Lecture notes in mathematics

C y clic Galols Extensions ° " of Commutative Rings Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest... ISBN 3-540-56350-4 Springer-Verlag Ber

Lecture Notes in Mathematics

C y clic Galols Extensions ° "

of Commutative Rings

Springer-Verlag

Berlin Heidelberg New York

London Paris Tokyo

Hong Kong Barcelona

Budapest

ISBN 3-540-56350-4 Springer-Verlag Berlin Heidelberg New York

ISBN 0-387-56350-4 Springer-Verlag New York Berlin Heidelberg

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O: Galots theory of commutative rinsa

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

§3 Kummer theory without the condition ,,p-I ~ R" 116

§4 The main result and Artin-Hasse exponentials 120

§6 Application: Generic Galois extensions 135

INTRODtI~I'ION

The s u b j e c t o f t h e s e n o t e s is a part o f c o m m u t a t i v e algebra, and is a l s o

c l o s e l y r e l a t e d t o c e r t a i n t o p i c s in algebraic n u m b e r t h e o r y and algebraic g e o m e t r y 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 t h e f o l l o w i n g : W h a t

is t h e c o r r e c t definition o f a Galois e x t e n s i o n ? 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 t h e field case}? And t h e m o s t f r u i t f u l 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 t h e r e any possibility o f describing a// Galois e x t e n s i o n s o f R with g r o u p G?

These q u e s t i o n s will be d e a l t with in c o n s i d e r a b l e generality In later c h a p t e r s ,

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 g e o m e t r i c a l s i t u a t i o n s , which m e a n s t h a t we c o n s i d e r m o r e special c o m m u t a t i v e rings: rings o f i n t e g e r s and rings o f f u n c t i o n s N o w algebraic n u m b e r t h e o r y as well as algebraic g e o m e t r y have their o w n refined m e t h o d s t o deal with Galois e x t e n s i o n s : in n u m b e r t h e o r y one s h o u l d name c l a s s field t h e o r y f o r instance Thus, the m e t h o d s o f t h e general

t h e o r y f o r Galois e x t e n s i o n s o f rings are always in c o m p e t i t i o n with t h e m o r e special m e t h o d s o f t h e discipline where they are applied It is hoped the reader will g e t a feeling t h a t t h e general m e t h o d s s o m e t i m e s also lead t o new r e s u l t s and provide an i n t e r e s t i n g a p p r o a c h t o old ones

Let us briefly review the d e v e l o p m e n t o f the subject Hasse {|949} s e e m s t o have been t h e first t o c o n s i d e r the t o t a l i t y o f G - G a l o i s e x t e n s i o n s L o f a given

n u m b e r field K He realized t h a t f o r finite abetian G this s e t admits a natural

abelian g r o u p s t r u c t u r e , i f one a l s o a d m i t s c e r t a i n " d e g e n e r a t e " e x t e n s i o n s L / K

which are n o t fields For example, t h e neutral e l e m e n t o f this g r o u p is t h e direct

p r o d u c t o f copies o f K, w i t h index s e t G This c o n s t i t u t e s t h e f i r s t f u n d a m e n t a l idea The s e c o n d idea, initiated by A u s l a n d e r and G o l d m a n {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 R o s e n b e r g (1965L is t o admit base rings R instead o f fields It is n o t so obvious w h a t t h e definition o f a G - G a l o i s 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 s h o u l d be, b u t once one has a g o o d definition (by t h e way, all g o o d definitions t u r n o u t t o be equivalent}, t h e n one a l s o o b t a i n s nice

f u n c t o r i a l i t y p r o p e r t i e s , stability under base c h a n g e f o r instance, and t h e t h e o r y runs a l m o s t as s m o o t h l y as f o r fields H a r r i s o n {1965} p u t t h e t w o ideas t o g e t h e r

and defined, for G finite abelian, t h e g r o u p o f all G - G a l o i s 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 m o d u l o G - i s o m o r p h i s m This g r o u p is n o w called the H a r r i s o n

g r o u p , and we d e n o t e it by H(R,G), Building on the general t h e o r y o f Chase, Harrison, and Rosenberg, and developing s o m e new t o o l s , we c a l c u l a t e in t h e s e

n o t e s t h e g r o u p H(R,G) in a fairly general setting

viii

The principal link b e t w e e n this t h e o r y and n u m b e r t h e o r y is t h e s t u d y o f ramification Suppose L is a G - G a l o i s e x t e n s i o n o f the n u m b e r field K, E a s e t o f finite places o f K, and R = Or,~: the ring o f ) - 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 - G a l o i s 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 b e l o n g t o E In m o s t applications, will be t h e s e t o f places over p The r e a s o n f o r this choice will b e c o m e a p p a r e n t when we d i s c u s s Z - e x t e n s i o n s below

We n o w d i s c u s s t h e c o n t e n t s o f t h e s e n o t e s in a little more detail

A f t e r 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

t h e c o n n e c t i o n w i t h n u m b e r t h e o r y , and Z - e x t e n s i o n s , we develop in Chap I a p

structure theory f o r Galois e x t e n s i o n s with cyclic g r o u p G = Cpn o f order pn, under the h y p o t h e s i s t h a t p - I e R and p is an o d d prime number For technical r e a s o n s ,

we also s u p p o s e t h a t R has no nontrivial i d e m p o t e n t s Since the H a r r i s o n g r o u p H(R, G) is f u n c t o r i a l 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 t h e 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 f o r t h e case G finite abelian, IGI - t e R The basic idea is simple If R c o n t a i n s a primitive p n - t h r o o t o f unity ~ {this

n o t i o n has t o be defined, o f course}, and p - t e R, t h e n Kummer t h e o r y is available

f o r Cp~-extensions o f R The s t a t e m e n t s o f Kummer t h e o r y are, however, m o r e

c o m p l i c a t e d t h a n in t h e field case: it is no l o n g e r true t h a t every C p n - e x t e n s i o n

