The theorem of the cube for principal homogeneous spaces

J.

Math. Kyoto Univ. (JMKYAZ)

12-1 (1972) 1-15

The theorem of the cube for principal homogeneous spaces By

Masayoshi MIYANISHI (Received, February 4, 1971)

O . T h e statement o f th e theorem.

S , (i= 1, 2 , 3 ) b e a proper flat S-prescheme o f finite presentation such that f i has a section e i and that fi*(ex,)-:11 "— =C s uniS-group prescheme o f finite preversa lly. Let G be a flat commutative sentation. For any subset I o f {1, 2, 3 }, w e den ote by X l= H X i the ie f fibre product o f X i , i E I , b y s1 the immersion X j--X {1,2 ,3} defined by fo r i E I and e • f o r i E {1, 2, 3} —I and by s . the immersion Xj --.7(1 defined by id x 1 fo r j E J and ei f o r j E I— J if .1( I. Let f i :

j a

A trivialization o f a Gx ,1,2 ,3 ,-torsor E with respect to e i , (i= 1, 2, 3) is a set of isomorphisms ce s l ( E ) - - G x fo r any subset I o f {1, 2, 3} such that for J ( I ,

4, (ceD=aj. 1

T h e set o f isomorphism classes of

trivializable Gx,,,,. 3 ,-torsors forms an abelian group which is denoted by X X 2 X X 3 , G). s We shall sprove the following

(i= 1, 2, 3) an d G be X 3 , G).= 0 i f G satisfies moreover

T h e theorem o f t h e c u b e . L e t f i :

as ab o v e. T h e n PH ( e i ,,,, e 3 ) (X 1 x X2 one of the f ollow ing conditions: (1) (2)

s

X

s

G is affine an d sm ooth ov er S. G is f inite and f lat o v e r S.

M a sa yosh i M iya n ish i

2

( 3 ) G is a n a b elia n schem e, S is quasi-com pact an d norm al and

f ( i = 1, 2, 3) are geometrically normal. I f G is the multiplicative group prescheme, G„,,5, this theorem is the ordinary theorem o f th e cu b e (cf. [1 ], [3 ], [6 1).

The notation and

definitions are those o f [4 ] a n d [5 ]. T h e co h o m o lo g ies should be understood to be (f.p.q.c.)-cohomologies unless otherwise mentioned.

1 . T h e fo rm a l non ram ifiedness o f t h e functor C orr,(X , X ). -

-

Let

1

2

f i : X —*S (i= 1 , 2 ) b e a proper S-prescheme such that f , has i

a section ei an d that fi* (0 x ,),

univrsally and let G be a commuta-

tive affine flat S-group prescheme o f finite presentation. W e d e fin e a (f.p.q.c.)-sheaf of abelian groups, Corr

.(2( ., X ) , on the site (Sch/S) , 1

2

p

b y the following split exact sequence,

Pr;-Fprl

-

PH (X ix X 2 / S, G)

PH(X 1 / S, G) x P H (X 2 / S, G )

O

-

>Corr,q(X , -

i

.

X2 )

Corr.g(X i , X 2 ) is called the functor of divisorial correspondences of type

G between X a n d X 2 and satisfies the following properties; 1

(1)

X i), w h e r e on the shoulders

Corr,(X1, X2) x

denote the base change b y S'—>S. (2)

Corr,(X1, X2)=—Corr

,Ç(X , X ) ( S ) is i

2

a d i r e c t summand

o f P H ( X j. x X 2 / S, G)=-ciP fH(X i x X2 / S , G )(S ) w i t h t h e complement def

PH(X i / S, G)(1) PH (X 2 / S, G). First of all, we shall prove

1. C orrg, (X , X ) is f o rm ally n o n -ram if ie d if G i s a sm ooth a ffin e com m utativ e S -g ro u p p resch em e o f f inite presentation and f i or f 2 is flat. Lem m a

1

2

P r o o f. W e m a y assume that S is affine and that

f i s flat. L e t i

Sz-----S p e c (A ), le t I be a square zero ideal o f A an d let S =S pec(A //).

