l l

rings are supposed commutative.

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7

I ~ t l u L U o n A n R - a l g e b r a S is c a l l e d separable if S is p r o j e c t i v e a s a m o d u l e o v e r
S ® R S ( t h e s t r u c t u r e is ( s ® t ) y = syt f o r y e S, s đ t ã S ® S ) .

I f one a d m i t s n o n -

c o m m u t a t i v e a l g e b r a s S, o n e h a s t o t a k e S® S °pp in t h e p l a c e o f S® S.
EmLmple. I f R c S is a f i e l d e x t e n s i o n o f f i n i t e d e g r e e , t h e n S is a s e p a r a b l e R - a l gebra iff the extension S/R

is s e p a r a b l e in t h e u s u a l s e n s e .

G a t o i s e x t e n s i o n s a r e a l w a y s s e p a r a b l e ; m o r e p r e c i s e l y , t h e r e is t h e f o l l o w i n g
e x t e n s i o n t o T h e o r e m 1.6:
21

Let S / R

b e an e x t e n s i o n o f rings, G a f i n i t e s u b g r o u p o f A u t ( S / R )

such that S ~ = R. T h e n the f o l l o w i n g are equivalent:
(i) S / R

is G-Galois

(ii) S is separable o v e r R, and f o r each n o n z e r o idempotent e • S and any c,
• G with c * z, there e x i s t s y • S with e ' o ( y ) * e . z ( y ) . ( N o t e that the last c o n dition is vacuously true i f S has no idempotents beside 0 and 1.)
Proof. See C h a s e - H a r r i s o n - R o s e n b e r g

(1965), Thin. 1.3. T h e l a s t c o n d i t i o n in (ii) is

a b b r e v i a t e d t o "if o * z, t h e n c a n d ~ a r e strongly distinct" in l o c . c i t .
To k e e p m a t t e r s s i m p l e , l e t us a s s u m e f r o m n o w o n t h a t S is c o n n e c t e d , i.e. S
h a s no i d e m p o t e n t s b e s i d e s 0 a n d 1. The f i r s t p a r t o f t h e M a i n T h e o r e m r u n s a s
follows:
T h e o r e m 9_9_. [ C h a s e - H a r r i s o n - R o s e n b e r g

(1965)] Let S / R

be a G-Galois e x t e n s i o n ,

H c G a s u b g r o u p , and let U = S n be the s u b a l g e b r a o f H - i n v a r i a n t e l e m e n t s . Then:
(i) U is separable over R
(ill S is, in the canonical way, an H - G a l o i s e x t e n s i o n o f U
(iiil H is the group o f all c • G which leave U pointwise f i x e d
(iv) I f H is a normal s u b g r o u p o f G, then U is, in the canonical way, a G / H Galois e x t e n s i o n o f R.
P r o o f . W e i n c l u d e m o s t o f t h e p r o o f , in o r d e r t o give t h e r e a d e r a b e t t e r f e e l i n g
f o r t h e t h e o r y . O u r a r g u m e n t is m a i n l y t h e o r i g i n a l o n e ( l o c . c i t . ) ; t h e c h a n g e s r e flect personal tastes and do not claim to be simplifications. Parts of the proof can
be understood without any knowledge about separable algebras.
(ii): C h o o s e x 1..... xn" Yl ..... Y~ • S a s in (ii') ( p r o o f o f 1.6). T h e n , a f o r t i o r i ,

~lxic(yl)

= ~o,id f o r a l l c • H. The f o r m u l a S n = U h o l d s b y d e f i n i t i o n . H e n c e S / U

is

H - G a l o i s b y Thm. 1.6.
(i): By (ii) a n d T h m . 1.6 (iii), S is p r o j e c t i v e o v e r U, h e n c e S® R S is p r o j e c t i v e o v e r
U® a U. R e c a l l i n g t h e d e f i n i t i o n o f s e p a r a b l e a l g e b r a s , w e s e e f r o m T h m . 2.1 t h a t
S is p r o j e c t i v e o v e r S® R S. H e n c e , b y a n e a s y a r g u m e n t , S is p r o j e c t i v e o v e r U® R U.
B u t U is a d i r e c t s u m m a n d o f S ( a s a U - m o d u l e , a n d h e n c e a s a U® U - m o d u l e ) ,
b y (ii) a n d L e m m a 1.10 c). H e n c e U is p r o j e c t i v e o v e r U® R U, q.e.d.

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(iii): W e r e p r o d u c e t h e d i r e c t a r g u m e n t o f Chase, H a r r i s o n , and R o s e n b e r g . Let
H ' = { o e G l a f i x e s U pointwise}. Then H c H ' and S n ' = S H = U. A p p l y i n g (ii)
a n d t h e d e f i n i t i o n o f G a l o i s e x t e n s i o n t o U a n d b o t h o f H, H ' , we o b t a i n t h a t

SeaS

is s i m u l t a n e o u s l y i s o m o r p h i c t o S ~n; a n d t o S Cn'~, which f o r c e s till = I H ' h

a n d hence H = H ' .
(iv): See loc.cit, p.23. A n o t h e r a p p r o a c h : Reduce by f a i t h f u l l y f l a t d e s c e n t t o t h e
c a s e S = R (~) and check d i r e c t l y t h a t S H is c a n o n i c a l l y i s o m o r p h i c t o S (G/n). By
t h e way: It is n o t d i f f i c u l t t o p r o v e a l s o (ii) by t h i s m e t h o d .
The c o n v e r s e o f t h i s t h e o r e m r e a d s as f o l l o w s f o r c o n n e c t e d g r o u n d r i n g s R.