S/R c a n be g o t t e n by " e x t r a c t i n g t h e p n - t h r o o t o f a unit o f R", b u t the o b s t r u c - tion is u n d e r c o n t r o l The p r o c e d u r e is n o w t o adjoin ~, t o R s o m e h o w (it is a lot

o f work t o make this precise}, use Kummer t h e o r y for the ring S o b t a i n e d in this way, and d e s c e n d again Here a very i m p o r t a n t c o n c e p t m a k e s its appearance

A G - G a l o i s e x t e n s i o n S/R is defined t o have normal basis, if S has an R - b a s i s o f the f o r m {y(x) [ y e G} for s o m e x e S Fo G = Cp~, t h e e x t e n s i o n s with n o r m a l 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 t h e s t r u c t u r e o f NB(R, Cpn), and o f H(R, Cpn)/NB(R, Cpn) In the field case, t h e l a t t e r g r o u p is trivial, b u t n o t in general K e r s t e n and Michali~ek {1988} were t h e f i r s t t o prove r e s u l t s for NB(R, Cpn) Our r e s u l t says t h a t

NB(R, Cpn) is " a l m o s t " i s o m o r p h i c t o an explicitly given s u b g r o u p o f S,*/(pn-th

powers}, and Hi R, Cpn )/NB(R, Cpn ) is i s o m o r p h i c t o an explicitly given s u b g r o u p

o f the Picard g r o u p o f S The d e s c r i p t i o n o f NB{R, Cp~} is basic f o r the c a l c u l a -

t i o n s 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 g e t a n o t h e r d e s c r i p t i o n o f NB(R,Cp~), this time as

a factor g r o u p o f S~/(p~-th powers} This is s o m e t i m e s more practical, as w i t -

n e s s e d by t h e lifting theorems which c o n c l u d e Chap II: If I is an ideal o f R, c o n - tained in t h e J a c o b s o n radical o f R, t h e n every C p n - e x t e n s i o n S o f R/I with normal basis is o f t h e f o r m S - T/IT, T e NB(R, Cp~)

In Chap Ill we s e t o u t t o c a l c u l a t e the o r d e r o f NB(R, Cp,~), w h e r e n o w

R O r [ p - i ] , K a n u m b e r field A l t h o u g h one a l m o s t never k n o w s t h e g r o u p s S * t l explicitly, which are c l o s e l y r e l a t e d t o the g r o u p o f units in t h e ring o f i n t e g e r s

o f K(~ n), one can n e v e r t h e l e s s do the c a l c u l a t i o n one w a n t s , by dint o f s o m e t r i c k s involving a little c o h o m o l o g y o f g r o u p s 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 t h e s t r e n g t h o f t h e m e t h o d by deducing the Galois t h e o r y

o f finite fields, and a piece o f local c l a s s field theory The main r e s u l t for n u m b e r fields K is t h a t with R as above, and n n o t " t o o small", the o r d e r o f NB(R, Cpn)

equals c o n s t p (1 +r2)n, where r 2 is h a l f the n u m b e r o f nonreal e m b e d d i n g s K C

as usual

The goal o f Chap IV is t o g e t an u n d e r s t a n d i n g , h o w far t h e s u b g r o u p

NB(R, Cp,~) d i f f e r s f r o m H(R, Cp,~), and a similar q u e s t i o n f o r Zp in t h e place o f

Cpn Here H(R, Zp) is t h e g r o u p o f Z p - e x t e n s i o n s o f 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 k n o w n t h a t all Z p - e x t e n s i o n s o f K are unramified o u t s i d e p, and hence already a Z p - e x t e n s i o n s o f R, which justifies t h e choice o f the ring R

1 + r 2 This was previously proved in a s p e -

We prove in IV §2: NB(R,Zp) - Zp

clal case by K e r s t e n and Michali~ek (1989) The r e s u l t is w h a t one e x p e c t s f r o m the f o r m u l a f o r INB( R, Cp~)t, b u t the p a s s a g e t o t h e limit p r e s e n t s s o m e subtleties The index qn = [H(R, Cpn):NB(R,Cp~)] is s t u d i e d in s o m e detail, and we s h o w t h a t q~ either g o e s t o infinity or is e v e n t u a l l y c o n s t a n t f o r n -* ~ The first case c o n -

j e c t u r a l l y never happens: we prove t h a t this case o b t a i n s if and only t h e f a m o u s

L e o p o l d t c o n j e c t u r e fails f o r K and p A n o t h e r way o f saying this is as f o l l o w s : NB(R, Zp) has finite index in H(R, 7p) if and only if the L e o p o l d t c o n j e c t u r e is t r u e

f o r K and p We give r e s u l t s a b o u t the actual value o f t h a t index; in particular, it can be d i f f e r e n t f r o m 1

A p a r t f r o m adjoining r o o t s o f unity, there is so far only o t h e r explicit way

o f g e n e r a t i n g large abelian e x t e n s i o n s o f a n u m b e r 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 s h o w in IV §S t h a t

l i p - e x t e n s i o n s obtained in t h a t way t e n d to have normal b a s e s over R = O r [ p - i ] , and a weak c o n v e r s e t o this s t a t e m e n t These r e s u l t s are in tune with t h e m u c h 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 c h a n g e o f scenario in Chap V There we c o n s i d e r f u n c t i o n fields

o f varieties over n u m b e r fields Such f u n c t i o n fields are a l s o called absolutely fini- tely generated fields over Q A f t e r s o m e p r e r e q u i s i t e s f r o m algebraic g e o m e t r y , we

s h o w a relative finiteness r e s u l t on C p n - G a l o i s coverings o f such varieties, which

is similar t o r e s u l t s o f Katz and Lang (1981), and we prove t h a t all Z p - e x t e n s i o n s

o f an a b s o l u t e l y finitely g e n e r a t e d field K already c o m e f r o m the g r e a t e s t n u m b e r field k c o n t a i n e d in K In o t h e r w o r d s : for n u m b e r fields k one does n o t k n o w h o w

many i n d e p e n d e n t Z - e x t e n s i o n s k has, unless L e o p o l d t ' s c o n j e c t u r e is k n o w n to p

be t r u e f o r K and p, b u t 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 l a s t c h a p t e r {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 c a s e t h e g r o u n d ring R c o n t a i n s a primitive p ~ - t h r o o t o f unity