The theorem of the cube

W e have to show th a t th e canonical morphism obtained from the base

change b y S >S, —

i :

X2)-->COrr(X1) X2)

COTT

is injective. L e t $ b e an element of C orr,(X , X ) su c h th a t i(e )= 0 . By definition, $ is representable by a Gx ,xx,-torsor E such that st(E ) i

2

(resp. 4 (E )) is a triv ia l Gx, (resp. Gx )-torsor and th a t E x S is also 2

a trivial Gy 1x x -torsor. 2

Then we should prove that E is itse lf a trivial

Gx, X x2 -torsor. Consider the following diagram, X2

w here g i a n d g2 are canonical projections and w h ere h = f ig i= f 2 g 2 . If g " is a quasi-coherent Ox „x -Module, the Leray spectral sequence for 2

1

f g g iv e s an exact sequence,

the composite morphism 3

1 -

-

-

i

i

> k f i * ( g i * . F ) - >R 1 h*(F) --

-

-

-

>fi*R g i* (F). i

If .F = h * g for some quasi-coherent e s -Module g , this sequence becomes >R 1 f1 *(f ig ) - - - - - >R 1 h *(0 g ) -

0

-

R i f2 * (f

where we used the flat base change theorem for f (cf. EGA, III (1.4.15)). Sin ce S is affine, this sequence is equal to an sequence, 2

0

-

>111 (Xi, f t g )

-

>I1-1 (X1 x X 2 , h*g)---).11'(X 2 , f i g ) .

Moreover this sequence splits because X a n d X2 have sections from S. 1

O n the other hand, w e h ave th e following commutative diagram from Lemma 2 below:

Masayoshi Miyanishi

4 0— > H (.2f1 L ie GcO lex s

111(X1, G)

1

C)

s* /r (X j. x X 2 , G)-- 1

x X 2 , Lie GO /0x, „x 2)-

1-11(ffi X 1 2 , C)

- 1

Cs

S2*

0--->H (X 2 , Lie GO /e x ) 1

i 2

H 1- (X 2 G )

>1/1 (X 2 , G ) i

s

the lines are exact an d th e left column splits, e defines an element $' o f 1/ (X 1 x X 2 , G ) such that i i ,2 (n = 0 and $ = 0 if and only if $'=0. q. e. d. Then the diagram chasing shows that $=0. where

1

Lemma 2 . L et G be a smooth affine commutative S-group prescheme,

' S be a S-prescheme quasi-com pact and quasi-separated ov er S an d le t S be a closed subprescheme def ined by a square-z ero Ideal .fir of Cs . T h e n w e hav e an ex act sequence. let f : X

—

0 ---> f * (Lie G O i r Ox) -

- - -

>f*(G) -

-

+f*(C) -

-

g

R f * (Lie GO Ste x )--->R V V G )--->R l

1

*

>

(G).

If f * (0 x ):"= ' s univ ersally , then f ( G ) > J ( G ) i s su rje c tiv e . Moreover, if S i s affine, w e hav e an ex act sequence, -

.

0- - > H (X, Lie GC)..f0 x )--->H 1 (X , G)- - > R 1 (X , G). 1

Os

P r o o f . W e shall

show that if S is affine, w e have an exact sequ-

ence,

0- - > (X , Lie G ® .1 0 x )-- - >G ( X ) - - - G ( X ) - - > Os H 1 (X , Lie GO .10 x )--->H 1 (X , G) - - > H 1 (k , G). os

The first exact sequence is obtained by localizing the above sequence. A n element $ of 11 (X , G ) can be given by a Cech-cocycle. Since 1