W a r n i n g : f o r n o n c o n n e c t e d R t h e s t a t e m e n t is m o r e involved, see Chase, H a r r i s o n ,
and R o s e n b e r g (1965).

2.3. Let R, S, and G be as in 2.2; let U c S be a separable R-subalgebra.
Then there is a subgroup H o f G with U = S It, and H is o f necessity the group o f all
ae G f i x i n g U pointwise.
F o r t h e proof, we r e f e r t o loc.cit. (The t h e o r y o f s e p a r a b i l i t y is u s e d in an e s s e n tial way.)

03 Punctmqal~y, and the Harrison

product

In t h i s s e c t i o n we s u m m a r i z e t h e p a p e r of H a r r i s o n (1965). Several p r o o f s are
omitted.
W e have a l r e a d y s e e n in §l t h a t any h o m o m o r p h i s m f : R -~ T o f c o m m u t a t i v e
r i n g s i n d u c e s a f u n c t o r "base e x t e n s i o n " f r o m t h e c a t e g o r y GAL(R,G) o f G - G a l o i s
e x t e n s i o n s o f R t o t h e c a t e g o r y GAL(S,G). W e now c o n s i d e r t h e s e c o n d a r g u m e n t
w i t h t h e aim o f e s t a b l i s h i n g f u n c t o r i a l i t y in G, t o o . F o r m o t i v a t i o n , c o n s i d e r a f i n i t e
g r o u p G and a f a c t o r g r o u p G/N. Then in t h e c l a s s i c a l c a s e t h e r e is j u s t one way
t o a s s o c i a t e a G / N - G a l o i s e x t e n s i o n w i t h a given G - G a l o i s e x t e n s i o n L / K : j u s t
t a k e L N / K . This w o r k s f o r r i n g s j u s t as well, by Thm. 2.2. It is i m p o r t a n t , however,
t o a l l o w g e n e r a l g r o u p h o m o m o r p h i s m s n: G -* H. Before giving t h e c o n s t r u c t i o n ,
l e t us b r i e f l y m e n t i o n t h e c a s e w h e r e n is t h e i n c l u s i o n o f G in H. This c a s e has
no c o u n t e r p a r t in c l a s s i c a l G a l o i s t h e o r y ; it will t u r n o u t t h a t in t h i s c a s e t h e
m a p ~*: GAL(R,G) -~ GAL(R,H) is given by a s o r t o f i n d u c t i o n p r o c e s s , as in r e p r e s e n t a t i o n t h e o r y , a n d even if S / R is a G - G a l o i s f i e l d e x t e n s i o n , 7t*(S/R) is never

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a field unless G ----H. E x t r e m e example: G ----e, and S is the(!) G - G a l o i s e x t e n s i o n
R o f R. Then ~ R will t u r n out t o be the trivial H - e x t e n s i o n o f R. Now we p r e s e n t
the general result.

Tlmorem 3_1- a) Let R be a commutative ring R, ~: G ~ H a homomorphism o f finite
groups. Then there is a canonical functor ~*: GAL(R,G) -~ GAL(R,H). I f ~ happens
to be a canonical surjection G -~ G / N , then ~*(S) = S ~ as in the above discussion.
(For the construction o f ~ ~, see the proof o f this theorem.)
b) The prescription "~ ~ ~ " preserves composition up to canonical isomorphism.
In other words: I f we let H(R,G) be the set o f isomorphism classes o f G-Galois extensions o f R, then H(R,G) is again functorial in G, and the prescription "~ ~-~ H(R,~)"
now preserves composition.
I~lamltAo~ The s e t H(R,G) j u s t defined is also called the Harrison set o f R and G.

Proof o f Thm. 3.1. We do a) and b) s i m u l t a n e o u s l y . First we define ~*. Let S
GAL(R,G). We s e t
~*S = M a p ~ ( H , S ) ( = {x: H ~ S[VgeG, h~H: x ( ~ ( g ) h ) = g ( x ( h ) ) } . )
The H - a c t i o n on ~*S is given by (h'~x)(h) = x(h.h') f o r x ~ M a p ~ ( H , S ) , h , h ' ¢ H .
The R - a l g e b r a s t r u c t u r e is defined " c o m p o n e n t - w i s e " , i.e. by the inclusion o f
M a p ~ ( H , S ) in M a p ( H , S ) = S ~s~. (It is immediate t h a t M a p ~ ( H , S ) is indeed a s u b algebra.)
One sees easily t h a t ~* is a f u n c t o r f r o m GAL(R,G) in the c a t e g o r y o f R - a l g e bras with action o f H. It remains to establish:
(i) If ~b: H -~ J is a n o t h e r g r o u p h o m o m o r p h i s m , t h e n we have a natural i s o m o r p h i s m ¢*(n*S) ,, (¢n)*S;

(ii) n * S / R is, with the given H - a c t i o n , indeed an H - G a l o i s extension.
We do (i) first, by exhibiting natural bijections
Map~(J, Map~(H,S))

~

M a p ~ ( J , S).