~n b u t n o t necessarily p - I e R It is a s s u m e d , however, t h a t p does n o t divide zero

in R Even t h o u g h Kummer t h e o r y fails for R, we may still a s s o c i a t e t o many

Cpn-extensions S/R a c l a s s ~0n(S) = [ u ] in R* m o d p~ - t h powers If R is normal,

S will be t h e integral c l o s u r e o f R in R[p-l,P~Z-u] The main q u e s t i o n is: W h i c h units u • R* may o c c u r here? In § 2 we e s s e n t i a l l y p e r f o r m a r e d u c t i o n t o t h e case

R p - a d i c a l l y c o m p l e t e Taking up a paper o f Hesse {1936), we t h e n a n s w e r our

q u e s t i o n 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 o u t t h a t t h e a d m i s - sible values u are precisely the values o f certain universal polynomials, with p a r a -

m e t e r s running over R Reduction m o d p also plays an e s s e n t i a l role, and for this

r e a s o n we have t o review Gatois t h e o r y in c h a r a c t e r i s t i c 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-

e x t e n s i o n o f a c e r t a i n universal p - c o m p l e t e ring c o n t a i n i n g ~ {but n o t p - l ) w a s

c o n s t r u c t e d , and we are n o w able t o see in detail h o w this e x t e n s i o n d e s c e n d s

d o w n t o a similar g r o u n d ring w i t h o u t ~,,, t o wit: the p - a d i c c o m p l e t i o n o f Z[X] This e x t e n s i o n is, r o u g h l y 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

M o s t 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 r e f e r e n c e s are indicated in the usual style: the c h a p t e r s are n u m b e r e d O, I, II VI, and a

r e f e r e n c e n u m b e r n o t c o n t a i n i n g O or a Roman numeral m e a n s a r e f e r e n c e within the same c h a p t e r All rings are supposed commutative {except, occasionally, an

e n d o m o r p h i s m ring), and with unity O t h e r c o n v e n t i o n s are s t a t e d where needed Earlier v e r s i o n s o f c e r t a i n p a r t s o f t h e s e n o t e s are c o n t a i n e d in t h e journal articles Greither {1989), {1991)

It is my p l e a s u r a b l e d u t y t o t h a n k my c o l l e a g u e s w h o have helped t o improve the c o n t e n t s o f t h e s e notes Ina K e r s t e n has influenced t h e p r e s e n t a t i o n o f earlier versions in many ways and provided valuable i n f o r m a t i o n Also, t h e helpful and detailed r e m a r k s o f several r e f e r e e s are appreciated; I like t o 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 o r g a n i z a t i o n o f the notes Finally, I am

g r a t e f u l f o r 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 N g u y e n Quang Do

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 a r e j u s t

f o r m a l s y m b o l s ) , w i t h m u l t i p l i c a t i o n given by (Xuo)(itu~) X.c(it)-uc~ ( X , i t e L ) The m a p j: L a G -) E n d r ( L ) is given b y

jC~u o) = (it ~ X'o(~)) ~ E n d r ( L )

~ p o s l U o n ~ 9_ j is a w e l l - d e f i n e d K - a l g e b r a homomorphism, which is bijective i f f

G is e m b e d d e d in A u t ( L / K ) and L / K is a G-Galois extension

Proof The f i r s t s t a t e m e n t is e a s y t o 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 o f 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 b i j e c t i v e

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

w e d with t h e obvious addition and multiplication (Note t h a t L ~, w i t h o u t b r a c k e t s ,

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 L 4 The m a p 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 l e f t

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

P r o o f The f i r s t 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 t h e matrix

(o(yt))oeG, l<i< n has full rank ( n o t e t h a t this is indeed a square matrix) The l a t t e r

c o n d i t i o n says t h a t the images o f all 0 e G are L - l e f t linearly independent in

E n d r ( L ) , or ( w h a t is t h e same) t h a t t h e map j o f 1.1 is injective Hence 1.4 f o l l o w s

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

e x a c t l y as in 1.3, is bijective, or ( w h a t is the same) an S - a l g e b r a i s o m o r p h i s m lSAtamplml: a) Galois e x t e n s i o n s 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 f o l l o w s : The a l g e b r a R (c) is again j u s t Map(G,R) with the canonical

R - a l g e b r a s t r u c t u r e , and t h e action o f G is given by index shift:

= f o r o G ,

It is an easy exercise t o prove t h a t in this case indeed S a = R and h is bijective

We shall see m o r e e x a m p l e s below

There e x i s t p l e n t y o f o t h e r definitions, or r a t h e r c h a r a c t e r i z a t i o n s , o f G - 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 rings Some o f t h e m are listed in the next t h e o r e m :

~ 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]: Let R c S b e c o m m u - tative rings, G c A u t ( S / R ) a f i n i t e s u b g r o u p such that S ~ = R T h e n the f o l l o w i n g conditions are equivalent:

(i) S / R is G-Galois (i.e p e r def.: h: S ®e S ~ S (~) is bijective);

(ii) h: S @1~ S * S (~) is surjective;

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

(ii') ~ (iii): W e f i r s t s h o w t h a t RS is 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 D e f i n e t h e t r a c e tr: S * R b y t r ( y ) = ~ , a c G o ( y ) (tr is w e l l - d e f i n e d s i n c e S ~ = R, a n d R - l i n e a r s i n c e

(iv) ~ (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 s i n g l e o * e~

By (iv), t h e ideal o f S g e n e r a t e d b y 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 e q u a l t o S O n e f i n d s h e n c e n o e IN a n d 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

(ii) S is s e p a r a b l e o v e r R, and f o r each n o n z e r o i d e m p o t e n t 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 v a c u o u s l y t r u e i f S has no i d e m p o t e n t s b e s i d e 0 and 1.)

