1976 Queen's at the Forms

a l g e b r a i c _ y a x i e t i e s

M a n f r e d K n e b u s c h

L e c t u r e s a t t h e Q u a d r a t i c Forms C o n f e r e n c e Queen's U n i v e r s i t y , K i n g s t o n / O n t a r i o

A u g u s t 1976

P r e f a c e

A r r o u n d 1964 C a s s e l s and P f i s t e r i n i t i a t e d a t h e o r y o f q u a d r a t i c f o r m s o v e r f u n c t i o n f i e l d s i n a r b i t r a r y many v a r i a b l e s c f . t h e l a s t c h a p t e r o f Lam's K i n g s t o n l e c t u r e s . The d i s c o v e r y t h a t i n t e r e s t i n g t h e o r e m s can be p r o v e d f o r q u a d r a t i c f o r m s over- s u c h c o m p l i c a t e d f i e l d s had a s t r o n g a p p e a l t o many a l g e b r a i s t s , and i n t h e l a s t 12 y e a r s t h e t h e o r y o f q u a d r a t i c f o r m s o v e r

f i e l d s has b e e n pushed f o r w a r d c o n s i d e r a b l y . B u t d e s p i t e t h e r e m a r k a b l e p r o g r e s s and t h e s o m e t i m e s ' v e r y i n g e n i o u s arguments t h i s a l g e b r a i c t h e o r y of q u a d r a t i c f o r m s o v e r f i e l d s i s t o my o p i n i o n n o t a f u l l y adequate r e s p o n s e t o t h e g e n e r a l hope o r b e l i e f t h a t a r i c h and i n t e r e s t i n g t h e o r y o f q u a d r a t i c f o r m s o v e r f u n c t i o n f i e l d s i s p o s s i b l e . There s h o u l d e x i s t an " a l g e - b r a i c g e o m e t r y of q u a d r a t i c f o r m s " t o u n d e r s t a n d q u a d r a t i c f o r m s o v e r f u n c t i o n f i e l d s i n much t h e same way as a l g e b r a i c g e o m t r y i s known t o be n e e d e d t o u n d e r s t a n d f u n c t i o n f i e l d s t h e m s e l v e s .

F o r t h i s a l g e b r a i c g e o m e t r y o f q u a d r a t i c f o r m s t h e c l a s s i - c a l l a n g u a g e of a l g e b r a i c g e o m e t r y i s u n s u i t e d . I n d e e d , t h e c l a s s i c a l l a n g u a g e assumes t h e p r e s e n c e of an a l g e b r a i c a l l y c l o s e d b a s e f i e l d . T h i s w o u l d s p o i l many i n t e r e s t i n g phenomena of q u a d r a t i c f o r m s , s i n c e -1 w o u l d be a s q u a r e i n e v e r y f u n c t i o n r i n g . Thus we a r e u r g e d t o use G r o t h e n d i e c k¹s l a n g u a g e o f scheme

We t h e n s h o u l d a l s o a d m i t schemes X w i t h 2 n o t a u n i t i n t h e r i n g $ ( X ) o f g l o b a l f u n c t i o n s . T h i s means t h a t we have t o d e v e l o p two t h e o r i e s , one f o r s y m m e t r i c b i l i n e a r f o r m s and one

f o r t r u l y q u a d r a t i c f o r m s , and t h a t we have t o s t u d y a l s o t h e r e l a t i o n s b e t w e e n t h e s e t h e o r i e s ( e . g . t e n s o r p r o d u c t s o f

s y m m e t r i c b i l i n e a r f o r m s w i t h q u a d r a t i c f o r m s , c f . ! MI,p.111 j f o r X a f f i n e ) .

Around 196? I s t a r t e d - n o t a l w a y s i n t h e r i g h t way J - s u c h a t h e o r y f o r s y m m e t r i c b i l i n e a r f o r m s o v e r schemes [ K ] . S i n c e t h e n good p r o g r e s s h a s b e e n made i n t h e l o c a l p a r t o f t h i s t h e o r y ( s y m m e t r i c b i l i n e a r f o r m s o v e r l o c a l r i n g s ) . B u t on t h e g l o b a l s i d e o n l y f e w c o n c r e t e r e s u l t s have b e e n o b t a i n e d .

The r e a s o n f o r t h i s j u s t seems t o be t h a t n o t t o o many m a t h e m a t i c i a n s h a v e s p e n t much e f f o r t i n t h i s a r e a . F o r example

up t o now nobody seems t o have computed t h e W i t t r i n g o f t h e n p

s p h e r e s Spec R [ x .. • • , x ^ ] / ( 53 x- - 1) o v e r t h e f i e l d P o f r e a l o n 0

numbers. I n r e c e n t y e a r s many p a p e r s have b e e n w r i t t e n on h e r m i t i a n and q u a d r a t i c f o r m s o v e r r i n g s w i t h i n v o l u t i o n , a

s u b j e c t w h i c h v i r t u a l l y i n c l u d e s t h e a f f i n e p a r t o f o u r t h e o r y . B u t most o f t e n t h e a u t h o r s have b e e n l e d b y a t o t a l l v d i f f e r e n t m o t i v a t i o n , f o r example s u r g e r y t h e o r y .

The a i m o f t h e s e l e c t u r e s i s t o s t i m u l a t e i n t e r e s t i n t h e g l o b a l t h e o r y o f s y m m e t r i c b i l i n e a r f o r m s o v e r schemes. Q u a d r a t i c f o r m s a r e i m p o r t a n t as w e l l b u t have n o t b e e n i n c l u d e d f o r l a c k o f t i m e . I t i s r e a s o n a b l e t o s t u d y i n s t e a d o f s y m m e t r i c b i l i n e a r

*) r e f e r e n c e s a t t h e end o f t h e l e c t u r e s .

**) I n [ K ] t h e d e f i n i t i o n o f t h e W i t t r i n g W(X) i s n o t t h e r i g h t one i f X i s n o t a f f i n e .

forrns more g e n e r a l l y h e r m i t i a n f o r m ? o v e r schemes w i t h i n - v o l u t i o n ( c f . e.g. t h e i n t r o d u c t i o n i n T K ^ ] ) , b u t I r e f r a i n e c ' f r o m t h i s e n l a r g e m e n t of t h e t h e o r y h e r e t o keep t h e c o m p l e x i t y of our n o t i o n s as l o w as p o s s i b l e .

I t p r o b a b l y i s a l o n g way t o p u s h our t h e o r y so f a r t h a t good a p p l i c a t i o n s c a n be made on a l a r g e s c a l e t o q u a d r a t i c f o r m s o v e r f u n c t i o n f i e l d s . Up t o now s u c h a p p l i c a t i o n s a r e v i s i b l e o n l y f o r c u r v e s , and h e r e t h e r o l e o f c o m p l e t e schemes

i s i m p r e s s i v e . But o f c o u r s e our t h e o r y s h o u l d have - as t h e u s u a l a l g e b r a i c g e o m e t r y - enough meaning b y h e r s e l f and a l l o w o t h e r a p p l i c a t i o n s . The l a s t c h a p t e r of t h e l e c t u r e s seems t o i n d i c a t e t h a t t h e W i t t r i n g s W(X) s t u d i e d h e r e a r e s i g n i f i c a n t t o u n d e r s t a n d t h e s e t of r e a l p o i n t s on an a l g e b r a i c v a r i e t y d e f i n e d o v e r t h e f i e l d P o f r e a l numbers. F o r a n o t h e r a p p l i c a - t i o n see my t a l k " R e a l c l o s u r e s of a l g e b r a i c v a r i e t i e s " a t t h i s c o n f e r e n c e . A f t e r a l l our l e c t u r e s may w e l l be r e g a r d e d a c h a p t e r o f a l g e b r a i c K - t h e o r y as f i r s t v i s u a l i z e d by G r o t h e n d i e c k i n h i s work on t h e Riemairn-Roch t h e o r e m .

The w r i t t e n v e r s i o n o f t h e l e c t u r e s i s q u i t e a b i t l o n g e r t h a n t h e o r a l v e r s i o n , s i n c e I had t o r e p l a c e a l o t of s k e t c h e s and h a n d w a v i n g b y s o l i d a r g u m e n t s . I f u r t h e r added two s e c t i o n s and t h r e e a p p e n d i c e s , marked by a s t e r i s k s , t o t h e m a t e r i a l o f t h e o r a l l e c t u r e s t o i l l u s t r a t e and t o r o u n d o f f t h e r e s u l t s d i s p l a y e d t h e r e . These s e c t i o n s and a p p e n d i c e s a r e n o t n e c e s s a r y f o r an u n d e r s t a n d i n g of t h e o t h e r s e c t i o n s . But t h e y a r e e a s i l y a c c e s s i b l e t o anybody who i s a c q u a i n t e d w i t h t h e m a i n p a r t s of

t h e l e c t u r e s .

I want t o t h a n k t h e s t a f f a t Queen's and i n p a r t i c u l a r G r a c e O r z e c h and P a u l o R i b e n b o i m f o r t h r e e v e r y p l e a s a n t weeks

a t K i n g s t o n and my a u d i e n c e i n t h e f i r s t two weeks f o r great- e n d u r a n c e .