(It is l e f t t o the reader t o verify t h a t ~ and ~ are J - e q u i v a r i a n t R - a l g e b r a h o m o m o r p h i s m s . ) Let c t ( y ) = y ( - ) ( e s ) for y in t h e l e f t hand side, i.e. ct(y)(j) = y ( j ) ( e s )
for j ~ J . Let ~(z)(j)(h) = z(~b(h)j) for z in the r i g h t hand side, h~H, j e J .
We check ~ is w e l l - d e f i n e d , i.e. ~(y)e Map~b~(J,S): Let j ~ J , geG. W e calculate:

txty)(@r(g).j)
=

y(~Im(g).j)te~)

=

(~(g)*y(j))(e
n)

= y(j)(enn(g) )
= g(y(j)(en))

(since y ~ M a p ~ . . . )
(def. of H - a c t i o n on Mapn(H,S))

(since y ( j ) ~ M a p x . . . )

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=

g (cx{y)(j)), q.e.d.

B0t is t h e i d e n t i t y : L e t y e Mapcv(J, M a p n ( H . S ) ) .

jeJ,

h ~ H . T h e n ([3~(y))(j)(h) =

0t(y)(~(h)j) = y(d2(h)j)(e H ) = ( h * y ( j ) ) ( e H ) ( s i n c e y e Map~b...), a n d t h e l a s t e x p r e s s i o n
e q u a l s y ( j ) ( h ) , q.e.d.
0t~ is t h e

identity: Let z e Map~b~(J,S), jeJ.

Then

(0t~(zl)(j) -- ~ ( z } ( j ) ( e H) --

z(d/(e u )j) = z ( j ) , q.e.d. This c o m p l e t e s t h e p r o o f o f (i).
(ii): W e w i l l give o n e a r g u m e n t f o r t h e g e n e r a l c a s e , a n d a n o t h e r f o r t h e s p e c i a l c a s e t h a t H is a b e l i a n .
N o t e f i r s t t h a t ~* c o m m u t e s w i t h f a i t h f u l l y f l a t b a s e c h a n g e , i.e. f o r any f a i t h f u l l y f l a t R - a l g e b r a T a n d any S in G A L ( R , G ) , t h e r e is a c a n o n i c a l H - e q u i v a r i a n t
i s o m o r p h i s m z * ( T ® S) - T~:~*S. By f a i t h f u l l y f l a t d e s c e n t , it t h u s s u f f i c e s t o f i n d
such a T with ~*(T®S)

a n H - G a l o i s e x t e n s i o n o f T. T a k i n g T = S a n d c h a n g i n g

n o t a t i o n , w e a r e r e d u c e d t o p r o v i n g : ~* o f t h e

trivial G-extension

R (~) is a n

H - G a l o i s e x t e n s i o n o f R. Let t: {e} -~ G, t': {e} -* H be t h e o b v i o u s m a p s . O n e
c h e c k s q u i t e e a s i l y : t*R is t h e t r i v i a l G - e x t e n s i o n R ¢~). Since ~L = t', w e o b t a i n :
7t*(R I~)) -, 7t*t*R

" (t'l*R

( b y (i))

R (H),
a n d w e a l r e a d y k n o w t h a t t h i s is i n d e e d an H - G a l o i s e x t e n s i o n , q.e.d.
The f o l l o w i n g nice a r g u m e n t f o r H a b e l i a n is d u e t o H a r r i s o n . W e f a c t o r z as
= ~y, w i t h y =

(idG,eH): G--> G x H , a n d ~ - - ( ~ , i d x ) : G × H --> H. T h e n y is a s p l i t

m o n o m o r p h i s m , a n d ~ is o n t o . I t is s u f f i c i e n t t o s h o w (ii) f o r % a n d f o r g, t a k i n g
i n t o a c c o u n t (i). F o r ~ -- % o n e s e e s d i r e c t l y t h a t z * S ,, S® R R Cn). w i t h t h e o b v i o u s
a c t i o n o f G × H , a n d o n e c a n c h e c k t h a t t h i s is a G × H - G a l o i s

e x t e n s i o n . F o r ~ ---- 5,

i.e. ~ o n t o , o n e c a l c u l a t e s f r o m t h e d e f i n i t i o n t h a t 5"S ---- S Ker(~), w h i c h is i n d e e d
a G a l o i s e x t e n s i o n w i t h g r o u p Im(6) by Thin. 2.2.
We now present Harrison's construction

w h i c h m a k e s t h e s e t H ( R , G ) i n t o an

a b e l i a n g r o u p if G is a f i n i t e abelian g r o u p . This w i l l t h e n b e c a l l e d t h e H a r r i s o n
g r o u p o f R a n d G. ( R e c a l l t h a t H ( R , G ) = GAL(R,G)/~,,). W e u s e w i t h o u t f u r t h e r
c o m m e n t t h e f o l l o w i n g e a s y f a c t : I f S, T ~ G A L ( R , G ) , t h e n S® R T w i t h t h e n a t u r a l
a c t i o n o f G x G , is a G x G - G a l o i s e x t e n s i o n o f R. L e t G b e f i n i t e a b e l i a n , t: {e} -~ G
b e t h e i n c l u s i o n o f t h e t r i v i a l g r o u p in G, ~: G×G ~ G t h e m u l t i p l i c a t i o n Ca h o m o m o r p h i s m ! ) , a n d j : G -> G t h e m a p g ~
I~flm~o~.

g-I (again, a homomorphism).