P r o o f 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

(ill S is, in the c a n o n i c a l 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 g r o u p o f all c • G which leave U pointwise f i x e d

(iv) I f H is a n o r m a l s u b g r o u p o f G, then U is, in the c a n o n i c a l way, a G / H - Galois e x t e n s i o n o f R

to be a canonical surjection G -~ G / N , then ~ * ( S ) = S ~ as in the above discussion

(For the construction o f ~ ~, see the p r o o f o f this theorem.)

In other words: I f we let H(R,G) be the set o f isomorphism classes o f G-Galois e x t e n - sions 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 d e f i n e d is a l s o c a l l e d t h e Harrison set o f R and G Proof o f Thm 3.1 W e do a) and b) s i m u l t a n e o u s l y F i r s t we define ~* Let S

(ii) n * S / R is, with t h e given H - a c t i o n , indeed an H - G a l o i s e x t e n s i o n

We do (i) f i r s t , by exhibiting n a t u r a l bijections

M a p ~ ( J , M a p ~ ( H , S ) ) ~ M a p ~ ( J , S)

( I t is l e f t t o t h e r e a d e r t o verify t h a t ~ a n d ~ 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 ) L e t c t ( y ) = y ( - ) ( e s ) f o r y in t h e l e f t hand side, i.e ct(y)(j) = y ( j ) ( e s )

f o r j ~ J Let ~(z)(j)(h) = z(~b(h)j) f o r z in t h e r i g h t hand side, h~H, j e J

W e c h e c k ~ is w e l l - d e f i n e d , i.e ~(y)e Map~b~(J,S): Let j ~ J , g e G W e c a l c u l a t e :

txty)(@r(g).j) = y(~Im(g).j)te~)

= (~(g)*y(j))(e n) (since y ~ M a p ~ )

= y ( j ) ( e n n ( g ) ) (def o f H - a c t i o n on M a p n ( H , S ) )

= g ( y ( j ) ( e n ) ) (since y ( j ) ~ M a p x )

10 c h a p 0

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

3 9 [ H a r r i s o n (1965)] a) With this definition, H ( R , G ) b e c o m e s an abelian group whose neutral e l e m e n t 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

L e t 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 L e t S, T, U e H ( R , G ) Then: ( S ' T ) ' U = ~t*(~t*(S®,T) o R U)

3.~ Given a commutative ring R, there exists a pro finite abelian group D R

Proof Every l e f t e x a c t 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 H a r r i s o n (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 t h e same time the uniqueness o f fl R O f t e n tq R is called t h e abelian f u n d a m e n t a l group o f R, or t h e 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 t h e g r o u p s Z / n Z f o r m

an inductive s y s t e m indexed by t h e divisibility lattice o f ~q, and v means P o n t r y a g i n dual See Harrison, loc.ctt

b) By t h e m e t h o d s o f t h e 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 t h e h o m o m o r p h i s m g e -, gk (G finite abelian), t h e n [k]*: H{R,G) -, H(R,G) is m u l t i p l i c a t i o n by k in t h e H a r r i s o n group In particular, H(R,G) is anni- hilated by t h e e x p o n e n t o f G, hence t o r s i o n T h e r e f o r e t h e P o n t r y a g i n dual in a) is indeed profinite

We w a n t t o s h o w t h a t the g r o u p D R has in many c a s e s an i n t e r p r e t a t i o n 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 a p p r o p r i a t e (infinite) e x t e n s i o n 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 e x t e n s i o n o f K More generally, J a n u s z (1965) has p r o v e d t h e e x i s -

t e n c e o f a separable closure f o r every c o n n e c t e d ring R This is by definition a c o n -

n e c t e d R - a l g e b r a R sep which is a filtered union o f G - G a l o i s e x t e n s i o n s o f R (G varies, o f c o u r s e ) , s u c h 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 t o w h a t t h e Galois g r o u p o f S / R is,

b e c a u s e o f t h e f o l l o w i n g 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 c h a p t e r , 7.3}: I f S / R is a G - G a l o i s e x t e n s i o n o f c o n n e c t e d rings, t h e n

A u t ( S / R ) = G As p r o v e d by Janusz (1966), the g r o u p ~/R = A u t ( R S ~ 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 c o n t a i n e d in RssP), hence T R is profinite (For a n o t h e r e x p o s i t i o n o f this material, see also D e M e y e r - l n g r a h a m (1971).)

14 c h a p 0

L e m m a 3.7 Let R be connected, S/R, T / R be two Galois extensions (with maybe dif- ferent 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 b y an i d e m p o t e n t

Fropomltion 3.8 [ H a r r i s o n (1965)] Let R b e c o n n e c t e d , S / R a G - G a l o i s e x t e n s i o n with

f i n i t e abelian g r o u p G T h e n 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 ( w h e r e i: H -* G is the i n c l u s i o n )

the f o l l o w i n g f o r m u l a s in E n d s ( U ) : el.o.e I - el.o o.e I for o • H , eloe i = 0 for o

n o t in H Let ~0 e E n d s ( U ) T h e n ei.~o.e I e E n d s ( S ) ; b y 1.6 (iii), there are s o • $