5

T he theorem of the cube

w e are dealing with th e (f. p. q. c.)-topology, E is given by a 6ech-coc y cle

giJ E G (U u) for U = { Ui } E Cov(X ), where Ui is an affine scheme which is faithfully flat over a n affine open set V i o f X, V V 1 =-X and where (A i = U 1

xUp

The im age of e in H (X, G ) is zero if and only if { i

is a C ech-coboundary. Then replacing II by a finer open cover of X, we may assume that there exists hi E G(C7i ) fo r a ll i such that gu =

on Chi for all id. L e t Ui -= Spec (A ) a n d le t rii =Spec (il i //i), I being a square-zero ideal of A . S i n c e G is smooth over S , there exists h 1 EG (U 1 ) for all i such that hi =h i modulo I . Let g i = g — h i d - h 1 . T h e n e u = 0 modulo h. Now we shall use the following i

Sublem m a. L e t G b e an af f ine sm ooth T -preschem e an d le t T be

a closed subprescheme o f T defined by a square-zero Ideal .fi" o f O T . Then w e hav e the f ollow ing ex act sequence, 0- - >T ( T , Lie GO 5 )--> G (T )---> G (1 ).

cr

a quasi-coherent CT-Algebra d and d is th e d irect sum of O r and the augmentation Ideal f , e., 0 7 , C V . L e t g b e an element o f G( T) such that g =0 modulo 5 . L e t g be defined by a n CT-Algebra homomorphism : d—*9T. T h e n go sends to 5 since the composite homomorphism OT — Z 'T factors through s i f / / . Since 5 is square-zero, go f def in e s a n 07-Module homomorphism Conversely, if i/ o f : f / j r — .5r is any CT-Module homomorphism, we can construct an CT -Algebra homomorphism composed with b y Cfil r = id o r a n d Cfil, f = the canonical projection jr — > j/ j . Then it is easy to see that = and $ -= 0. On the other hand, Hom o ( i / f 1 J O HOM O rC itl , ( T ) =T ( T , Lie G ® 5 ) sin ce / L e is locally free CT-Module. P r o o f . Since G is affine o v e r T , G is given by

o

o

2

a

r

.2

2

r

2

(, 7

q. e.d.

6

Masayoshi Miyanishi N ow w e shall g o back to th e proof o f Lemma 2.

From th e sublem m a, there exists a n element vu of T ( U , L ie GO f e u , i ) determined Os uniquely by 6 . T h e n v u is a Cech-cocycle o f C (U , Lie Gas/Ox), -

Os

hence defines a n element C o f H ( X, Lie G O .F0x) w hich goes to E. 1

Os

E is a Cech-coboundary if an d only if C comes from an element of g e G (g ) and let { Vi l be a n affine open cover of X . T hen gly , comes from g i of G(V 1), since G is sm ooth over S. T h en fo r a n y i, j, g i — gi corresponds w ith an

C (X ) b y th e following morphism 6 : L e t

elem ent v u o f F( Vu , Lie GOE5 0 x ) and I v u l is a Cech-cocycle of C ( , Os Lie G O .s/ex). Hence fo u l defines an element C of H ( X, Lie G(D.SI0x). Oso s Then 6 is a morphism which sends g to C. If E defines a n element C '

1

1

which comes from g EG(X ) b y th e morphism 6 above, we can see easily fro m th e definition that E is a Cech-coboundary. Conversely, if Ç is a Cech-coboundary, replacing 2I by finer cover, th ere ex ist g ' i EG (U i ) for a ll i , w h ich is in turn coming from T (U i , Lie (G )® J 9x), such that '

5-

Os g ';)= 0 modulo I,

g i.i= e i g ; o n U15 f o r a ll i, j. Since g u(= g i g i fo r a ll i, j. Hence {g' i } defines a n elem ent g ' of G M , w hich is easily seen to giv e C b y 6 . H ere w e note th at 1-Pi (X, Lie G®5(9x) i