The H a r r i s o n p r o d u c t S . T o f S, T c GAL{R,G) is d e f i n e d t o b e

S'T

=

t~*(S®RT} e G A L ( R , G ) .

By f u n c t o r l a l i t y , t h e H a r r i s o n p r o d u c t [S.T] o f t w o i s o m o r p h i s m c l a s s e s [ S ] , [ T ]
H ( R , G ) is a w e l l - d e f i n e d e l e m e n t o f H ( R , G ) . W e s h a l l o f t e n a b u s e n o t a t i o n a n d

write S e H(R,G) etc.

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39.
[ H a r r i s o n (1965)] a) With this definition, H(R,G) becomes an abelian
group whose neutral element is (the class of) the trivial extension R(C)/R.

b) I f x: G ~ H is a homomorphism f r o m G to another abelian group H, then
n*: H(R,G) ---, H ( R , H ) is a group homomorphism.

Proof. a) This is a r a t h e r f o r m a l a r g u m e n t e x p l o i t i n g t h e f u n c t o r i a l i t y p r o p e r t i e s .
Let us b e g i n b y s h o w i n g a s s o c i a t i v i t y o f t h e H a r r i s o n p r o d u c t . Let S, T, U e H(R,G).
Then:

(S'T)'U

=

~t*(~t*(S®,T) o R U)

=

tl*((tl*®id~*)(S®To U)

=

(lz(vxid))*(SoToU)

(3.1. b))

=

( v ( i d × p ) ) * ( S o T o U)

(this is j u s t t h e a s s o c i a t i v i t y o f G)

=

S'(T.U)

(same calculation backwards).

In t h e s a m e m a n n e r , one p r o v e s c o m m u t a t i v i t y : S.T = p*(S® T) = (~tx)*(S® T) ( w h e r e
x: G × G -~ G x G is t h e i n t e r c h a n g e i s o m o r p h i s m ; ~ = t~x since G is c o m m u t a t i v e ) .
N o w (VT)*(S®T) - p*x*(S®T), a n d x * ( S o T )

- T ® S by a d i r e c t a r g u m e n t . This

finally gives S.T =, T.S.
To see t h a t E = fl Galois e x t e n s i o n o f R w i t h g r o u p {e}. F o r any S ~ H(R,G), w e t h e n get:

S'E

=

tl*(Soat*R)

-

tl*(idGxt)*(So~R)

=, i d G * ( S ® a R )

(since tl(idaxt) = idG)

-

(direct argument).

S

Finally, we use j t o c o n s t r u c t an inverse o f S e H(R,G). W e need a s m a l l auxiliary f o r m u l a , t o wit: S ® S (ms a G × G - G a l o i s e x t e n s i o n ) is i s o m o r p h i c t o A ' S ,
w h e r e A: G -, G × G is t h e d i a g o n a l e m b e d d i n g . E P r o o f o f t h e l a t t e r f o r m u l a : T h e r e
is an i s o m o r p h i s m cz: Map(G, S)
, MapA(GxG, S) w h i c h m a p s (f: G -- S) t o
((g,h) v---, g ( f ( g - t h ) ) ). F u r t h e r m o r e , MapA(GxG, S) is j u s t A*S. On t h e o t h e r hand,
t h e r e is t h e c a n o n i c a l i s o m o r p h i s m h: S® S ~

Map(G,S) f r o m t h e d e f i n i t i o n o f

G - G a l o l s e x t e n s i o n . O n e t h e n c h e c k s t h a t t h e c o m p o s i t e ~h is G x G - e q u i v a r i a n t . ]
N o w we c l a i m t h a t j * S gives an inverse t o S in t h e H a r r i s o n group. W e c a l c u l a t e :

S'o*S

=

t.t*(Soo*S)

= ~t*((ida*oj*)(SoS))
=, ~t*(idGxj )*A*(S )

"

(t~(ldxj)A)*(S)

-

(t~)*(S)

=

t*(E*(S))

-

t*(S ~) =

w i t h E: G -~ {e} t h e o b v i o u s m a p

t*R = R
The p r o o f o f b) is quite similar, and h e n c e o m i t t e d .

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By Thm. 3.1 a n d 3.2, w e have, f o r a n y c o m m u t a t i v e r i n g , a f u n c t o r H(R,--) f r o m t h e

c a t e g o r y o f f i n i t e a b e l i a n g r o u p s t o t h e c a t e g o r y o f a b e l i a n g r o u p s . The m o s t i m p o r t a n t g e n e r a l p r o p e r t y o f t h i s f u n c t o r is t h e f o l l o w i n g :
~ m
3.3. The f u n c t o r H(R,--) is left exact, i.e.: 0 --> J ~ G ~ G' --> 0 is a short
exact sequence o f f i n i t e abelian groups, then the induced sequence
0

~

H(R,J)

~

H(R,G)

-~

H(R,G')

is exact.