16 chap 0

p h i s m f : tit a ~ A u t ( S / R ) = G Taking E = S, we g e t t h e i d e n t i t y f o r f ~ , hence

v a ( f ) = S This was t h e e a s y case The p r o b l e m a r i s e s w h e n S is n o t c o n n e c t e d But t h e n , 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 a n d 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 t o find a p r e i m a g e o f U u n d e r u s , b u t this we c a n 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 f l R is i s o m o r p h i c to the abelianization T n / [ T R , T R ]

P r o o f By Thm 3.6, t h e p r o f i n i t e g r o u p TR/[qrR,qr ~] p r o - r e p r e s e n t s t h e 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 t h i s f u n c t o r by definition It is w e l l - k n o w n

F o r t h e g e n e r a l case, one e x h i b i t s f o r S c H(R,G) a canonical m a p

O~s/l~: T®Rrc*S , n * ( T ® R S ) ;

one c h e c k s t h a t t h e definition o f ~ is c o m p a t i b l e 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 , which r e d u c e s t h e p r o o f t o t h e c a s e w h e r e S is a trivial G - e x t e n s i o n This c a s e can easily be d o n e 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 h o r n o m o r p h i s m

P r o o f The H a r r i s o n p r o d u c t w a s d e f i n e d w i t h t h e help o f tl* (~: G × G " G t h e

m u l t i p l i c a t i o n map.) By 3.10, the m a p T® R - c o m m u t e s w i t h ~* F r o m this t h e c l a i m

f o l l o w s easily

3 J 2 Let T / R be an H-Galois extension o f connected rings, H abelian Then

isomorphic to H ( R , H )

t h a t ~ is a g r o u p h o m o m o r p h i s m If ~(~) is trivial, t h e n by 3.3 already ~*T is trivial, w h e r e ~b: H -~ I m ( ~ ) is t h e s u r j e c t l o n defined by ~ But ~b*T is (up t o c a n o - nical i s o m o r p h i s m ) a s u b a l g e b r a o f T, and T is c o n n e c t e d Hence necessarily d~*T = R, and ~ is the trivial h o m o m o r p h i s m

For e a c h ~ e H o m ( H , G ) we have T e a ~ ' ~ T - ~ * ( T ® a T ) by 3.8, and already

T® T I T is t h e trivial /-/-extension Hence t h e c o m p o s i t e map ( T ® - ) ~ is zero

To c o n c l u d e t h e p r o o f , a s s u m e S e H(R,G) s u c h t h a t T® a S is the trivial G - e x - tension By 3.8, we can find an inclusion i: G' c G o f a s u b g r o u p and a c o n n e c t e d

G ' - G a l o l s e x t e n s i o n S' s u c h t h a t S - i*S' Then T® a S - / * ( T e a S ' ) , and by 3.3,

T e a S ' / T is the trivial G ' - e x t e n s i o n This Implies (by 3.4) t h a t there e x i s t s a

T - a l g e b r a h o m o m o r p h i s m T ® a S ' -~ T, hence an R - a l g e b r a h o m o m o r p h i s m ~0: S' -~ T Since S ' is c o n n e c t e d , ~o m u s t be injective by 3.7 Since S' is s e p a r a b l e over

R (every Galois e x t e n s i o n is separable!), we may use t h e main t h e o r e m 2.3 and obtain: S ' - ~b*T f o r s o m e s u r j e c t l o n H -~ G' P u t t i n g t h i n g s t o g e t h e r , we o b t a i n

S = i~'S ' = i*t~*T = ~*T f o r ~ = i~: H -* G

§ 4 R m n l f l c a ~ n

W e have n o t seen many e x a m p l e s o f Galois e x t e n s i o n s yet The p u r p o s e o f this s e c t i o n is t o s e t up a t r a n s l a t i o n machinery which a l l o w s us t o find many Galois e x t e n s i o n s o f rings o f a n u m b e r - t h e o r e t i c a l kind These r e s u l t s are quite well k n o w n t o e x p e r t s , b u t it s e e m s c o n v e n i e n t t o give p r o o f s here

N ~ I f K is an algebraic n u m b e r field (i.e a finite e x t e n s i o n o f Q), t h e n O K

d e n o t e s t h e ring o f i n t e g e r s in K Example: OQ = Z

l ~ m m ' t : I f L / K is a G - G a l o i s e x t e n s i o n o f n u m b e r fields, t h e n G o p e r a t e s also on the r i n g O L, and (OL) c = O L N K = O x

The remark suggests a possibility of O L / O K being a G-Galois extension of rings S o m e care is necessary, as is s h o w n by the following simple example:

T l m o ~ m 4 L [ c f A u s l a n d e r - B u c h s b a u m (1959)] Let L / K be a G-Galois extension

o f algebraic n u m b e r fields, S and S' as above Then O L , s , / O r , s is a G-Galois e x t e n - sion i f and only i f L / K is u n r a m i f i e d at all finite places which are not in S

u n r a m i f i e d over K outside S and infinity For the injectivity o f or, it s u f f i c e s that R is

an integrally closed domain with f i e l d o f quotients K

20 chap 0

The a c t i o n m a k e s s e n s e , since t h e R - a l g e b r a e n d o m o r p h i s m o': X ~-* ~X o f R [ X ]