—

l

i

i

-

g

Os

H L ,(X , Lie G o fo x ). g com es from a n elem ent o f G (X ) if and os only if 6( g)= O. T he remaining parts follows from th e sublemma. If f * ( e s universally, G(X)—>C(X) is surjective since G(X )= Horn s (Spec( f * 0 x ), G ), C ( X ) Hom3(Spec(f * ex), G ) a n d sin ce Spec (f*(

)) is a closed subprescheme of Spec(f * (e x )) defined by a square-

q. e. d.

zero Ideal. Lemma 3 . L et G, X 1 a n d X 2 b e a s i n L em m a 1.

T h e n the unit

section e o f Corr,(X 1, X 2 ) is representable by an open im m ersion. P r o o f . L e t T b e a n y S-prescheme an d let Z2- =( T, a ) x ( S , e ) Cor :§(Xl, X 2) ,

T--C orr,(X i, X 2). W e have to prove that fo r a n y S-morphism N am ely, if t is a p o in t o f T such that a(t) an open s e t o f T. ZT is

7

The theorem of the cube

,)-7=

= 0, then Spec (Or, t) E ZT * ) .

Z1- x Spec (CT, ) and t E Since ZS pe c(CT, L e t A = OT, . Then a Z sp e c(Or , ,), w e m a y replace T by Spec (CT, ). defines a n element $ o f Corr,NX , T, X2, T). F in a lly w e m a y assume th a t T = S . B y (f p. q. c.)-descent, w e m ay replace A by its completion w ith respect to its maximal ideal nt. In fact, if g is S p ec ( 4 ) , the t

t

t

i

morphism Corr,ç(Xi, X2)(S)---Corr,(Xi, X2) (S')

is injective because :S" is fa ith fu lly flat and quasi-compact o v e r S and C orr, (X i, X 2 ) i s a (f. p. q. c.)-sheaf. L e t A = A/Inn a n d l e t Sn= T hen by virtue of Lemma 1, the canonical morphism S p e c (A .). .

-1-1

n

COrll Z(Xl , n 3

X 2,

n)

-

÷ Corr (X13 X2)

i s injective, w h e re S -= Spec (A / n t) a n d C =G x s S • S i n c e t i s zero, E„ = $ modulo (at') i s z e r o . $ is representable uniquely up to isom orphism s by a Gx, x -torsor x

E such that s tE and . 1 E are trivial.

The sections 6„ : ( X 1 X

2

Then E = lim E x S n i s trivial. S

E x S n w h ic h tr iv ia liz e E x S c a n b e n

chosen so that the following diagram is commutative for a n y n > m, X Sn

(X 1X

(X x X2)m i

Then there exists a section 6 : Therefore E is trivial.

X 1X

E x S.

by virtue of EGA, III (5. 4. 1.).

Thus S p e c (A )( Z.q

L e m m a 4 . Let f i : such that f i has a section

.

e

.

d

.

S (i= 1, 2, 3) be a proper flat S-prescheme

a n d t h a t f i* (e x ) 0 s univ ersally and let G be a smooth affine commutative S-group prescheme of finite presentae i

i

-

I ') In fact, C orrg(X i , X 2 ) is a functor of finite presentation since PH-functors are so and Corrfsl( X i , X 2 ) is a direct summand of a PH -fu nctor, (cf. [4] or SG A D , E xp V / , (10. 16)). Th en the fact that Spec (0 2% 0 c Z T implies that there exists an affine open set U o f t such that UOEZT. (

Masayoshi Miyanishi

8

tion.

T hen any S -m orphism f: X -->Corr,q(X , X ) w hich sends the sec1

3

2

tion e t o the unit section e of Corrg(X , X ) factors through the unit section, i.e., f = e •f . i

3

2

3

P r o o f . L et Z =(X 3 , f ) x (S, e).