Remark. H ( R , - ) is u s u a l l y n o t r i g h t e x a c t ; o n e c a n e s t a b l i s h a r e l a t i o n b e t w e e n t h e
obstruction to right-exactness

and certain Brauer groups. We shall not pursue this.

Proof o f 3.3. Since ~bi is t h e z e r o m a p , (kbi)* h a s a l s o t o b e t h e z e r o m a p . H e n c e
~*i* = 0. F o r t h e o t h e r p a r t s o f t h e p r o o f , w e n e e d a l e m m a .
L e m m n 3.4. S e G A L ( R , G ) is trivial i f f there exists an R-algebra homomorphism

S ~ R. (Such an e is called an a u g m e n t a t i o n . )

Proof o f L e m m a 3.4. The t r i v i a l e x t e n s i o n R ~ ) o b v i o u s l y h a s a u g m e n t a t i o n s , n a m e l y ,
t h e c a n o n i c a l p r o j e c t i o n s t o R. S u p p o s e on t h e o t h e r h a n d t h a t E: S -~ R is an
augmentation.

Define an R - a l g e b r a

homomorphism(!)

@: S -* R t~l b y 9(Y) =

(e(oy))ae G. I t is e a s y t o c h e c k t h a t ~0 is a l s o G - e q u i v a r i a n t , h e n c e an i s o m o r p h i s m
by 1.12.
W e c o n t i n u e in t h e p r o o f o f Thm. 3.3. S u p p o s e S e H ( R , G ) w i t h i*S t r i v i a l .
Then, by t h e l e m m a , i*S h a s a n a u g m e n t a t i o n E i n t o R. R e c a l l t h a t i*S--- MaPi(G, S).
We can construct

ct: S ~ i*S b y l e t t i n g , f o r y • S,

an R-algebra homomorphism

a n d ~ e G: u(y)(o) ---- c(y} if o e I m ( i ) , a n d ~ ( y ) ---- 0 o t h e r w i s e . H e n c e S i t s e l f h a s
an a u g m e n t a t i o n e~, a n d h e n c e is t r i v i a l , b y t h e l e m m a .
To c o n c l u d e t h e p r o o f , w e s u p p o s e T e H ( R , G ) s u c h t h a t kb*T ( - T Ker(~b)) is
t r i v i a l , i.e. ~*T ,, R ~6'~. R e c a l l R (~'~ ** ~ c ' ~ G '

fG ''R' w h e r e {fG,} is t h e c a n o n i c a l

b a s i s m a d e u p o f i d e m p o t e n t s . I d e n t i f y T ker(kb) w i t h R ~') a n d c o n s i d e r t h e R - a l g e b r a A = T.fe, ( w h e r e e ' is t h e n e u t r a l e l e m e n t o f G'}. N o w o n e c h e c k s , b y r e d u c t i o n
t o t h e c a s e T --- R (~) ( f a i t h f u l l y f l a t d e s c e n t ) , t h a t A is c a n o n i c a l l y a J - G a l o i s
e x t e n s i o n o f R. ( N o t e J ,, K e r ( ~ ) . ) W e d e f i n e an R - a l g e b r a

T--, i~A b y

homomorphism

6:

~(y)(o) = o(y}.fe, (y e T, o e G).
To s e e t h a t t h i s is w e l l - d e f i n e d , o n e h a s t o v e r i f y t h a t ~(y) is i n d e e d in MaPi(G,A).
F o r ~ e J w e have ~(y)(i(~).o) = (i(x)o)(y).fe, = z ( 0 ( y ) - f e , ) ( s i n c e r e ' is f i x e d b y ~),
a n d t h e l a s t e x p r e s s i o n is x ( ~ ( y ) ( o ) ) . F u r t h e r m o r e , B is G - e q u i v a r i a n t . A g a i n b y 1.12,
is an i s o m o r p h i s m , a n d T is in t h e i m a g e o f i*, q.e.d.
This t h e o r e m h a s an i m p o r t a n t c o n s e q u e n c e :

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3.~. Given a commutative ring R, there exists a pro finite abelian group D R
such that the functor H(R,--): {finite abelian groups} ~
{abelian groups} is prorepresented by fiR' i.e. naturally isomorphic to Homcont(tqR,--).
Proof. Every left exact f u n c t o r f r o m finite abelian g r o u p s t o abelian g r o u p s is
p r o - r e p r e s e n t a b l e . For details, see Harrison (1965), p r o o f o f Thm. 4.
i l a m ~ l m , a) The p r o o f gives at the same time the uniqueness o f fl R. O f t e n tqR is
called the abelian fundamental group o f R, or the abelianized absolute Galois group
o f R. Explicitly one has fl R = p r o j . l i m ( H ( R , Z / n Z ) V ) , where the g r o u p s Z / n Z form
an inductive s y s t e m indexed by the divisibility lattice o f ~q, and v means Pontryagin
dual. See Harrison, loc.ctt.