T a b l e o f C o n t e n t s . Pap;e

C h a p t e r I . D e f i n i t i o n o f t h e G r o t h e n d i e c k - W i t t r i n g L ( X ) and

• t h e V i t t r i n g W(X)

§ 1 B i l i n e a r s p a c e s 7

§ 2 S u b b u n d l e s 20

§ 5 M e t a b o l i c s p a c e s 26

§ 4 The G r o t h e n d i e c k - W i t t r i n g L ( X ) 34

§ 5 D e f i n i t i o n o f W(X) 40

§ 6 F u n c t o r i a l i t y . 44

§ 7 The r a n k homomorphism 49

C h a p t e r I I . L o c a l t h e o r y

§ 1 C o n n e c t i o n w i t h W i t t ' s t h e o r y . . 50

§ 2 The s i g n e d d e t e r m i n a n t 53

§ 3 O r t h o g o n a l b a s e s 56

§ 4 G e n e r a t o r s and r e l a t i o n s f o r W(A) 60

§ 5 The p r i m e i d e a l s o f W(A) 63

Page

§ 6 N i l p o t e n t and t p r s i o n e l e m e n t s 69

§ 7 A c l o s e r l o o k a t s i g n a t u r e s 71 G u i d e t o t h e l i t e r a t u r e : L o c a l t h e o r y 77

C h a p t e r I I I . The p r i m e i d e a l t h e o r e m

§ 1 D i v i s o r i a l schemes 80

§ 2 Consequences o f t h e p r i m e i d e a l theorem * • . 82

§ 3 B i l i n e a r c o m p l e x e s 89

§ 4 E u l e r c h a r a c t e r i s t i c s • 97

§ 5 P r o o f o f t h e p r i m e i d e a l t h e o r e m , p a r t I . • 100

§ 6 P r o o f o f t h e f u n d a m e n t a l lemma 106

*§ 7 An example: P r o j e c t i v e s p a c e s 114

*§ 8 A s e m i l o c a l - g l o b a l p r i n c i p l e 119

C h a p t e r I V . S p a c e s o f r a n k one

§ 1 The group o f s q u a r e c l a s s e s Q(X) 125

§ 2 E x p l i c i t d e s c r i p t i o n o f s p a c e s o f r a n k one • 127

§ 3 D e t e r m i n a n t s 131

§ 4 The u n i t s o f V ( X ) 136

C h a p t e r V. A l g e b r a i c schemes o v e r P.

§ 1 F a c t o r i z a t i o n o f l o c a l s i g n a t u r e s 139

§ 2 S i g n a t u r e s and r e a l p o i n t s 144

§ 3 P r o o f o f t h e theorem o f Craven-Rosenberg-Ware 150

§ 4 C u r v e s o v e r P 154

Page

* A p p e n d i x 1. L e v e l and h e i g h t o f a non r e a l

c o m m u t a t i v e r i n g 161

* A p p e n d i x 2. The p r i m e i d e a l s o f L ( X ) 165

* A p p e n d i x 3. A b s t r a c t W i t t r i n g s 168 G u i d e t o t h e l i t e r a t u r e : G l o b a l t h e o r y 172

R e f e r e n c e s 176

G h a p t e r I D e f i n i t i o n o f t h e G r o t h e n d i e c k - W i t t r i n p ; L ( X ) and t h e W i t t r i n g ; W(X).

5 1 B i l i n e a r s p a c e s

L e t X be an a r b i t r a r y scheme, n o t n e c e s s a r i l y s e p a r a t e d , i . e . a "prescheme" i n t h e t e r m i n o l o g y o f G r o t h e n d i e c k¹ s b l u e b o o k s f E G A ] . We d e n o t e b y o r s i m p l y b y © t h e s t r u c t u r e s h e a f

of X. Thus f o r e v e r y open s e t Z c X t h e s e t o f s e c t i o n s <&(Z) o f 6 o v e r Z i s a c o m m u t a t i v e r i n g w i t h 1 and f o r an open Z' c Z t h e r e s t r i c t i o n map f -» f J Zf f r o m .6(Z) t o ^ ( Z1) i s a r i n g horac- morphism (mapping 1 t o 1 ) . F o r x a p o i n t o f X we d e n o t e b y S t h e s t a l k of S a t x,

6 = l i m S( Z ) Z3x

w i t h Z r u n n i n g t h r o u g h t h e open n e i g h b o u r h o o d s o f x. T h i s i s a l o c a l r i n g whose m a x i m a l i d e a l w i l l be d e n o t e d b v rr> .

More g e n e r a l l y we r e g a r d - as nowadays u s u a l - any s h e a f F o f a b e l i a n g r o u p s o v e r X as a f u n c t o r Z H F ( Z ) f r o m t h e c a t e - g o r y o f open s e t s o f - X 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 morphisms o f t h e f i r s t c a t e g o r y a r e t h e i n c l u s i o n maps Zf -* Z e x i s t i n g whenever Zf i s c o n t a i n e d i n Z. Thus f o r e v e r y p a i r o f open s e t s Z',Z w i t h Z^f c Z we h a v e a " r e s t r i c t i o n homomorphism"

f r o m F ( Z ) t o F ( Z^f) w h i c h we d e n o t e u s u a l l y b y f -> f | z^f. That t h i s f u n c t o r F i s a s h e a f means b y d e f i n i t i o n t h a t t h e f o l l o w i n g

- I l l -

c o n d i t i o n h o l d s t r u e :

S h e a f c o n d i t i o n : I f Z i s an open s e t i n X, and |Z | T i s an open c o v e r i n g o f Z, and i f { f^a!^ap j i ^{g 3} f a m i l y ⁰ f s e c t i o n s

f f F ( Z ) w i t h f | z n Z^{Q e} f^Q| z n Z^D whenever Z n Z^Q i s non

a ct ot.¹ ct p p' a n a p

empty, t h e n t h e r e e x i s t s a u n i q u e s e c t i o n f f F ( Z ) o v e r Z w i t h f I Z = f f o r a l l cu

1 a a

N o t i c e t h a t t h i s c o n d i t i o n i m p l i e s F ( 0 ) = C f o r t h e empty s e t 0, and t h a t f o r a f u n c t o r F w i t h F ( 0 ) = 0 t h e c o n d i t i o n

"Z^ n Zp n o n empty" above c a n b e d r o p p e d .

Our s h e a f F o f a b e l i a n g r o u p s i s c a l l e d , an ©-module i f e v e r y F ( Z ) i s an ( u n i t a r y ) ©(Z)-module and t h e r e s t r i c t i o n homo- morphisms F ( Z ) -» F ( Z ' ) a r e c o m p a t i b l e w i t h t h e r i n g homomorphism

©(Z) -» © ( Z1) i n t h e o b v i o u s s e n s e . Then e v e r y s t a l k F = l i m F ( Z )

x€Z

i s a module o v e r t h e l o c a l r i n g © . A homomorphism n:F P f r o m

x i

F t o a n o t h e r ©-module F^ c o n s i s t s o f a system o f ©(Z)-module homomorphisms

a^z : F ( Z ) -> F ^ Z )

s u c h t h a t f o r e v e r y p a i r o f open s e t s Zf c Z t h e d i a g r a m F ( Z ) — > P¹( Z )

r e s

F ( Z ' ) — > F ( Z ' )

w i t h t h e r e s t r i c t i o n maps a s t h e v e r t i c a l a r r o w s commutes.

I n t h i s way we o b t a i n t h e " c a t e g o r y o f ©-modules". C l e a r l y a homomorphism a : F -» F ^ i s an i s o m o r p h i s m i n o u r c a t e g o r y i f and o n l y i f a l l -are b i j e c t i v e . I t f o l l o w s e a s i l y f r o m t h e s h e a f c o n d i t i o n t h a t a i s a l r e a d y an i s o m o r p h i s m i f t h e i n d u c e d

©^.-homomorphi sms

x x 1x

on t h e s t a l k s a t a l l p o i n t s x o f X a r e b i j e c t i v e .

I f Z i s a f i x e d open s u b s e t o f X t h e n we o b t a i n f r o m e v e r y

©^-module F an S^-module P| Z b y r e s t r i c t i n g t h e f u n c t o r F t o t h e c a t e g o r y o f open s u b s e t s o f Z, c a l l e d t h e r e s t r i c t i o n o f t h e

©^x-module F t o Z.

A f t e r t h e s e p r e l i m i n a r i e s we come t o o u r f i r s t d e f i n i t i o n . D e f i n i t i o n 1. A v e c t o r b u n d l e E on X i s an ©-module E w h i c h i s l o c a l l y f r e e o f f i n i t e r a n k , i . e . f o r e v e r y p o i n t x o f X t h e r e e x i s t s an open n e i g h b o u r h o o d Z o f x s u c h t h a t t h e ©z-module E|Z

n

i s i s o m o r p h i c t o t h e f r e e ©^-module f o r some n a t u r a l number n . We c a l l n t h e l o c a l r a n k o f E a t x and c a l l t h e l o c a l l y

c o n s t a n t f u n c t i o n x -» n on X t h e r a n k r k E o f E. I f X i s c o n n e c t e t h e n r k E o f c o u r s e i s a c o n s t a n t .