T hen by L em m a 3 , Z i s an

C orr s ( X I, X 2) ( ;

open subprescheme o f X 3 which contains e 3 ( S ) . To complete the proof,

w e have to show th a t Z = X . I f Z * X 3 , take any closed point x of X3 — Z and let s = f 3 ( x ) . Then f : X3, 4 C o r tl( )( Xi, „ X , ) does not Therefore we are refactor through the unit section e of the latter. 3

5

duced to consider the case w here S

-

s

2

5

Spec (k ), w h ere k is a field.

By

(f.p.q.c.)-descent, w e m a y assume th a t k is alg eb raically clo sed . I f G

is connected, C orrg(X i, X 2) is rep resen tab le b y a S-group prescheme locally o f finite type o ver S. In fact, Corrg(Xl, X2) i s th e kernel of th e S-homomorphism (sr, st) : PH (X x X / S, G).— PH (X / S, G) x PH 1

2

1

(X / S, G ), where PH(T / S , G), T =X x X , X o r X 2 is representable 1

2

2

1

b y a S-group prescheme lo cally o f fin ite type over S. I f G is etale, C o rr,(X i , X 2) is rep resen tab le b y a n etale S-group prescheme. (For these results, see [41.) In general, G h a s a connected component G o such G/G o t h a t is etale and satisfies the following exact sequence, 0

-

-

-

)•COrr,T a i ,

X2) —*Corr.qIG°(X15 X2))

j

° whence th e connected component C o r r (X 1., X2) of the unit section of ° C o r r ( X , X 2 ) is rep resen tab le a n d co in cid es w ith Corrg(Xi, X2) . ° Therefore C o rr,(X 1 , X 2 ) is se p a ra te d o v e r S. T h en the unit section -

i

e is a closed im m ersion . Then Z i s a closed and open subprescheme o f X . However since f3 * (0 x ) 1-: es, X 3 is connected by Zariski's conq. e. d. nectedness theorem. Therefore X3=Z. 3

.

-

2 . The p ro o f o f the theorem. The first case. L et E b e a

Gx ,, , ,-torsor representing a n elem ent o f PH ,,,,,, 1

(

2 3

( X X X2 X X 3 / S ,G ). Then E defines a S-morphism 1

s

S

: X 3 — > C orrg(X 1, X2)

e3)

9

T he theorem of the cube

which sends the section e t o the unit section e o f Corr.(X1, X2). T h e n e factors through the unit section e by virtue o f Lemma 4. Moreover E considered as a G(x, x )x(X xx ) torsor defines a n element 3

x3

s

s2

3

3

-

o f PH((X i. x X 3 ) x (X2 X X 3 )/ X3 , G ) w h ich is in turn isomorphic to X,

S

the direct sum,

PH(X i x X3/ X3, G)OPH(X2x X3 / X3, G scorrg(Xi, x2)(x3). )

The components o f 77 by this decomposition a re st 3 (E), s

which are a ll zero.

Hence

is z e ro . Then E is trivial.

3

( E ) and e q. e. d.

A s this proof shows, if Corr,g(Xi, X2) = 0, the proof of the theo-

rem becomes almost trivial. T h e following result shows that the difficulty of the proof of the theorem comes from the torus part of G. X 1 —* S(i =1, 2 ) b e a proper f l a t S-pres-

Proposition 5 . Let

cheme o f f inite presentation such that f i h a s a section e fi* (X iY :=

1

a n d that

e s an d le t G b e a smooth affine commutative S-group pres-

cheme of finite presentation.

Suppose th at the semi-simple rank of G is

z ero at every Point o f S . T hen C o r r (X i , X 2 ) =-0.