b) By the m e t h o d s o f the p r o o f o f 3.2, one can see w i t h o u t difficulty: if [k]:
G -, G is the h o m o m o r p h i s m g e---, gk (G finite abelian), then [k]*: H{R,G) -,
H(R,G) is multiplication by k in the Harrison group. In particular, H(R,G) is annihilated by the e x p o n e n t o f G, hence torsion. Therefore the Pontryagin dual in a) is
indeed profinite.
We w a n t t o show t h a t the g r o u p D R has in many cases an interpretation as a
(profinite) g r o u p o f a u t o m o r p h i s m s o f an appropriate (infinite) extension o f R.
If R - - K is a field, the required g r o u p is A u t ( K a b / K ) , where K ab is the maximal
abelian Galois extension o f K. More generally, Janusz (1965) has proved the existence o f a separable closure for every c o n n e c t e d ring R. This is by definition a c o n nected R - a l g e b r a R sep which is a filtered union o f G-Galois e x t e n s i o n s o f R
(G varies, o f course), such t h a t every c o n n e c t e d Galois e x t e n s i o n S / R is e m b e d dable in R sep. There is no ambiguity here as to what the Galois group o f S / R is,
because o f the following r e s u l t (see C h a s e - H a r r i s o n - R o s e n b e r g (1955), Cot. 3.3
or this chapter, 7.3}: If S / R

is a G-Galois e x t e n s i o n o f c o n n e c t e d rings, then

A u t ( S / R ) = G. As proved by Janusz (1966), the g r o u p ~/R = Aut(RS~P/R) is a filtered
projective limit o f finite g r o u p s {more precisely: o f Galois g r o u p s o f Galois e x t e n sions contained in RssP), hence T R is profinite. (For a n o t h e r exposition o f this
material, see also D e M e y e r - l n g r a h a m (1971).)
Our next objective is the following result.

3.6. Let R be connected, and T R = Aut(RSeP/R) as in the preceding discussion. Then there is a natural isomorphism
~: Hom.0nt(5*R,--}

-~ H(R,--)

o f functors f r o m finite abelian groups to abelian groups.
We need a few preparations for the proof.

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L e m m a 3.7. Let R be connected, S/R, T/R be two Galois extensions (with maybe different g r o u p s ) , and f: S - - ~ T an R - a l g e b r a h o m o m o r p h i s m . T h e n Ker(f) is g e n e r a ted by an idempotent.
P r o o f . I t is w e l l k n o w n t h a t an i d e a l I o f S is g e n e r a t e d b y a n i d e m p o t e n t i f f l is
f i n i t e l y g e n e r a t e d a n d I z = I. [ O n e d i r e c t i o n is c l e a r . I f I is f i n i t e l y g e n e r a t e d a n d
I.I = I, t h e n b y B o u r b a k i , A l g . c o m m . , II §2 n o 2 c o r . 3, t h e r e e x i s t s e e I s u c h t h a t
( 1 - e ) l = O. I t f o l l o w s e a s i l y t h a t e z = e a n d I = e S . ~ B o t h o f t h e s e p r o p e r t i e s o f
I may be tested by a faithfully flat base extensions,

hence we may, and shall,

a s s u m e t h a t b o t h S a n d T a r e t r i v i a l , i.e. a f i n i t e p r o d u c t o f c o p i e s o f R, c o n s i d e r e d j u s t a s R - a l g e b r a s . H e r e t h e d e s i r e d p r o p e r t y o f K e r ( f ) is e a s i l y o b t a i n e d b y
explicit calculations with canonical idempotents.
RemRrk. I n t h e p r o o f , w e o n l y u s e d t h a t S a n d T b e c o m e i s o m o r p h i c t o a p r o d u c t
of copies of the ground ring after suitable faithfully flat base change.

Fropomltion 3.8.

[ H a r r i s o n (1965)] Let R be c o n n e c t e d , S / R a G-Galois e x t e n s i o n with

f i n i t e abelian g r o u p G. Then there e x i s t s a s u b g r o u p H C G and a c o n n e c t e d H - G a l o i s
e x t e n s i o n U / R such that S - i*U (where i: H -* G is the inclusion).
P r o o f . Since R is c o n n e c t e d , e v e r y f i n i t e l y g e n e r a t e d p r o j e c t i v e R - m o d u l e

has a

w e l l - d e f i n e d r a n k (~ ~q). F r o m t h i s it f o l l o w s t h a t t h e R - a l g e b r a S c a n b e d e c o m posed as a product of finitely (at most rank(S)) algebras without proper idempot e n t s , i.e. t h e r e a r e i r r e d u c i b l e o r t h o g o n a l i d e m p o t e n t s

e 1..... e n in S w h o s e s u m

is 1, t h e s e t M o f i r r e d u c i b l e i d e m p o t e n t s e q u a l s {e I ..... en}, a n d S = ~):=ln e l S , w h e r e
all e~S are c o n n e c t e d .
G o p e r a t e s o n M. Since t h e s u m o v e r a n y G - o r b i t o n M is G - i n v a r i a n t , h e n c e
an i d e m p o t e n t o f R, t h e o p e r a t i o n o f G o n M m u s t b e t r a n s i t i v e . Pick e I e M, a n d
l e t H b e t h e S t a b i l i z e r o f e I in G. Since G is a b e l i a n , H is t h e s t a b i l i z e r o f e v e r y
e e M. L e t U = e l S . By t r a n s i t i v i t y o f G o n M , a l l e S (e ~ M ) are i s o m o r p h i c t o U
a s R - a l g e b r a s , s o S - U ~M> ( n o n c a n o n i c a l l y ) , U is c o n n e c t e d , a n d H a c t s o n U b y
R-automorphisms.