Remark. A c t u a l l y t h e word " v e c t o r b u n d l e " h e r e i s an abuse o f l a n g u a g e . V e c t o r b u n d l e s i n t h e p r o p e r s e n s e a r e d e f i n e d i n [ E G A , I I § 1.7]. The l o c a l l y f r e e ©-modules o f f i n i t e r a n k a r e t h e s h e a f s o f s e c t i o n s o f t h e s e h o n e s t v e c t o r b u n d l e s and c o r r e s - pond w i t h them i n a u n i q u e way.

I f X i s a f f i n e , i . e . X ^ Spec ( A ) w i t h A t h e c o m m u t a t i v e r i n g © ( X ) , t h e n e v e r y v e c t o r "bundle E o v e r X i s u n i q u e l y d e t e r - m i n e d b y t h e A-module P := E ( X ) , s i n c e E i s c e r t a i n l y " q u a s i - c o h e r e n t^{1 1} [ E G A I § 1 . 4 ] . P i s 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 A - m o d u l e . We o b t a i n i n t h i s way an e q u i v a l e n c e o f t h e c a t e g o r y

o f v e c t o r b u n d l e s o v e r X and t h e c a t e g o r y o f 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 m o d u l e s o v e r A , c f . [ B b , I I §

F o r E a v e c t o r b u n d l e o v e r an a r b i t a r y scheme X we d e n o t e b y E * t h e d u a l v e c t o r b u n d l e o f E , d e f i n e d as f o l l o w s . E * ( Z ) i s t h e s e t o f homomorphisins f r o m t h e ©^-module E| Z t o t h e ~>^-module

©z. ^ ny s u c h homomorphism a:E| Z -* ©^z c a n be m u l t i p l i e d , w i t h

s e c t i o n s o f © ( Z ) i n an o b v i o u s way, and E * ( Z ) i s an © ( Z ) module.

F u r t h e r m o r e f o r Z¹ c Z t h e homomorphism a y i e l d s a homomorphism a' f r o m E| Z' t o ©^z» "by l o o k i n g o n l y a t t h e open s u b s e t s o f Z¹. T h i s map a r t a* i s t h e r e s t r i c t i o n map f r o m E * ( Z ) t o E * ( Z ' ) . I f E| Z s- t h e n a l s o E*| Z a- ©§. Thus E * i s i n d e e d a g a i n a

v e c t o r b u n d l e . F o r a f f i n e open s e t s Z o f X we c l e a r l y c a n i d e n t i f y E*(Z) w i t h t h e s e t o f © ( Z ) - l i n e a r maps f r o m E ( Z ) t o © ( Z ) ,

E*(Z) = H o m^{S ( Z )}( E ( Z ) , © ( Z ) ) .

We a l s o h a v e

E * = Horn. ( E ,© )

x © ^x x* x^y

X

f o r any x i n X .

F o r u i n E ( Z ) and a i n E * ( Z ) we d e n o t e b y <u,a> t h e element a^{< z}( u ) o f © ( Z ) . { R e c a l l t h a t a y i e l d s i n p a r t i c u l a r a map a.r? from

E ( Z ) t o (&(Z)|. C o n s i d e r now t h e d u a l v e c t o r b u n d l e E** o f E*.

Then we c a n e a s i l y e s t a b l i s h a homomorphism K:E -» E** i n a u n i q u e way s u c h t h a t

<a,K(u)> « <u,cv>

f o r u i n E ( Z ) , a i n E * ( Z ) . F o r any x i n X t h e i n d u c e d map K :E -» ( E * * )

X X ^{v y}x

c o i n c i d e s w i t h t h e w e l l known i s o m o r p h i s m f r o m t h e f r e e -

module E . o n t o i t s b i d u a l . Thus k: i s an i s o m o r p h i s m . We u s u a l l y i d e n t i f y E and E**.

We a l s o i n t r o d u c e f o r a v e c t o r b u n d l e E o v e r X t h e s h e a f E E d e f i n e d , b y

(E xx E ) ( Z ) := E ( Z ) x E ( Z )

w i t h o b v i o u s r e s t r i c t i o n maps. The s t a l k o f E E a t a p o i n t x i s E x E w h i c h j u s t i f i e s o u r f i b r e p r o d u c t n o t a t i o n t o some

J\. x

e x t e n t .

D e f i n i t i o n 2. A s y m m e t r i c b i l i n e a r f o r m B on t h e v e c t o r b u n d l e E i s a morphism

B:E E -» S

i n t h e c a t e g o r y o f s h e a v e s o v e r X, s u c h t h a t f o r e v e r y open Z t h e map

Bz : E ( Z ) x E ( Z ) -* G ( Z )

i s a s y m m e t r i c b i l i n e a r f o r m on t h e &(Z) module E ( Z ) .

C l e a r l y t h e n B a l s o y i e l d s a s y m m e t r i c b i l i n e a r f o r m B on t h e f r e e S -module E f o r e v e r y x I n X.

x x ^u

B i n d u c e s f o r each open Z map co^z : E ( Z ) -* E * ( Z )

as f o l l o w s . F o r u i n E ( Z ) and v i n E ( Z ' ) , Z^f c Z, t h e homomor- p h i s m m^z( u ) : E J Z -> maps v o n t o B^z, ( u j z^f, v ) . These maps t o g e t h e r

c o n s t i t u t e a homomorphism cp f r o m t h e b u n d l e E t o t h e d u a l b u n d l e E*. F o r Z a f f i n e

coz:E(Z) -> H o m6 ( z )( E ( Z ) , © ( Z ) )

i s t h e u s u a l l i n e a r map f r o m t h e p r o j e c t i v e module E ( Z ) o v e r S ( Z ) t o i t s d u a l module a s s o c i a t e d w i t h t h e b i l i n e a r f o r m B^.

I n g e n e r a l a n y homomorphism ro f r o m E t o E* h a s an a d j o i n t homo- morphism cp*:E E** —22 > E*. The s y m m e t r i c b i l i n e a r f o r m s on E c o r r e s p o n d i n t h e way i n d i c a t e d above u n i q u e l y w i t h t h e homomorphisms cp:E -» E* w h i c h a r e s e l f ad j o i n t , co = cn • We c a l l B n o n d e g e n e r a t e i f co i s a n i s o m o r p h i s m .

A p a i r (E,B) c o n s i s t i n g o f a v e c t o r b u n d l e E and a symme- t r i c b i l i n e a r f o r m B on E w i l l be c a l l e d a b i l i n e a r b u n d l e . I f B i s known t o b e n o n d e g e n e r a t e i t w i l l be c a l l e d a b i l i n e a r s p a c e . We o b t a i n a c a t e g o r y o f b i l i n e a r b u n d l e s b y a d d i n g t h e f o l l o w i n g d e f i n i t i o n .

D e f i n i t i o n 5« A morphism f r o m a b i l i n e a r b u n d l e ( E ^ B ^ ) t o a

b i l i n e a r b u n d l e (E^B^ i s a homomorphism a f r o m t h e v e c t o r b u n d l e E„ t o E⁰ s u c h t h a t

B^?( a ( u ) , a ( v ) ) = B ^ C ^ v ) f o r any open s e t Z and any u,v i n E ^ ( Z ) .

H e r e we s i m p l y w r o t e a ( u ) , a ( v ) i n s t e a d o f o.r/(iO> a^Cv) and B- i n s t e a d o f B. Such n o t a t i o n a l s i m p l i f i c a t i o n s w i l l be made q u i t e o f t e n , a s l o n g a s no c o n f u s i o n i s t o be f e a r e d .

We now s h a l l g i v e o t h e r i n t e r p r e t a t i o n s o f t h e n o t i o n

" b i l i n e a r s p a c e " i n two s p e c i a l c a s e s .

Example 1. Assume X i s a f f i n e , X = S p e c ( A ) . Then a b i l i n e a r b u n d l e (E,B) o v e r X i s u n i q u e l y d e t e r m i n e d b y t h e p a i r ( P , f )

c o n s i s t i n g o f t h e p r o j e c t i v e A-mbde P := E ( X ) and t h e s y m m e t r i c b i l i n e a r f o r m

$ = Bv : P x P -* A.

We c a l l s u c h a p a i r (P,&) a b i l i n e a r A-module, and we c a l l ( P , 0 a b i l i n e a r s p a c e o v e r A i f ft i s non de?:enei\ate, i . e . i f

$ i n d u c e s an i s o m o r p h i s m f r o m t h e A-module P t o t h e d u a l A- module P* = Hom^(P,A). The c a t e g o r y o f b i l i n e a r b u n d l e s o v e r X i s e q u i v a l e n t t o t h e c a t e g o r y o f b i l i n e a r modules o v e r A, and u n d e r t h i s e q u i v a l e n c e , d e s c r i b e d above, t h e b i l i n e a r s p a c e s o v e r X c o r r e s p o n d w i t h t h e b i l i n e a r s p a c e s o v e r A.

I n a n a n a l o g o u s way we s h a l l a l w a y s t r a n s f e r w i t h o u t f u r t h e r comment n o t i o n s d e v e l o p e d f o r b i l i n e a r b u n d l e s and s p a c e s o v e r X t o b i l i n e a r m o d u l e s and s p a c e s o v e r A i f

X = S p e c ( A ) .