It is sufficient to prove that Corr (X , X ) = O. W e m a y assume th at S is affine, S = Spec(A ). Since Corr,q(Xi , X ) is a functor locally o f finite presentation (cf. [4 1 ), w e m ay assume that A is a local r in g . By (f. p. q. c.)-descent, we can replace A by its completion  with respect to the maximal ideal ni. L e t k= A / in and let Spec (k). Suppose we have shown that Corr sG (X i ,„ X 2 , , ) = 0 . Then by Lemma 1, C orr(X i,n , X ,) = O, whence one deduces Corr,g (Xi, X2) = 0, using the argument of the proof o f Lemma 3. Now we shall show b y induction on the unipotent rank o f G , that C o r r ( X i ,„ X2 ) = O. By (f. p. q. c.)-descent, we m ay assume that k is perfect. Then G , has a composition series, P ro o f.

1

2

2

2

8

0

=G o

C

•

•

•

G,, G ,

Masayoshi Miyanishi

10

Ga , k the additive group prescheme over k for i = 0,

such that Gi, i/Gi n — 1.

For an exact sequence, 0--->G -- > G exact sequence of group functors, 1

0— > Corr Gs (X1 1

---> C o rr(X i

, s5

X2 , s ) -

- - -

0

i + 1

> COrr Gs i +I (XL

,

w e have an

s) X2, s)

s X 2 , )• )

Therefore if Corr .(X ,„ X , ) = 0, w e are done by induction on n. In the case w h ere G, -=Ga , Corr.NXi, s, X2, s) = PH (X1, x X2, 5/s) Ga)/ 6 .9

2

i

5

-

PH (X i, WS, G ) x PH (X 2 , /s,L i e ( P i c (X a

5

X Lie (Pic ( X2, s)) =

0

(c f.

1

,

x X2, ))/L ie s

(Pic (X1, s))

Da.

q. e. d.

Corollary 6 . L et G, X 1 a n d X 2 be as in Proposition 5.

Then we have

PH (X / S , G) x PH(X / S, 1

2

PH(X i x X / S , G). 2

Therefore

11 (X x X , G)1" --=-' 1

1

G)( D H (X2, G)/H (S , G), 1

2

1

where H 1 (S , G ) is considered a s a subgroup of H i (X i, G)C)11 -1 (X2, G) by the injectiv e homomorphism E— qf

— ftE).

P r o o f . O bvious by definition. See [41

and [51.

Corollary 7 . L et G be as in Proposition 5 and let A be an abelian

scheme over S.

Then we have

H 1 (A ,

Extis_ g r (A, G)C111 1 (S , G).

Let f A and eA b e the structure morphism and the unit section of A resp ectively. T h en w e have, P ro o f.

11

The theorem of the cube

PH (A / S, G),0 H l (S , G).

H 1 (A ,

Exes_gr(A, G). T h e re fo re it is s u ffic ie n t to show th a t PH(A/S, T ake any elem ent C o f PH(A / S, G). Ç is representable by a GA-torsor E su ch th at e'IlE i s trivial. L e t 7r b e the multiplication of A . Then the GA.A-torsor 6(E)=7 -c*E— prIE— prE is trivial since C o ril(A , A )= 0 fro m Proposition 5. T h e n E h a s a structure of com m utative group S-prescheme w ith a section of e'AE as unit section E and i s an extension of A b y G (cf. [21, (1.3.5.)). The extension class o f E is deter-

m ined uniquely by Ç. S e n d in g Ç to the extension class of E , one can define a homomorphism 0 w h ic h is the inverse of the canonical homoExtls _„(A , G)---->PH(A/ S, G).

morphism 3 . The

q. e. d.

proof o f th e th e o re m . The second case.

Let x , s ( i = i , 2 ) b e a proper flat S-prescheme o f finite presentation such that f h a s a section e an d that f i * ((9 x)= C s universally , let G be a fin ite flat commutative S-group prescheme of finite presentation and let D (G ) b e its Cartier dual. W e sh all recall the following i

i

Lem m a 8 .

5 (

[

i

C o r r (X 1 , X 2 )..1._' Horn s_ r (D (G ), Corr

11).

g

(X 1 , X2)).