W e n o w c l a i m t h a t U / R is H - G a l o i s .

F i r s t o f a l l , H e m b e d s i n t o A u t ( U ) . ( I f 0 e l l is i d e n t i t y o n U = e t S , o n e s e e s ,
a g a i n s i n c e G is t r a n s i t i v e o n M, t h a t o is i d e n t i t y o n a l l e S (e ~ M ) , h e n c e o ---- id.)
Next, G/H operates transitively and without fixed points on M, whence [G:H] =
JMI. F r o m t h e d e f i n i t i o n o f G a l o i s e x t e n s i o n , o n e h a s r a n k a ( S ® S) = r a n k R ( S ) - [ G t ,
h e n c e r a n k R ( S ) = fGt. T h e r e f o r e r a n k a ( U ) = IHt ( u s e S • U~M)). T h u s w e k n o w
t h a t U [ H ] a n d E n d R ( U ) a r e f i n i t e l y g e n e r a t e d p r o j e c t i v e o f t h e s a m e r a n k o v e r R,
a n d it s u f f i c e s t o s h o w t h a t t h e m a p j : U [ H ] - ~

E n d R ( U ) (cf. Thin. 1.6) is onto.

U is e m b e d d e d in E n d ~ ( U ) via l e f t m u l t i p l i c a t i o n s . W i t h t h i s i d e n t i f i c a t i o n , o n e h a s
the following formulas

in E n d s ( U ) :

not in H. Let ~0 e E n d s ( U ) .

Then

el.o.e I -----el.o ----o.e I for o • H , eloe i = 0 for o
ei.~o.eI e Ends(S);

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by 1.6 (iii),there are s o • $


§3

t5

w i t h ~.oeGSa'O = el.~.e r M u l t i p l y i n g t h i s e q u a t i o n w i t h e I f r o m b o t h s i d e s , w e
o b t a i n b y t h e a b o v e r e m a r k s : ~'o•H (eisa)'°'et = el"~'er I f w e r e s t r i c t b o t h s i d e s t o
U, w e g e t t h e e q u a t i o n ~.o•H(eisG).o ---- q~ in E n d R ( U ) , h e n c e j is o n t o .
It r e m a i n s t o s h o w S = i*U. T h i s is d o n e b y t h e u s u a l m e t h o d : D e f i n e a m a p

~: S - - * i*U by [3(s)(o) = e i o ( s ) . E x a c t l y a s in t h e p r o o f o f 3.3 o n e s e e s t h a t ~ is a
G-equivariant R-algebra homomorphism,

h e n c e a n i s o m o r p h i s m by 1.12.

Q.E.D.

R e m a r k . I f R = K is a f i e l d , t h e n o n e c a n s h o w t h a t t h e c o n n e c t e d H - e x t e n s i o n

U

is a l s o a f i e l d . U is c a l l e d t h e " K e r n k S r p e r " ( c o r e f i e l d ) o f S / K in H a s s e (1949).

Proof o f 3.6. W e d e f i n e f o r e a c h f i n i t e a b e l i a n g r o u p G a m a p v~ : H o m c o ~ t ( T R , G )
- - - H ( R , G). R e c a l l R s~p is a s e p a r a b l e c l o s u r e o f R. F o r a n y c o n t i n u o u s h o m o m o r p h i s m f: Ta = A u t ( R ~ P / R )

v~(f)

•

G, w e d e f i n e

= f e~(E) • H ( R , G ) ,

w h e r e E is a n y a b e l i a n G a l o i s e x t e n s i o n o f R, c o n t a i n e d in R s~p, s u c h t h a t f f a c t o r s t h r o u g h a h o m o m o r p h i s m f ~ : A u t ( E / R ) --* G. ( T h e e x i s t e n c e o f s u c h an E
f o l l o w s , s i n c e f is c o n t i n u o u s , a n d G is a b e l i a n . ) O f c o u r s e w e m u s t s h o w t h a t t h i s
is i n d e p e n d e n t o f t h e c h o i c e o f E: if E ' is a n o t h e r , w. I. o. g. E c E ' c R ~ p , t h e n
b y t h e M a i n T h e o r e m (§2), t h e G a l o i s g r o u p o f E ' m a p s o n t o t h e G a l o i s g r o u p o f
E ( c a l l t h i s e p i m o r p h i s m ~ ) , a n d E is t h e f i x e d f i e l d o f K e r ( ~ ) in E ' . F r o m f E ' =

f ~ n a n d E = ~ * E ' w e g e t fE,*E' = fE*E.
N o w w e p r o v e t h a t va p r e s e r v e s s u m s . L e t us w r i t e G a d d i t i v e l y f o r t h i s . L e t
f, g e Hom o,t(TR,G), and let • denote the Harrison product. We obtain:
va(f) ' v~(g)