Example 2. L e t Y b e an a l g e b r a i c s u b s e t o f t h e a f f i n e s p a c e CN o r t h e p r o j e c t i v e s p a c e PN( c ) w h i c h i s d e f i n e d o v e r "", i . e .

can be d e s c r i b e d as t h e s e t o f z e r o s o f a s y s t e m of p o l y n o m i a l s r e s p . f o r m s w i t h c o e f f i c i e n t s i n P. On Y we have an i n v o l u t i o n yh* y mapping a p o i n t y t o t h e p o i n t y w h i c h h a s a f f i n e r e s p . p r o j e c t i v e c o o r d i n a t e s c o m p l e x c o n j u g a t e t o t h o s e o f y. F o r

any s u b s e t A o f Y we d e n o t e b y X t h e image o f A under t h e com- p l e x c o n j u g a t i o n y n y. W i t h Y t h e r e i s a t t a c h e d a scheme X i n t h e f o l l o w i n g way. The p o i n t s o f X a r e t h e s u b s e t s T u T o f Y w i t h T r u n n i n g t h r o u g h t h e Z a r i s k i - c l o s e d i r r e d u c i b l e s u b s e t s o f Y, i . e . t h e c l o s e d s u b s e t s o f Y w h i c h a r e d e f i n e d o v e r IF- and " R - i r r e d u c i b l e " . F o r e v e r y Z a r i s k i - o p e n s u b s e t W o v e r Y we d e f i n e a s u b s e t W o f X as f o l l o w s . W c o n s i s t s o f a l l p o i n t s x = T U T o f X w i t h W n (T U T) n o n empty { w h i c h i m p l i e s W n T non empty}. These s e t s W a r e b y d e f i n i t i o n t h e open s u b s e t s of X. Then t h e s t r u c t u r e s h e a f ©^x = © i s d e f i n e d as f o l l o w s . <9(W)

i s t h e r i n g o f a l l r e g u l a r f u n c t i o n s f :W -> C w h i c h a r e compat- i b l e , w i t h c o m p l e x c o n j u g a t i o n : f( y ) = fTyT f o r a l l y i n W. I n t h i s way we have o b t a i n e d a l l " r e d u c e d a l g e b r a i c schemes o v e r P^{1 1} w h i c h a r e a f f i n e r e s p . p r o j e c t i v e .

We now assume f o r s i m p l i c i t y t h a t Y i s c o n n e c t e d and c o n s i d e r a c o m p l e x a l g e b r a i c v e c t o r b u n d l e p:F -> Y w h i c h i s

" d e f i n e d o v e r IR". By t h i s we mean r o u g h l y t h e f o l l o w i n g . Y h a s a c o v e r i n g iW^I b y Z a r i s k i - o p e n s u b s e t s W , a l l s t a b l e u n d e r c o m p l e x c o n j u g a t i o n , and t h e r e e x i s t s a f a m i l y { \ai o f b i j e c t i v e mappings

& Ct CI

s u c h t h a t t h e d i a g r a m s

commute and t h e t r a n s i t i o n reaps

a r e o f t h e f o r m

( y , z ) ( y^fu^{a p}( y ) z )

w i t h f u n c t i o n s Uap;^a n' ^ g G L ( n , c ) a l l whose c o e f f i c i e n t s l i e i n K \ n Wp).

We c a n i n t r o d u c e a C - a n t i l i n e a r i n v o l u t i o n T:F

c o v e r i n g t h e i n v o l u t i o n y on Y s u c h t h a t o v e r e v e r y W t h e

I

p ~ (W ) c o r r e s p o n d s v i a X t o t h e s t a n d a r d i r i v o l u -^* t i o n (y,z)*-* (y*z") on W^ x Cn. I n t h i s way an a l g e b r a i c v e c t o r b u n d l e o v e r Y w h i c h i s d e f i n e d o v e r !K may be c o n s i d e r e d as a p a i r (F,T) c o n s i s t i n g o f a c o m p l e x a l g e b r a i c v e c t o r b u n d l e P

o v e r Y and a " l o c a l l y t r i v i a l " C - a n t i l i n e a r i n v o l u t i o n on F c o v e r i n g t h e c o m p l e x c o n j u g a t i o n on Y.

We now d e f i n e an S-^-module E as f o l l o w s * F o r e v e r y open s e t W o f Y s t a b l e u n d e r c o m p l e x c o n j u g a t i o n , t h e ft(W)-module E(W) c o n s i s t s o f a l l s e c t i o n s s:W -» p ~ (W) s u c h t h a t f o r any W

m e e t i n g W t h e map

X o (s|w n w ):W n w -> p~¹(w n w ) -^->( w n w )>< r^{T l}

a ¹ or a ¹ a a'

has c o o r d i n a t e f u n c t i o n s l y i n g i n fc(W n W ) . N o t i c e t h a t t h i s a r e p r e c i s e l y t h e r e g u l a r s e c t i o n s s o f t h e c o m p l e x v e c t o r b u n d l e F o v e r W w h i c h h a v e t h e p r o p e r t y

s ( y ) = T ( s ( y ) >

f o r e v e r y y i n W. U s i n g t h e maps \^ we see t h a t E| W^ i s i s o - m o r p h i c t o (©y )ⁿ. Thus E i s a v e c t o r b u n d l e o v e r X. I t i s o n l y

a

an e x e r c i s e t o v e r i f y t h a t we o b t a i n i n t h i s way an e q u i v a l e n c e f r o m t h e c a t e g o r y o f a l g e b r a i c v e c t o r b u n d l e s o v e r I w h i c h a r e d e f i n e d o v e r ^ t o t h e c a t e g o r y o f v e c t o r b u n d l e s o v e r X. I t i s f u r t h e r e a s i l y v e r i f i e d t h a t t h e s y m m e t r i c b i l i n e a r f o r m s

B:E x x E -» &x

c o r r e s p o n d u n i q u e l y w i t h t h e r e g u l a r maps p:F x y F ~» C

on t h e c l a s s i c a l f i b r e p r o d u c t F Xj F h a v i n g t h e f o l l o w i n g p r o p e r t i e s :

a) F o r e v e r y y i n X t h e r e s t r i c t i o n o f p t o p~ ( y ) x p ~ ( y ) i s

—1 a s y m m e t r i c b i l i n e a r f o r m on t h e C - v e c t o r space p"~ ( y ) . b ) F o r u,v i n p ~ ( y )

P(TU,TV) = p ( u , v ) .

M o r e o v e r B i s n o n d e g e n e r a t e i f and o n l y i f o u r b i l i n e a r f o r m s on t h e v e c t o r s p a c e s p ~ ( y ) a l l a r e non d e g e n e r a t e .

We r e t u r n t o o u r g e n e r a l t h e o r y . Two b i l i n e a r modules ( E . j B . ) and ( Ep, Bp) o v e r X c a n be added and m u l t i p l i e d . The

<§rtl&igonal sum ( E ^ B ^ ) a ( E², B²) i s d e f i n e d as the v e c t o r bundle E^ © Eg having s e c t i o n modules

® E²) ( Z ) :- E¹( Z ) $ E²( Z )

equipped with tike f o l l o w i n g b i l i n e a r form B « ± B^i

Bz (ui ® u2, v1 ® v2) » B/ J( u1, v1) + B2( u2, v2)

f o r s e c t i o n s ^ui »^vi i n E^CZ) and u², v² i n E²( Z ) . S i m i l a r l y the tensor product ( E ^ B ^ ) ® ( E², B²) i s defined as the v e c t o r bundle E^ .&£ E2 equipped with the b i l i n e a r form .B^ ® B2. R e c a l l t h a t the ftanctor

ZtU E⁴( Z ) % ( 2 ) ^E 2^{( Z)}

toes not n e e e s s a r i l y f u l f i l l the sheaf c o n d i t i o n , and that

*1 ®$ ^E2 ^{i s t n e s l 3}^^{a f} a s s o c i a t e d to t h i s "presheaf^{1 1}. But f o r Z an a f f i n e open subset of X we have

( E1 ®q Eg) (Z) - E^C Z) ®e > (z ) E2( Z) .

B^ ® B2 can be c h a r a c t e r i z e d as the unique b i l i n e a r form B on X which f u l f i l l s

• B^z(n^ ® u^v^, ® v²) = B^{1 z}( u¹, v^{/ )}) B²2;(u²,v²)

f o r a f f i n e open sets Z and s e c t i o n s u ^ v ^ i n E ^ ( Z ) , and u², v² i n E²( Z ) . The s t a l k of our b i l i n e a r blondle ( E ^ B ^ ) ® ( E², B²) at a p o i n t x i s j u s t the G>^x-module E ^ ®^ E ^ equipped with the b i l i n e a r form B ^^x ® B ^ .