P r o o f • a ) L e t ~ = : Y T h e n S = R ( n ; u ) = R ~ o t R e $ o~-lR For l_<i_<n-1, ~ i - 1

is a unit in R (Reason: I f ~n d e n o t e s a primitive n t h r o o t o f unity in C, t h e n ( 1 - ~ a ) t l - ~ a Z ) ' ' ( 1 - ~ : - l ) = ( ( X ' Z - 1 ) / t X - 1 ) ) I x = I = n, hence (1-~/) is a unit in

l [ ~ , n - l ] , a n d t h e r e is a ring h o m o m o r p h i s m f r o m Z [ ~ , n - i ] t o R which m a p s ~

t o ~.) F r o m t h i s one s e e s t h a t t h e fixed ring o f 0 in S is R, since 0 o p e r a t e s o n

t h e cyclic s u m m a n d giR as m u l t i p l i c a t i o n by ~i

f i r s t f a c t o r is a unit in S b e c a u s e ~ is {recall ~n = u) The s e c o n d f a c t o r is a l s o a

u n i t in R, since again all ~i_~j ( 0 < i < j < n) a r e units in R H e n c e by definition, S i R

m u l t i p l i c a t i o n H e n c e S-T is t h e fixed ring o f Ker(~) in $®R T, a n d Ker(~) is g e n e -

r a t e d b y t h e pair (o,o -1) e C a ×C a F r o m t h i s one c a l c u l a t e s d i r e c t l y t h a t S T is t h e

R - s p a n o f {1®1, ~®[3 ~n-l®{Sn-t} O b v i o u s l y t h e n - t h p o w e r o f ¢¢®{3 e q u a l s uv, and f r o m t h i s t h e r e q u i r e d i s o m o r p h i s m is easily o b t a i n e d

c) W e h a v e t o s h o w : R(n; u) is t r i v i a l i f f u is a n - t h p o w e r in R* N o w o b v i -

o u s l y t h e R - a l g e b r a h o m o m o r p h i s m s E: R(n; u) -, R a r e in b i j e c t i o n w i t h t h e s e t

o f n - t h r o o t s o f u in R ( o r R*) H e n c e , t h i s s e t is n o n e m p t y i f f t h e r e e x i s t s o n e

s u c h E, a n d t h i s is e q u i v a l e n t t o t h e t r i v i a l i t y o f R(n; u) b y L e m m a 3.4 Q.E.D The q u e s t i o n a r i s e s : w h a t is t h e c o k e r n e l o f i~? In t h e p r o c e s s o f a n s w e r i n g

~ m m $,2 The isomorphism classes o f discriminant modules ( o f type n over R) f o r m

an abelian group D i s c ( R , n ) in a natural way

Proof T h e m u l t i p l i c a t i o n o f d i s c r i m i n a n t m o d u l e s is g i v e n b y (M,cp).(M',cp') =

( M ® R M ' , ~ ® ~o') I n o r d e r t h a t t h e m a p ~ ® ~ ' m a k e s e n s e , i t is n e c e s s a r y t o i d e n -

t i f y ( M ® R M ' ) ®n w i t h M®n®s M '®n, a n d R® a R w i t h R O b v i o u s l y , t h e c l a s s o f

(R, id a) g i v e s a n e u t r a l e l e m e n t , a n d a s s o c i a t i v i t y u p t o i s o m o r p h i s m is c l e a r I f (M,~o) is g i v e n , w e m a y c o n s t r u c t a n i n v e r s e b y c o n s i d e r i n g ( M * , ~ *-1) ( h e r e ( - ) *

g r o u p r i n g R [ C ], w e d e f i n e f o r 0 _< i < n:

M "~ = { x e M l a x = ~ t x }

= { x E M I C o p e r a t e s o n x v i a t h e c h a r a c t e r X}

e m p l o y e d w o u l d lead us t o o far afield at t h e m o m e n t H e n c e we j u s t s t a t e t h e

s i m p l e s t c a s e here For a p r o o f , see VI §1 or the r e f e r e n c e s given there

T l m t w l m t S.S L e t p b e a n y p r i m e a n d R a ring o f c h a r a c t e r i s t i c p L e t P: R " R b e the h o m o m o r p h i s m o f additive g r o u p s (!) g i v e n b y P ( x ) = x p - x , x ~ R T h e n t h e r e is

§ 6 Normal bases and GaJois module structure

The m a t e r i a l o f t h i s s e c t i o n is b a s i c f o r s e v e r a l c h a p t e r s o f t h e s e N o t e s Much o f it is s t a n d a r d , a n d may a l s o be f o u n d in C h a s e - H a r r i s o n - R o s e n b e r g (1965) The m o t i v a t i n g q u e s t i o n is: Given a G - G a l o i s e x t e n s i o n S / R ( o f c o m m u t a t i v e

r i n g s as a l w a y s ) , w h a t c a n b e s a i d a b o u t t h e s t r u c t u r e o f S a s an R [ G ] - m o d u l e ?

I t is c l e a r t h a t t h e o p e r a t i o n o f G o n S m a k e s S i n t o a l e f t R [ G ] - m o d u l e In

G a l o i s t h e o r y o f f i e l d s , it is a c l a s s i c a l r e s u l t t h a t f o r e v e r y G - G a l o i s e x t e n s i o n

L / K , L is f r e e c y c l i c o v e r K[G], w h i c h m e a n s in o t h e r w o r d s t h a t t h e r e is a K - b a - sis o f L o f t h e f o r m { o x l x e G}, f o r s o m e x E G Such a b a s i s is t r a d i t i o n a l l y c a l l e d

6.1 For any G-Galois extension S / R of commutative rings, the R[G]-module

S is inoertible, i,e finitely generated projective of constant rank 1

homomorphism r~ of finite abelian groups, one has a commutative diagram

Proof L e t S, T • H{R,G} T h e n S ® a T is a G x G - G a l o i s e x t e n s i o n , a n d i t is c l e a r

t h a t picc~×o{S® a T} = p i c { S ) ® R p i c ( T ) ( p i c a × a is a d h o c n o t a t i o n f o r pic: H{R,G×G) * Pic(R[G×G]), a n d w e i d e n t i f y R[G×G] w i t h R[G]® a R[G] a s u s u a l ) W e a p p l y

I ~ f l a l t t o m A G - G a l o i s e x t e n s i o n S / R has a normal basis, i f S is f r e e c y c l i c a s a n

R [ G ] - m o d u l e (G is a f i n i t e g r o u p , S a n d R c o m m u t a t i v e r i n g s , a s a l w a y s )