Lem m a 9 . C o r r , ( X 1, X2) is f orm ally non-ramified. -

P r o o f . L e t S b e an affine schem e a n d le t S b e a closed subschem e o f d e fi S ned b y a square-zero ideal. W e have only to show th at the canonical morphism C o r r (X i,

X2) - >

Corr Gs (X i , X 2 )

is injective. T his follows from the commutativity of the diagram, H om s_„(D (G ), C orr s (X i , X2)) --->lIom s, _„(D (C ), C o rr ,,(Xi., X2)) -

f corrs(X„ x2)(D(G))--->Corr(X1, X 2)(D(G)),

12

M asayoshi M iyanishi

where Corr (X , X ) = C o rrg . (X , X2). s

i

2

,

q. e.d.

i

Lem m a 1 0 . T h e u n it se c tio n e o f C o rr (X i , X 2) is representable

by an open an d closed immersion.

It is sufficient to show that i f u: D(G)— >Corr s (X i , X 2 ) is an y homomorphism o f S-groups and H is the kernel o f u , then the set Z = Is E S; H s =D (G ),} is an open and closed set of S. However since the unit section of Corr (XI, X ) is representable b y an open and closed immersion (cf. [1 1 ), H i s an open and closed subgroup prescheme o f D(G), hence it is finite and flat. Then the rank o f each fibre of H is lo cally constant, whence the required result folq. e. d. lows easily. P ro o f.

s

2

Now Lemma 4 is an easy consequence of Lemma 1 0 if G is understood a finite flat commutative S-group prescheme o f finite presentation in Lemma 4. Then one can prove the second case of the theorem following word for word the proof for the first case. Proposition 1 1 . L et G, X 1 a n d X 2 be as ab o v e . I f both f

f

2

i

an d

a re geom etrically norm al, C o r r (X i , X ) = 0. 2

Since C orrg(X l, 2(2 ) is a (f.p.q.c.)-sheaf, w e have only to show that C orrg(X i, X 2)= 0 i f S is an affine scheme. Moreover since C oril(X i , X 2 ) is a functor locally of finite presentation over S (cf. [41), w e m ay assume that the affine ring A o f S is a local rin g. We may replace A by its com pletion with respect to the maximal ideal m . I f w e could prove that Corr,(X1, X )(k )= 0 , w here k = A /m , the proof w ill b e co m p leted, usin g th e argu m en t of th e proof o f Lemma 3. Therefore we shall show th a t Corrg(X , X ) = 0 i f S i s the spectrum In th is case o f a fie ld k. W e m a y assume k algebraically closed. C orrs(X i, X ) = Homk_„(Alb (X ), Pie9 k ) w h ic h is torsion free (cf. DJ, P. 1 5 5 ) . Then Coril(Xi, X2)=Hom1_ r(D(G), Corr s (X l , X2))= 0. q.e.d. P ro o f.

2

i

2

i

2

r2 /

g

T he theorem of the cube

13

Corollary 1 2 . L e t G , X i. and 2 ( 2 be as in Proposition 11.

Then

S, G),CDPH(X 2 / S , G).

PH(X i x X 2 / S, Therefore /P (X l X X 2,

G)a...- H 1 (X 1 , G)(DH 1 (X 2, G)/H 1 (S , G)

w here IP(S , G ) is considered a s a subgroup of H 1 (X 1, G )0H 1 (X2, G) by the injective homomorphism defined as in Corollary 6. Corollary 1 3 . L et G be as above an d le t A be an abelian scheme

ov er S.

Then --= ' Ex t _g r (A, G)011 1 (S, G). H 1 (A, G):_s -

P ro o f.

The same reasoning as for Corollary 7.