=

f E * E • gE*E

(E c R sev G a l o i s / R , l a r g e e n o u g h )

= ~ * ( f e * E ®~ g e * E )

=

(~*(feìgE)*A*)(E)

=

(fe+gE)*E"

(def. of ã )

(A*E =

Eđ E, cf. p r o o f o f 3,2)

t h e l a s t s t e p b e i n g j u s t i f i e d b e c a u s e ~ (fE xgE ) A c o i n c i d e s w i t h f E + gE"
The p r o o f o f t h e n a t u r a l i t y o f v¢ in G is s i m i l a r ( a n d e a s i e r ) , a n d w e o m i t it.
W e s h o w v¢ is i n j e c t i v e : S u p p o s e f * E is t h e t r i v i a l G - e x t e n s i o n

( w h e r e f is in

H o m ( A u t ( E / R ) , G ) , E / R a b e l i a n c o n n e c t e d G a l o i s e x t e n s i o n ) . W e c a n f a c t o r f in
t h e f o r m i~b, w i t h ~b s u r j e c t i v e a n d i i n j e c t i v e . By 3.3, a l r e a d y ~b*E is t r i v i a l , i.e.
E K e r ( ~ ) is a p r o d u c t o f c o p i e s o f R, a s an R - a l g e b r a , b u t a l s o c o n n e c t e d

(since

E is). T h e r e f o r e w e m u s t have E K e r ( ~ ) = R, a n d f o r r e a s o n s o f r a n k : C o k e r ( ~ ) is
t r i v i a l . H e n c e f is t h e z e r o h o m o m o r p h i s m .

v¢ is s u r j e c t i v e : S u p p o s e w e a r e g i v e n a G - G a l o i s e x t e n s i o n S / R . I f S is c o n n e c t e d , t h e n w e m a y a s s u m e S c R sep, a n d t h e r e is a n a t u r a l c o n t i n u o u s e p i m o r -

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p h i s m f : tit a ~ A u t ( S / R ) = G. Taking E = S, we get the identity for f ~ , hence
v a ( f ) = S. This was the e a s y case. The p r o b l e m arises when S is not c o n n e c t e d .
But then, t h a n k s t o 3.8, we find an inclusion i: H c G o f a s u b g r o u p H and a
c o n n e c t e d H - G a l o i s e x t e n s i o n U / R w i t h i*U - S. By n a t u r a l i t y o f v, it t h e n s u f fices to find a preimage o f U under u s , but this we can do since U is c o n n e c t e d .
Q.E.D.
C a r o i l m 7 3.9. F o r any c o n n e c t e d ( c o m m u t a t i v e )

ring R, the abelian f u n d a m e n t a l

g r o u p fl R is isomorphic to the abelianization Tn/[TR,TR].
P r o o f . By Thm. 3.6, the profinite g r o u p TR/[qrR,qr ~] p r o - r e p r e s e n t s the f u n c t o r

H(R,--). The g r o u p f i r p r o - r e p r e s e n t s this f u n c t o r by definition. It is w e l l - k n o w n
f r o m c a t e g o r y theory t h a t a p r o - r e p r e s e n t i n g object o f a given f u n c t o r is unique
up to i s o m o r p h i s m .
At the end o f this section, we give s o m e r e s u l t s which show how the f u n c t o rialities o f the t w o a r g u m e n t s in H(R,G) interact.
l~aoalUoa

3.IO. Let f : R ~ T b e a h o m o m o r p h i s m o f ( c o m m u t a t i v e ) rings, and n:

G ~ H a h o m o m o r p h i s m o f f i n i t e g r o u p s . T h e n the diagram

H(R,G)

,

To-- I
H(T,G)

H(R,H)
t

,

T®--

H(T,H)

is c o m m u t a t i v e .
P r o o f . If S is f l a t over R, the p r o p o s i t i o n f o l l o w s directly f r o m the definitions.
The reader will p r o b a b l y n o t e t h a t we used the r e s u l t (for S / R faithfully flat) in
the p r o o f o f Thm. 3.1 already,

For the general case, one exhibits for S c H(R,G) a canonical map
O~s/l~: T®Rrc*S

,

n*(T®RS);

one c h e c k s t h a t the definition o f ~ is c o m p a t i b l e with f a i t h f u l l y f l a t b a s e change,

which r e d u c e s the p r o o f to the case where S is a trivial G - e x t e n s i o n . This case can
easily be done directly.
C o r o l l a r y 3.11. I f R, S, G are as in 3.10, and G is abelian, then the m a p T ® R--:
H(R,G) ~ H ( S , G ) is a g r o u p hornomorphism.
P r o o f . The H a r r i s o n p r o d u c t was defined w i t h the help o f tl* (~: GìG --" G the

multiplication map.) By 3.10, the map Tđ R- c o m m u t e s with ~*. From this the claim
f o l l o w s easily.

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