T h e r e a r e t h e f o l l o w i n g e q u i v a l e n t f o r m u l a t i o n s o f t h i s v e r y i m p o r t a n t d e f i -

n i t i o n :

a) S is f r e e o v e r RIG] ( N o t e t h a t t h e r a n k o f S o v e r R is u n i q u e l y d e f i n e d ) b) There exists x ~ S such that {cxlc e G } is an R-basis of S (such a basis,

or s o m e t i m e s x by itself, is called a normal basis of S over R.)

c) pic(S) is trivial

I}eflmltloL N B { R , G ) denotes the set of all i s o m o r p h i s m classes of G-Galois exten-

sions S / R w h i c h have a n o r m a l basis B y definition, N B ( R , G ) is a subset of H(R,G)

C o r o l l m r y 6 4 I f G is abelian, then N B ( R , G ) is a subgroup o f H ( R , G )

Galois d e s c e n t is a f r a g m e n t o f t h e t h e o r y o f f a i t h f u l l y flat d e s c e n t So far,

we only have been using a "trivial part" o f this technique, useful in t e s t i n g w h e -

t h e r a given map is an i s o m o r p h i s m , say There is more t h a n t h a t t o d e s c e n t t h e -

ory Namely, d e s c e n t t h e o r y is also a means o f constructing c e r t a i n m o r p h i s m s and,

m o s t i m p o r t a n t o f all, o b j e c t s over a ring R which are p r e v i o u s l y only given over

a f a i t h f u l l y f l a t e x t e n s i o n S Briefly, one w a n t s t o solve t h e e q u a t i o n S ® R X - Y

f o r X (X m i g h t be an R - m o d u l e , an R - a l g e b r a ) This s e c t i o n is a b o u t this

c o n s t r u c t i v e p a r t o f d e s c e n t t h e o r y , limited t o t h e c a s e o f Galois descent, i.e S / R

a Galois e x t e n s i o n The r e s u l t s will be u s e d f r e q u e n t l y in later c h a p t e r s

The m o t i v a t i n g question, t h e r e f o r e , is: Given a G - G a l o i s 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), and an S - m o d u l e (or: S - a l g e b r a ) N, when is N, up t o i s o m o r - phism, o f t h e f o r m S® R M, f o r an R - m o d u l e M (or: R - a l g e b r a M ) ? I f 4: S®R M - - * S® a M ' is a h o m o m o r p h i s m o f S - m o d u l e s ( S - a l g e b r a s ) , when does kb have the f o r m

S ® @ ? Galois descent gives a complete answer to both questions; the second one

is slightly easier

C o m m m t i o ~ T o avoid repetition, the w o r d "R-object" in this section is s u p p o s e d to

m e a n consistently either " R - m o d u l e " or "R-algebra" or "R-algebra with action of a given group C by R-automorphisms" All statements are m e a n t simultaneously for these three kinds of objects Accordingly, the w o r d " R - m o r p h i s m " m e a n s either

" R - m o d u l e hem." or "R-algebra hem." or "C-equivariant R-algebra hem." There exists of course a categorical f r a m e w o r k encompassing all these cases and m u c h more, but for our purposes a m o r e d o w n - t o - e a r t h approach is preferable For the general theory, the reader m a y consult Grothendieck (1959)

DefinltioL Let S / R be a G-Galois extension

a) An S - m o r p h i s m f : A -~ B b e t w e e n t w o S - o b j e c t s is called o-linear ( f o r

s o m e o e G), if f ( s a ) = o ( s ) f ( a ) for all s e S, a e A

b) A d e s c e n t d a t u m • = (¢ba)ae G on s o m e S - o b j e c t B is a family o f R - a u t o -

m o r p h i s m s ~a o f B s u c h t h a t : Ca is a - l i n e a r f o r all a e G, and Oa¢x = Oar f o r all

a, r e G

E x a m p l e I f B = S ® R A f o r s o m e R - o b j e c t A, t h e n t h e r e is t h e s o - c a l l e d trivial

d e s c e n t d a t u m (¢a)oeG defined by dga(s® a) = a(s)® a f o r s c S, a ~ A

7.L Let B b e an S - o b j e c t , and dp a d e s c e n t d a t u m on it T h e n A = B 0 (which equals by definition { b e B l O a ( b ) = b f o r all a ~ G}) is an R - o b j e c t ; the c a n o n i c a l

m a p ct: S® R A -, B i n d u c e d by A , B is an i s o m o r p h i s m , and the trivial d e s c e n t d a t u m

on S ® R A c o r r e s p o n d s via o~ to the git, e n d e s c e n t d a t u m ~P

Proof It is Immediate t h a t A Is Indeed an R - o b j e c t (recall t h e above c o n v e n t i o n )

The p o i n t is t o s h o w t h a t a is bijective The last s t a t e m e n t o f t h e t h e o r e m is t h e n

a direct c o n s e q u e n c e

Let T be any f a i t h f u l l y flat R - a l g e b r a such t h a t S r (= T ® R S ) is t h e trivial

G - e x t e n s i o n o f T iT = S is a possibility.) It t h e n s u f f i c e s t o s h o w t h a t C~r:

Sr®t~A r ~ B r is an i s o m o r p h i s m N o w since t e n s o r i n g with T p r e s e r v e s kernels,

A r is precisely t h e fixed ring o f all T® Ca' oeG Hence we may c h a n g e n o t a t i o n

and a s s u m e t o begin with: S / R is t h e trivial G - e x t e n s i o n : S = R (~) = ~ a e ~ e a ' R , where t h e e o are t h e s t a n d a r d i d e m p o t e n t s , and G a c t s via x*e o = e a x - t ( o , r e G) Then B likewise s p l i t s in t h e f o r m B = ~ a e a Ba w i t h B o = eaB, and e a c h B a is an