Another way t o put the d e f i n i t i o n s of B^ x B² and B^ ® B² i s the f o l l o w i n g . The n a t u r a l map from the tensor product

E * E * t o t h e b u n d l e (E^ ^ E p ) * is an i s o m o r p h i s m s i n c e t h e E ^ a r e l o c a l l y f r e e . We i d e n t i f y E * *y E * w i t h (E^ ^ E⁰) * i n . t h i s way. S i m i l a r l y we i d e n t i f y (E^ rr E p ) * w i t h E * ^ E * ,

r e g a r d i n g t h e l i n e a a ? f o r m s on E . J Z as l i n e a r f o r m s on (E^. & ^E2 'i ^Z w h i c h a r e z e r o on E. l z | ( i, j ) = (1,2) o r * ( 2, 1 ) 1 . L e t

J

rn. : E ^ ^ K be t h e homomorphism a s s o c i a t e d w i t h . Then B^ i Bp i s t h e b i l i n e a r f o r m c o r r e s p o n d i n g t o

p

O N

\ 0 r^{0 p} / E¹ * E^p -* E * ^ E *

and B^ Bp i s t h e b i l i n e a r f o r m c o r r e s p o n d i n g t o cp^ cpp:E^ E p -» E * * E * .

From t h i s d e s c r i p t i o n o f B^ x Bp and B^ ^ Bp i t i s e v i d e n t t h a t t h e s e f o r m s a r e non d e g e n e r a t e i f b o t h B^ and Bp a r e non degener- a t e .

We now d i s c u s s a s p e c i a l t y p e o f b i l i n e a r s p a c e s o v e r X.

L e t ( a - -) be a s y m m e t r i c n x n - m a t r i x w i t h c o e f f i c i e n t s a- - i n t h e r i n g ©(X). We t a k e t h e f r e e b u n d l e Sⁿ, and d e n o t e b y

e ^ , . . . , eⁿ t h e s t a n d a r d b a s i s o f &ⁿ, i . e . t h e g l o b a l s e c t i o n s e_^ = (O^...^^!^... 0 )

i

i n <9ⁿ(X) = ' S( X )ⁿ. We i n t r o d u c e on ©ⁿ a s y m m e t r i c b i l i n e a r f o r m B as f o l l o w s . L e t Z be open i n X , and l e t

u = ( u^v. . . , uⁿ) c e ( Z )ⁿ v = ( vv. . . , vn) € ® ( Z )n

be s e c t i o n s i n ©ⁿ( Z ) . We p u t

n

B7( u , v ) := 7 (a- . Z) u-v,.

C l e a r l y B i s t h e u n i q u e b i l i n e a r f o r m on $n w i t h B(e„. ,e_.) = a.: ..

We a b b r e v i a t e t h i s b i l i n e a r b u n d l e (E,B) b y t h e m a t r i x ( a - . ) . C l e a r l y ( a . •) i s a s p a c e i f and o n l y i f t h e d e t e r m i n a n t o f t h i s

r o

m a t r i x i s a u n i t i n © ( X ) . I n t h i s way we o b t a i n up t o i s o m o r p h i a l l f r e e s p a c e s , i , e . a l l s p a c e s (E,B) w i t h a f r e e v e c t o r b u n d l e E s- r<)n.

Two f r e e s p a c e s ( o r f r e e b i l i n e a r b u n d l e s ) (^a^~) and (s-[-j a r e i s o m o r p h i c i f and o n l y i f t h e r e e x i s t s an n x n - m a t r i x S w i t h c o e f f i c i e n t s i n ©(X) and d e t e r m i n a n t a u n i t o f - (X) s u c h t h a t .

(a! .) = S\^H.)S.

The p r o o f i s t h e same a s t h e u s u a l p r o o f o v e r f i e l d s .

A " d i a g o n a l " f r e e b i l i n e a r b u n d l e (a- 0 , a- - = a..^; ft..., \ri.l'.

a l s o b e d e n o t e d b y <a^ ,.. •, aⁿ> . We have

<a1,,..,an> = <av)> x ... i <&n>:

The t e n s o r p r o d u c t o f two d i a g o n a l f r e e b i l i n e a r b u n d l e s

<a,p...,aⁿ> and •.•,!>> i s i s o m o r p h i c t o

§ 2 S u b b u n d i e s .

In. t h i s s e c t i o n we d e v e l o p some e l e m e n t a r y l i n e a r a l g e b r a d e a l i n g w i t h " s u b b u n d i e s " o f a b i l i n e a r s p a c e .

L e t E be a v e c t o r b u n d l e o v e r X. Assume f o r e v e r y open s u b s e t Z o f X t h e r e i s g i v e n an 6 ( Z ) - s u b m o d u l e V ( Z ) o f E ( Z ) ,

and t h a t f o r open s e t s Z' c Z t h e r e s t r i c t i o n homomorphism f r o m E ( Z ) t o E ( Z^f) maps V ( Z ) i n t o V ( Z^f) . I f t h e n t h e n f u n c t o r

V:Z V ( Z ) on t h e c a t e g o r y o f open s u b s e t s o f X f u l f i l l s t h e s h e a f c o n d i t i o n ( c f . § 1) we c a l l V an & j -submodule o f E . D e f i n i t i o n . We c a l l a n &-submodule V o f E a s u b b u n d l e o f E , i f V i s l o c a l l y a d i r e c t summand o f E . T h i s means t h a t e v e r y p o i n t x o f X h a s an open n e i g h b o u r h o o d Z s u c h t h a t

(*) E| Z as (V| Z) W w i t h W a s u i t a b l e ©£ -submodule o f E| Z.

C l e a r l y t h e n V and E/V a r e v e c t o r b u n d l e s , s i n c e d i r e c t summands o f l o c a l l y f r e e -modules o f f i n i t e r a n k a r e a g a i n l o c a l l y f r e e . On t h e o t h e r hand, i f we o n l y know t h a t E / V i s a v e c t o r b u n d l e t h e n we have a s p l i t t i n g (*) o v e r e v e r y a f f i n e open s u b s e t Z o f X# I n d e e d , t h e c a n o n i c a l p r o j e c t i o n f r o m E| Z o n t o ( E / V) | z h a s a s e c t i o n , s i n c e ( E / V) | z c o r r e s p o n d s t o a p r o - j e c t i v e module o v e r ® ( Z ) .

We now r e g a r d a b i l i n e a r b u n d l e ( E , B ) o v e r X w i t h

a s s o c i a t e d homomorphism eo:E -» E * . F o r e v e r y (c~submodule V o f E

we d e f i n e a n o t h e r &-submodule V as f o l l o w s . I f Z i s an open s u b s e t o f X t h e n V^X( Z ) c o n s i s t s o f a l l s e c t i o n s s i n E ( Z ) s u c h t h a t s j Z ' i s o r t h o g o n a l t o V ( Z^f) w i t h r e s p e c t t o B^, f o r e v e r y open Z1 c Z. I t s u f f i c e s f o r t h i s t o know t h a t t h e germ s o f

c . x

s a t e v e r y p o i n t x o f Z i s o r t h o g o n a l t o V . Thus e v e r y s t a l k ( V^x) c o n s i s t s o f a l l t i n E w h i c h a r e o r t h o g o n a l t o V , i . e .

X X X

( v ^x)^x = ( v^x)^A.

The &-module V1 i s t h e k e r n e l o f t h e homomorphism E -2 > E* —i l - > V*

w i t h i * t h e d u a l o f t h e i n c l u s i o n homomorphism i f r o m V t o E,

*)

as i s i m m e d i a t e l y v e r i f i e d . J

Assume now B i s n o n d e g e n e r a t e% and V i s a s u b b u n d l e o f E.

Then CD i s an i s o m o r p h i s m and i * i s an e p i m o r p h i s m . Thus r i n d u c e s an i s o m o r p h i s m

a : E / V^X -ZL-> V*,

and i n p a r t i c u l a r V^x i s a g a i n a s u b b u n d l e o f E.

We i d e n t i f y ( E / V ) * w i t h t h e &-submodule o f E* whose s e c t i o n s o v e r any open Z c X a r e t h e l i n e a r f o r m s X:E| Z -> w h i c h v a n i s h on v| Z. By d e f i n i t i o n V1 i s t h e i n v e r s e image o f ( E / V ) * u n d e r t h e i s o m o r p h i s m cp:E — E * . Thus we o b t a i n f r o m an i s o m o r p h i s m

p : Vx — > ( E / V ) * .

*) i * means r e s t r i c t i o n o f t h e l i n e a r f o r m s on E t o V.

Th e i s o m o r p h i s m s a and (3 c o r r e s p o n d t o b i l i n e a r maps

( 1 ) V ( E / V1) -> &

( 2 ) V¹ x ^x ( E / V ) -> ©

w h i c h a r e p e r f e c t d u a l i t i t e s . These b i l i n e a r maps a r e o f c o u r s e j u s t t h e maps o b t a i n e d f r o m B b y " r e s t r i c t i o n " i n t h e o b v i o u s way. S i n c e V¹ i s a g a i n a s u b b u n d l e o f E we a l s o have a p e r f e c t d u a l i t y

(3) V11 xx E / V1 ->

i n d u c e d b y B. V i s an ©-submodule o f V1 1. C o m p a r i n g (1) and ( 3 ) we s e e t h a t a c t u a l l y V = V1 1. L e t us summarize o u r o b s e r v a t i o n s . P r o p o s i t i o n 1. L e t V b e a s u b b u n d l e o f t h e b i l i n e a r space ( E , B ) ,

and l e t ep:E — ^ > E* d e n o t e t h e i s o m o r p h i s m f r o m E t o E* a s s o c i - a t e d w i t h B.

i ) V¹ i s a s u b b u n d l e o f E. T h e r e e x i s t s a u n i q u e i s o m o r p h i s m a f r o m E / V¹ t o E* s u c h t h a t t h e d i a g r a m

E E*

E / V1 — — > V*

a

w i t h c a n o n i c a l v e r t i c a l s u r j e c t i o n s commutes.

i i ) T h e r e e x i s t s a u n i q u e i s o m o r p h i s m p f r o m V"¹ t o ( E / V ) * s u c h t h a t t h e d i a g r a m

V¹ — > ( E / V ) *

E co -> E*

w i t h c a n o n i c a l v e r t i c a l i n j e c t i o n s commutes, i i i ) Vx l = V.