4 . The p roof of the th eo rem . The th ird case. We shall prove the following Proposition 14. L e t S b e a quasi-compact normal prescheme, A

be an abelian schem e ov er S an d le t f : X 1 ->S (i= 1 , 2 ) be a proper flat i

geometrically normal S-prescheme such that f

h as a section e i an d that

f i*(0 x i )- = s universally. L et Corrs (X 1 , X 2 ) b e th e se t o f all isom orp h ism classes of A x sx x2 -to rso r E such that $ 0 such that ne = O. Consider an exact sequence of (f. p. q. c.)-sheaves, j.

2

M asayoshi M iyanishi

14

where „A is a finite flat commutative S-group prescheme. Denote by A ( T ), H ( T , „A ) and I-4(T , A ) the kernels of A ( T ) - H 1(S , A ) respectively, A (S ), H 1 ( T „A ) H 1 ( S , ,A ) an d H ( T, w h ere T should be replaced by X i, X 2 or X1 X X 2 and w here e * is the homomorphism canonically deduced from e a n d e . T hen w e have the 0

i

2

l

following commutative diagram.

I

I

11 ---L--*Hô(X i, „A)

A0(X1)

s4 s

I

Ao(x, x x2)

I

p

r

H (x1 x

>H10(X1, A )

1st

t X 2 , nA )-

$.1 ïpr*2s

A)

.11(.X 1 X X 2 , A )

- - J/ 10. (X 1 X X 2 , A ) *

2

In (X 2 , „A ) ---11 --*H 1- (X 2, A ) n >- .1 1 1,-(X , A) 2

0

t o

w here the lines are exact and two left columns are sp lit exact. i i 2 ( ) fo r som e elem ent 77 o f H 0 (X 1 x X , A). S in ce ne-=0,

Let 7 O. T h en ii(V1)==eiii2(72)— O. A ls o 77 =-- s i(V ) an d 772 = 4(77)• Therefore 771= j i ( C i ) and 772 = j2(C2). Put C = p rt (C )-1-prt(C ). Then q.e. d. j12(C)=- 17. Hence e = O. T h u s C orrt(X i, X ) =0. 1-

2

R

s

i2 (7 7 2 )—

1

2

2

Corollary 1 5 . L e t S a n d f i ( i = 1 , 2 , 3 ) be a s in th e statement

of the theorem and let A be an abelian scheme over S. Suppose moreo-

v er that S is a quasi-compact normal preschem e and that f ( i = 1 , 2, 3) is geometrically n o rm a l. Then i

P H (e,,e2,e3)(X 1 X X 2 X X 3,

P r o o f . Easy from

Proposition 14.

A )=0 .

15

T he theorem of the cube

Corollary 1 6 . L e t S be a quasi-com pact regular preschem e and let

A an d B be abelian schem es ov er S.

Then

11 (B , A )=H (B, A )„,27_'- Ex tls _g r (B , A ) 0 11 (S , A ). 1

,

1

1

P r o o f . T h e first isomorphism is d u e to M . R ayn au d ([6 ]). T h e s e c o n d isomorphism is proved a s i n Corollary 7 a n d Corollary 13, using

Corollary 15 a n d [ 2 ] , Exp. VII, (1.3.5). UNIVERSITY OF BRITISH COLUMBIA AND KYOTO UNIVERSITY

Bibliography [1] [2] [3] [4] [5] [6]

M. Demazure, J. Giraud and M. Raynaud, Schémas abéliens, Séminaire d'Orsay, 1967/68. A. Grothendieck, Biextensions de faisceaux de groupes, S G A 7, Exp. VII, Mimeographed note from I. H. E. S. S . L a n g , Abelian varieties, Interscience publishers. M . M iyanishi, O n the cohom ologies of com m utative affine group schemes, J. M ath. Kyoto Univ., 8 (1968), 1-39. , S u r la prem ière cohom ologie d'un préschém a affine en groupes commutatifs, Jap. J. Math., 38 (1969), 51-60. M. Raynaud, Faisceaux am ples sur les schém as en groupes et les espaces homogènes, Lecture Notes, Springer, 1970.