R - o b j e c t One c h e c k s t h a t t h e x - l i n e a r a u t o m o r p h i s m Cr o f B is given by a family

( f o ( r ) ) o e a o f i s o m o r p h i s m s f a ( r ) : B a Bar-1 , and t h e s e i s o m o r p h i s m s s a t i s f y t h e

c o n d i t i o n

f o r - { P ) f a (r) = f o {pr) (a,x,p e G)

T h e r e f o r e t h e B a are c a n o n i c a l l y i s o m o r p h i c t o one R - o b j e c t A' (take e.g A ' = BI),

so B b e c o m e s identified with ~[~aeG A ' , and the d e s c e n t d a t u m • n o w o p e r a t e s j u s t

by index shift One t h e n obtains A = d i a g ( A ' ) c A '(a) = B, and it is n o w obvious

t h a t a: R ( a ) ® ~ A * B is an i s o m o r p h i s m , q,e.d

One a l s o needs t o d e s c e n d m o r p h i s m s This w o r k s as f o l l o w s

P r o p o a l U o n 7 9 Let A 1, A 2 b e two R - o b j e c t s , g: S ® R A l ~ S ® R A ~ an S - m o r p h i s m , and (~Pa(i))aec the trivial d e s c e n t d a t u m on S e R A t (i = 1,2), T h e n g is o f the f o r m S® f f o r s o m e R - m o r p h i s m f : A l ~ A 2 i f f ¢ <o~)g = g ¢ o ~l) f o r all a e G M o r e o v e r , f is

u n i q u e i f it exists

P r o o f The u n i q u e n e s s o f f is immediate f r o m t h e f a c t t h a t S is f a i t h f u l l y f l a t

over R

If g = S ® f , t h e n one c h e c k s t h e f o r m u l a ¢ ~ z ) g = g~0¢t~ directly, j u s t using

the definitions The o t h e r implication is s h o w n as f o l l o w s : A 2 e m b e d s into S®~ A z

by f a i t h f u l f l a t n e s s Moreover, A z is c o n t a i n e d in t h e fixed object A o f all Od2)

30 chap 0

oe G If A w e r e strictly larger than A 2, then the inclusion t: A 2 -~ A w o u l d not be

an isomorphism, contradicting the fact that S®t: S ® R A 2 ~ S ® R A - S ® a A 2 is an

i s o m o r p h i s m by 7.1 Since A t is fixed under all ~t>, our hypothesis implies that

g ( A I} is fixed under all ~ 0 ~2), hence contained in A 2 Let f glAv This gives a well-defined R - m o r p h i s m f: A I -~ A2, and S ® f = g

A s an application, w e prove:

Propomltion 7.3 Let S / R be a G-Galois extension of connected rings Then A u t ( S / R )

is equal to G

Proof W e s h a l l u s e t h e S - i s o m o r p h i s m h: S® a S - - - S (~) N o w S® R S c a r r i e s t h e trivial d e s c e n t d a t u m ( ~ o ) = (o® i d s ) a e ~, and d e s c e n d i n g a l o n g (~Po) gives b a c k S,

as we have s e e n in t h e p r o o f o f 7.2 Let us t r a n s p o r t ( ~ o ) via h t o a d e c e n t d a t u m ( ~ o ' ) on S ~a) A s h o r t c a l c u l a t i o n s h o w s q ~ o ' ( ( x ~ ) t ~ a ) = (o(xo-l~))~e a-

c a l l y isomorphic to Zp, the sequence (o)ne N corresponding to l ~ Z p

L e t R b e a c o m m u t a t i v e ring The group Hi R, Zp) o f Zp-extensions o f R

is d e f i n e d as t h e p r o j e c t i v e limit o f t h e s y s t e m

" n~ H(R, Cp2) n : H(R, Cp)

Thus, the e l e m e n t s o f H(R, Zp) are s e q u e n c e s ( A ) h e ~ with A ~ H(R, Cpn) and

~n*(An+l) = A n f o r all neIN Such s e q u e n c e s are also c a l l e d coherent, or towers

* A

l ~ m a r k Since n ( a÷l ) is j u s t t h e fixed ring o f Ker(Tr a) in An l ( b e c a u s e ~ is

o n t o ! ) , one may r e g a r d A as a subring o f A + t f o r all n, and it m a k e s s e n s e t o

Define x(x) t o be ~n(x) This is i n d e p e n d e n t o f t h e choice o f n, t h a n k s t o the

c o m p a t i b i l i t y c o n d i t i o n n~(A,÷ 1) = A (If Ao~ is c o n n e c t e d , one can even s h o w

t h a t Aut(Aoo/R) - lin~_ Cpa, cf 7.3.) This justifies t h e t e r m i n o l o g y " Z p - e x t e n s i o n "

Since projective limits p r e s e r v e m o n o m o r p h i s m s , t h e g r o u p NB(R, Zp) is a

s u b g r o u p o f H(R, Zp) We t h u s may define P(R, Zp) = H(R, Zp)/NB(R, Zp) Caution:

it is n o t c l e a r w h e t h e r P(R,Z v) - lirn(_P(R, Cp~) If all g r o u p s NB(R, C p ) happen

t o be finite, t h e n this i s o m o r p h i s m does hold, since t h e derived f u n c t o r lirn~ 1~ is zero on ( NB( R, Cp n))n Cf J e n s e n (1972)

The t o p o l o g y on Zp - lim<_ Cp is the p r o f i n i t e t o p o l o g y , and induced f r o m the p r o d u c t t o p o l o g y on ] - I ~ Cp In the c o n t e x t o f §3, we t h e n have the f o l - lowing r e s u l t :

I.¢.,mm 8.t- For connected rings R, there is a canonical isomorphism H(R, Zp) *

H o m c o t ( D a , Z p) (which is the same as Homcont(ur'a.Zp) by Cor 3.9)