We assume now t h a t ( E , B ) i s a b i l i n e a r b u n d l e and t h a t F i s .an & - s u b m o l u l e o f E w h i c h i s a g a i n a v e c t o r b u n d l e . We f u r t h e r assume t h a t t h e b i l i n e a r f o r m B | F i s non d e g e n e r a t e * The f o r m B may be d e g e n e r a t e . L e t co:E -* E* and * : F ~«^-> F* d e n o t e t h e homo- morphisms a s s o c i a t e d w i t h B and B F , and l e t i : F -» E d e n o t e t h e i n c l u s i o n homomorphism f r o m F t o E. As o b s e r v e d above P ¹ i s t h e k e r n e l o f t h e homomorphism i*pco f r o m E t o F*, h e n c e a l s o t h e k e r n e l o f t h e homomorphism p := * o i *⁰ rp f r o m E t o P. More c o n -

c r e t e l y , i f u i s a s e c t i o n o f E o v e r some a f f i n e open s e t Z t h e n b y d e f i n i t i o n p ( u ) i s t h e u n i q u e s e c t i o n v o f F o v e r Z w i t h

B(u,w) = B(v,w) f o r a l l w. i n F ( Z ) .

S i n c e • = i * o c p0i we have p o i = id-p. Thus E i s t h e d i r e c t sum o f F and F¹. M o r e o v e r i o p i s a p r o j e c t i o n o p e r a t o r on E w i t h image F , t h e " o r t h o g o n a l p r o j e c t i o n " f r o m E o n t o F . We s u m m a r i z e : P r o p o s i t i o n 2. Assume F i s an S-submodule o f a b i l i n e a r b u n d l e

( E , B ) , f u r t h e r t h a t F i s a v e c t o r b u n d l e and B | F i s non d e g e n e r a t e , Then E i s t h e d i r e c t sum o f F and F¹. I n p a r t i c u l a r F i s a s u b -

b u n d l e o f E. We h a v e a c a n o n i c a l o r t h o g o n a l p r o j e c t i o n f r o m E

o n t o F.

D e f i n i t i o n . We c a l l a morphism a f r o m a b i l i n e a r b u n d l e (E„ ,B„) t o a b i l i n e a r b u n d l e ( E⁰, B⁰) an i s o m e t r y i f a i s i n - j e c t i v e ^J and co(E^) i s a s u b b u n d l e o f Ep.

P r o p o s i t i o n 5+ I f B^ i s non d e g e n e r a t e t h e n e v e r y morphism a f r o m (E^,B^) t o ( E ^ B ^ ) i s an i s o m e t r y .

P r o o f . By P r o p o s i t i o n 2 i t s u f f i c e s t o show t h a t a i s an i n - f e c t i v e homomorphism f r o m E^ t o Ep. T h i s w i l l be t r u e i f we know t h a t t h e maps a r E . -» E0 on t h e s t a l k s a r e i n f e c t i v e .

| A p p l y t h e s h e a f c o n d i t i o n ! } Now i f a. ( u ) « 0 f o r some u i n E.

X I X

t h e n

B^{1 x}( u , v ) = B^{2 x}( a^x( u) , a^x( v ) ) = 0 f o r e v e r y v i n E. . Thus u = 0.

I

-/V

Assume now (E,B) i s a b i l i n e a r s p a c e .

D e f i n i t i o n . A t o t a l l y i s o t r o p i c s u b b u n d l e V o f (E,B) i s a sub- b u n d l e V of E s u c h t h a t B i s z e r o on V x^- V. I n o t h e r t e r m s^| VX => V.

A t o t a l l y i s o t r o p i c s u b b u n d l e V o f (E,B) w i l l a l s o be c a l l e d a s u b l a R r a n g i a n o f ( E , B ) . i " L a g r a n g i a n s " w i l l be i n t r o - d u c e d i n t h e n e x t s e c t i o n . } F o r e v e r y s u c h s u b l a g r a n g i a n t h e G- module V^x/V i s a s u b b u n d l e o f t h e v e c t o r b u n d l e E/V, s i n c e V¹ i s l o c a l l y a d i r e c t summand o f E. From B we o b t a i n i n an e v i d e n t

*) T h i s means t h a t a ^ E ^ C Z ) -» E p ( Z ) i s i n f e c t i v e f o r e v e r y open. Z c X.

way a s y m m e t r i c b i l i n e a r f o r m IT on V^X/ V s u c h t h a t t h e n a t u r a l p r o j e c t i o n p : V^X -> V^X/ V i s a morphism f r o m t h e b i l i n e a r b u n d l e

(v\ B | V^X) t o ( v^xA, E ) .

P r o p o s i t i o n 4. (V^X/V,B") i s a b i l i n e a r s p a c e .

P r o o f . L e t cp:E > E* d e n o t e a g a i n t h e homomorphism a s s o c i a t e d w i t h B and cp:V^X/V ( V^X/ V ) * t h e homomorphism a s s o c i a t e d w i t h IT.

Ve o b t a i n f r o m t h e d e f i n i t i o n s a c o m m u t a t i v e d i a g r a m

0

-> v-

^o

0 -> ( E / V¹) * -> ( E A ) *

—> ^(v

^x

^{/v)* -»}

⁰

w i t h e x a c t rows and i s o m o r p h i s m s p and y as d e s c r i b e d i n Pro- p o s i t i o n 1 i i . Thus a l s o co i s an i s o m o r p h i s m .

§ 3 M e t a b o l i c s p a c e s

We now i n t r o d u c e a s p e c i a l t y p e o f b i l i n e a r s p a c e s . L e t

(U,p) b e a b i l i n e a r b u n d l e . ' S t a r t i n g w i t h £ we d e f i n e a s y m m e t r i c b i l i n e a r f o r m B on t h e v e c t o r b u n d l e U & U* as f o l l o w s . F o r Z

an open s e t i n X and s e c t i o n s u^,Up i n U ( Z ) , u^j,u£ i n U*(Z) B(u^+u* ^Up+u"^) := pCu^pUp) + <u^u^> + <Up,u*>.

Thus B c o i n c i d e s w i t h 6 on U x^- U, i s z e r o on U* x.. U*, and i s t h e n a t u r a l p a i r i n g on U x ^ U* and U* x-^ U. L e t *o:U -» U* be t h e homomorphism a s s o c i a t e d w i t h p.- Then t h e homomorphism

U (P U* -* (U *> U * ) * •* U* U

a s s o c i a t e d w i t h B i s g i v e n b y t h e m a t r i x

(

^{cp i d \} _{i d 0 / .}

T h i s homomorphism i s a l w a y s an i s o m o r p h i s m and t h u s B i s non d e g e n e r a t e . We d e n o t e t h e s p a c e (U U*,B) b y M(U,p). A s p a c e i s o m o r p h i c t o some M(U,p) w i l l b e c a l l e d s p l i t m e t a b o l i c .

{ " M e t a b o l i c s p a c e s " w i l l b e i n t r o d u c e d below.}

E x a m p l e . I f (U,p) i s a f r e e d i a g o n a l b i l i n e a r b u n d l e

<a>|,..., a^p> > a^ € & ( X ) , t h e n c l e a r l y

On any v e c t o r b u n d l e U we c a n i n t r o d u c e t h e b i l i n e a r fori;;

B = C, and o b t a i n a s p a c e M(U,0) w h i c h we d e n o t e b y H ( U ) . A s p a c e i s o m o r p h i c t o some H(U) w i l l be c a l l e d h y p e r b o l i c . P r o p o s i t i o n 1. P o r any s y m m e t r i c b i l i n e a r f o r m a on a v e c t o r b u n d l e U t h e s p a c e M(U,2a) i s i s o m o r p h i c t o H ( U ) . I n p a r t i c u l a r

a l l s p l i t m e t a b o l i c s p a c e s a r e h y p e r b o l i c i f 2 i s a u n i t i n & ( X ) . T h i s f o l l o x t f s i m m e d i a t e l y f r o m t h e f o l l o w i n g more g e n e r a l

f a c t . ( P u t 6 = 2a, Y = - a ) .

Lemma. L e t p:U x^ U -* & be a s y m m e t r i c b i l i n e a r f o r m on t h e v e c t o r b u n d l e U and l e t y:U U -* © be an a r b i t r a r y ( n o t n e c e s s a r i l y s y m m e t r i c ) b i l i n e a r f o r m on U. L e t d e n o t e t h e b i l i n e a r f o r m d e f i n e d b y

Y | ( U , V ) - Yz( v , u )

(u,v € U ( Z ) , Z open i n X ) . Then

M(U,p) - MCU^p+v+Y*).

P r o o f . L e t cp:U -» U* d e n o t e t h e homomorphism "u -» y ( - , u ) "

a s s o c i a t e d w i t h Y- Then t h e a u t o m o r p h i s m u + u* u + u* + co(u)

o f t h e v e c t o r b u n d l e U & U* |u f U ( Z ) , u* f U * ( Z ) , Z open i n X}

i s an i s o m e t r y f r o m M ( U , P + Y + Y ) o n t o M ( U , p ) . I n d e e d , d e n o t i n g t h e b i l i n e a r f o r m o f t h e f i r s t s p a c e b y Bf and t h e b i l i n e a r f o r m of t h e s e c o n d s p a c e b y B, we have f o r s e c t i o n s u,v i n U ( Z ) , u*,v*

i n U * ( Z ) :

B(u+u*+co(u) ,v+v*+co(v)) = 6 ( u , v ) + <u,ro(v)> + <v,n(u)> +

+ <U,V*> + <V,U*> a P(U,V) + y ( u *^v) + Y^Cu^v) 4- <U,V*> + <V,U*>

= B*(u+u*,v+v*)•

q. e. d.

I f 2 i s n o t a u n i t i n & ( X ) t h e n n o t e v e r y s p l i t m e t a b o l i c

^ Q ] c a n n o t be h y p e r b o l i c , s i n c e f o r a n y g l o b a l s e c t i o n e o f a h y p e r b o l i c

s p a c e ( E^$B ) * H(U) we h a v e B ( e , e ) £ 2 6 ( X ) . N e v e r t h e l e s s s p l i t m e t a b o l i c s p a c e s a r e " s t a b l y h y p e r b o l i c " b y t h e f o l l o w i n g p r o - p o s i t i o n .

P r o p o s i t i o n 2. P o r a n y b i l i n e a r b u n d l e (U,p) M( U,p) j. M(U.,-p) - H(U) i M( U, - p ) .

P r o o f . We c o n s i d e r t h e space E :« M(U,R) i r i ( U , - p ) . L e t U^ and Up b e two c o p i e s o f U. We t h i n k o f M(U,p) a s t h e b u n d l e

^ U* and o f M(U,-p) a s t h e b u n d l e Up <r U*. Th"s E = ( U ^ ^ U*) x (Up p U J ) .

U* and t h e d i a g o n a l A o f U^ & Up a r e b o t h s u b l a g r a n g i a n s o f E . The b i l i n e a r f ornr B o f E g i v e s a p e r f e c t d u a l i t y b e t w e e n A and U*. Thus A U|!j i s a b i l i n e a r s u b s p a c e o f E and

A * U* « H(A) a H ( U ) . A c c o r d i n g t o § 2, P r o p . 2 ,

E - H(U) x (A ^ U * )1.

One know has t o compute (A ^ U * )^x. T h i s b i l i n e a r space t u r n s out t o be i s o m o r p h i c t o M(U,-p). We l e a v e t h e d e t a i l s as an e x e r c i s e ( o r c f . [K,p.19 f ] ) .

D e f i n i t i o n . We c a l l a b i l i n e a r s p a c e a n i s o t r o p i c i f i t has no t o t a l l y i s o t r o p i c s u b b u n d l e d i f f e r e n t f r o m z e r o . O t h e r w i s e we c a l l t h e s p a c e i s o t r o p i c .

I f V i s a m a x i m a l t o t a l l y i s o t r o p i c s u b b u n d l e o f a s p a c e ( E , B ) t h e n c l e a r l y t h e s p a c e ( V^X/ V , E ) s t u d i e d a t t h e end o f § 2 i s a n i s o t r o p i c .

P r o p o s i t i o n ^. I f X i s a f f i n e t h e n e v e r y b i l i n e a r s p a c e E o v e r X hSs a d e c o m p o s i t i o n

Such a d e c o m p o s i t i o n of E w i l l be c a l l e d a W i t t d e c o m p o s i - t i o n . I n g e n e r a l t h e i s o m o r p h i s m c l a s s o f t h e a n i s o t r o p i c p a r t E^q i s n o t u n i q u e l y d e t e r m i n e d b y E . An example o v e r a l o c a l r i n g c a n be e x t r a c t e d f r o m [ K , S a t z 9«3*8]. A l s o t h e i s o m o r p h i s m c l a s s of t h e s p l i t m e t a b o l i c p a r t M i s i n g e n e r a l n o t u n i q u e l y d e t e r - m i n e d b y E . H e r e c o u n t e r - e x a m p l e s a r e q u i c k l y o b t a i n e d . F o r

example o v e r any scheme X

E = E x M o

w i t h Eq a n i s o t r o p i c and M s p l i t m e t a b o l i c .

as i s e a s i l y v e r i f i e d .

P r o p o s i t i o n 3 f o l l o w s i m m e d i a t e l y f r o m t h e f o l l o w i n g more g e n e r a l p r o p o s i t i o n .

P r o - p o s i t i o n 3 a> Assume X i s a f f i n e . L e t ( E , B ) be a s p a c e

o v e r X and V a s u b l a g r a n g i a n o f ( E , B )^# L e t U b e a s u b b u n d l e o f E w i t h

E & V S U.

( S u c h a s u b b u n d l e e x i s t s , s i n c e X i s a f f i n e . ) Then U and V a r e i n d u a l i t y under B , hence

(U ^ v,B| U e V ) % M(U,B| U ) .

M o r e o v e r t h e s p a c e ( U ^ V )^x i s i s o m o r p h i c t o (vV"V,E), hence b y § 2, P r o p . 2

( E , B ) - (Vx/V,5) x K ( U , B | U ) .

P r o o f . The a s s e r t i o n t h a t U and V a r e i n d u a l i t y i s t r i v i a l . W i t h G ( U (P V) ¹ we have

E » ( U P V) x G.

From t h i s we deduce

V1 ^ V x G.

Thus c l e a r l y ( G , B J G ) i s i s o m o r p h i c t o ( VX/ V , B ) .

q. e. d.

D e f i n i t i o n . A s u b b u n d l e V o f a s p a c e ( E , B ) i s c a l l e d a

L a g r a n g i a n i f V ^X = V . A s p a c e w h i c h h a s a L a g r a n g i a n i s c a l l e d m e t a b o l i c .

C l e a r l y any s p l i t m e t a b o l i c space K ( L , R ) L a s t h e L a g r a n g i a n U*, hence i s m e t a b o l i c .

Prom P r o p o s i t i o n 3a we o b t a i n i m m e d i a t e l y t h e f o l l o w i n g two s t a t e m e n t s about m e t a b o l i c s p a c e s and L a g r a n g i a n s .

C o r o l l a r y 1. E v e r y m e t a b o l i c space o v e r an a f f i n e scheme i s s p l i t m e t a b o l i c .

C o r o l l a r y 2. L e t V be a s u b l a g r a n g i a n o f a space (E,B) o v e r any scheme X. Then

i ) r k E 2 r k V.

i i ) V i s a L a g r a n g i a n i f and o n l y i f r k E = 2 r k V.

I n d e e d , i t s u f f i c e s t o check t h e s e s t a t e m e n t s i ) , i i ) o v e r a f f i n e open s u b s e t s of X, where t h e y a r e e v i d e n t by P r o p o s i t i o n 3 a.

A l r e a d y o v e r an e l l i p t i c c u r v e t h e r e e x i s t s i n f i n i t e l y many m e t a b o l i c s p a c e s w h i c h a r e n o t s p l i t m e t a b o l i c , c f . f K ,

§ 13.1].

We d i s c u s s t h e b e h a v i o u r o f m e t a b o l i c s p a c e s x^ith r e s p e c t t o t e n s o r p r o d u c t s .

P r o p o s i t i o n 4. I f V i s a L a g r a n g i a n o f t h e s p a c e (E,B) t h e n f o r any o t h e r space (E',B') t h e submodule V ft E¹ o f E * E^f i s a L a g r a n g i a n o f (E,B) *> ( Ef ,B1) • F o r a s p l i t m e t a b o l i c space M(U,e) more e x p l i c i t l y

M(U,0) •* ( E^f , B¹) *M(<tJ,0) ® (E»,B')).

I n p a r t i c u l a r

H(U) r ( E^f, B^f) - H(U E¹) •

P r o o f . V & P i n j e c t s i n t o E 5> E^f s i n c e E^f i s l o c a l l y f r e e . (E/V) ? E^f c a n "be i d e n t i f i e d w i t h E * E^f/'V ^ E^f. Thus V * E^! i s a s u b b u n d l e o f E E1. C l e a r l y V ^ Ef i s t o t a l l y i s o t r o p i c . S i n c e

r k ( V * Ef) = r k V * r k E' = ^ r k ( E E') V # E must b e a L a g r a n g i a n .

We now c o n s i d e r t h e c a s e

(E,B) = M(U,B) « ( U U*,B).

Then t h e s u b b u n d l e U* ^ E' o f E E' i s a L a g r a n g i a n w h i c h under B # B¹ i s i n d u a l i t y w i t h U E^!. S i n c e E E^f i s t h e d i r e c t sum of U •<* E¹ and U* ^ E¹ we i n d e e d have

(E,B) * ( E \ B!) a- M(U * E',p «> Bf) .

As an example we c o n s i d e r t h e space

L e t e^^ep be a b a s i s o f K(<1>) w i t h v a l u e m a t r i x under t h e b i l i n e a r f o r m o f M(<1>). Then e ^ e ^ - e g h a s t h e v a l u e m a t r i x

( 0 -1 ) • ^{T h u s}

M(<1>) ~ <1,-1>.

M u l t i p l y i n g t h i s r e l a t i o n w i t h an a r b i t r a r y s p y c o (E,B) wc o b t a i n b y t h e p r e c e d i n g p r o p o s i t i o n 4 t h e f o l l o w i n g

C o r o l l a r y . (E,B) j. (E,-B) a M ( E , B ) .

T h i s c a n a l s o e a s i l y he v e r i f i e d i n a d i r e c t way.

§ 4 The G r o t h e n d i e c k - W i t t r i n p ; L ( X ) .

Ve now o f t e n d e n o t e a s p a c e (E,B) b y a s i n g l e l e t t e r E f o r s h o r t . We l o o k a t t h e s e t B i l ( X ) o f i s o m o r p h i s m c l a s s e s ( E ) o f b i l i n e a r s p a c e s E o v e r X. On B i l ( X ) we have an. a d d i t i o n and a m u l t i p l i c a t i o n g i v e n b y t h e o r t h o g o n a l sum and t h e t e n s o r

p r o d u c t o f s p a c e s . I n t h i s way B i l ( X ) i s a c o m m u t a t i v e s e m i r i n g . I t h a s t h e i s o m o r p h i s m c l a s s o f t h e space <1> « (&^,m) as u n i t

element with x-^ G^- -» fc-^ t h e m u l t i p l i c a t i o n on

We now c o n s i d e r t h e G r o t h e n d i e c k - r i n g K B i l ( X ) of t h e

s e m i r i n g B i l ( X ) . T h i s i s a w e l l known c o n s t r u c t i o n . The e l e m e n t s of K B i l ( X ) a r e f o r m a l d i f f e r e n c e s [ E ] - f P ] o f c l a s s e s [ E ] , T F ] of s p a c e s E,F w i t h t h e r u l e t h a t two d i f f e r e n c e s [ E ^ ] - [ F ^ J and [ E p ] - [ F p ]a r e e Q ual ^ a n ( i o n l y i f

E,, i Pg i G ft E2 x F1 x G

f o r some s p a c e G o v e r X. R e c a l l t h a t t h e map ( E ) -» [ E ] ' f r o m B i l ( X ) t o K B i l ( X ) i s a u n i v e r s a l map f r o m B i l ( X ) t o a r i n g , i . e . f o r e v e r y s e m i r i n g homomorphism X : B i l ( X ) -* A t o a r i n g A t h e r e

e x i s t s a u n i q u e r i n g homomorphism B i l ( X ) -» A s u c h t h a t t h e d i a g r a m

B i l ( X ) > K B i l ( X )

A

*) Of c o u r s e [ E l i s i d e n t i f i e d w i t h [ E ] - [ o l .

commutes.

L e t o d e n o t e t h e a d d i t i v e subgroup o f K B i l ( X ) g e n e r a t e d b y t h e e l e m e n t s [ E ] - [ H ( V ) ] w i t h E m e t a b o l i c and V a L a g r a n g i a n o f E. I f E^ and E^ a r e m e t a b o l i c s p a c e s w i t h L a g r a n g i a n s

and t h e n E^ i i s m e t a b o l i c w i t h t h e L a g r a n g i a n ^ V^.

From t h i s r e m a r k we o b t a i n i m m e d i a t e l y t h a t o i s t h e s e t o f d i f f e r e n c e s [ E ] - [ F ] o f m e t a b o l i c s p a c e s E and F h a v i n g i s o - m o r p h i c L a g r a n g i a n s ,

n i s an i d e a l o f K B i l ( X ) . I n d e e d i f E i s m e t a b o l i c w i t h L a g r a n g i a n V and F i s an a r b i t r a r y s p a c e , t h e n a c c o r d i n g t o

§ 3 P r o p . 4

[ F ] ( [ E] - r H( V ) ] ) = [F ® E l - [ H ( F - V ) ] , and F ^ V i s a L a g r a n g i a n of F E.

We now d e f i n e t h e G r o t h e n d i e c k - W i t t r i n p : L ( X ) o f t h e scheme X as t h e r i n g K B i l ( X ) / a . The image o f a c l a s s

[ E ] € K B i l ( X ) i n L ( X ) w i l l a g a i n be d e n o t e d b y [ E j . I n L ( X ) we have b y d e f i n i t i o n

[ E ] = [ H ( V ) ]

f o r E a m e t a b o l i c space w i t h L a g r a n g i a n V.

P r o p o s i t i o n 1. I f X i s a f f i n e t h e n o = 0, hence L ( X ) = K B i l ( X ) . P r o o f . L e t (E,B) be a space w i t h L a g r a n g i a n V. Then, as ex-

p l a i n e d i n § 3, E = U P V w i t h some s u b b u n d l e U w h i c h i s d u a l t o V w i t h r e s p e c t t o B. We have

(E,B) - M(U,B|U) = H ( U , p ) , a n d , a g a i n b y § 5,

M(U,p) x M(U,-p) s- H(U) ^x K ( U , - p ) . Thus i n K B i l ( X )

[ E , B ] = [ M ( U , p ) l = THCU)] = [ H ( V * ) 1 = [ H ( V ) 1 . T h i s p r o v e s ft - 0.

F o r h y p e r b o l i c s p a c e s we have t h e f o l l o w i n g g e n e r a l f a c t . P r o p o s i t i o n 2. L e t V be a v e c t o r b u n d l e o v e r a scheme X and V be a s u b b u n d l e o f V, Then i n L ( X )

[H(V')] = [ H ( V ) ] + [H(V/W)].

P r o o f . We work i n t h e b i l i n e a r s p a c e H(V) = V r V*. L e t W¹ d e n o t e t h e submodule W¹ n V* of V*. T h i s i s t h e s h e a f of l i n e a r f o r m s on V w h i c h a r e z e r o on W, i . e . t h e k e r n a l o f t h e r e s t r i c - t i o n homomorphism V* -» V*. On t h e o t h e r hand we o b t a i n from t h e

e x a c t sequence

0 -» W -» V -* V/W -* 0 o f v e c t o r b u n d l e s a d u a l e x a c t sequence

0 -> (V/W)"* -> V* -* W* -* 0.

Thus we have c a n o n i c a l i s o m o r p h i s m s

(V/W)* W, v*/w W*.

I n p a r t i c u l a r W^f i s a s u b b u n d l e o f V*. C l e a r l y U ;= W W'

- m o -

i s a t o t a l l y i s o t r o p i c s u b b u n d l e o f H ( V ) . I t has t h e r a n k r k W + r k W^f = r k V + r k ( V / W ) * = r k V.

Thus U i s a L a g r a n g i a n o f H ( V ) . We have i n L ( X )

[H( V ) ] = [H(U)] = [H(W)] + [H(W')] - [ H O O ] + [ H ( ( V / W ) * ) j «

= H(W) + H(V/W), as c l a i m e d above.

We now c a n p r o v e t h e f o l l o w i n g weak a n a l o g u e o f t h e W i t t

• d e c o m p o s i t i o n i n § 3 f o r an a r b i t r a r y scheme X. (The W i t t d e - c o m p o s i t i o n i n § 3 c o u l d o n l y be done o v e r a f f i n e X, c f . L 3, P r o p . 3a.)

Theorem 3* L e t V be a s u b l a g r a n g i a n o f a s p a c e (E,B) o v e r X.

Then we have i n L ( X ) t h e e q u a t i o n ( c f . S 3 f o r n o t a t i o n s ) [ E , B ] * [ H ( V ) ] +

[ W v , f f ] .

N o t i c e t h a t f o r V a maximal s u b l a g r a n g i a n o f (E,B) t h e space ( V^X/ V, E ) ±^B a n i s o t r o p i c .

P r o o f . We work i n t h e s p a c e

(F,B') := (E,B) x ( V^x/ V, - E ) . L e t

a : V^x -> E <* V^x/ V

d e n o t e t h e " d i a g o n a l i n j e c t i o n " o f V¹ i n t o t h i s s p a c e . The s u b - module <x(V^x) o f P i s c l e a r l y t o t a l l y i s o t r o p i c . We want t o show

t h a t a ( V^x) i s a L a g r a n g i a n o f ( P , B^f) . I t s u f f i c e s t o c h e c k t h i o v e r a f f i n e open s u b s e t s o f X, and t h u s we assume now t h a t X i t s e l f i s a f f i n e . As e x p l a i n e d i n § 3 ( p r o o f o f P r o p . 3a) we have' d e c o m p o s i t i o n s

E = (V U) i G, V¹ a V «r G

w i t h U d u a l t o V u n d e r B. The s e c o n d d e c o m p o s i t i o n y i e l d s a c a n o n i c a l i s o m o r p h i s m f r o m G o n t o V^x/V. We have

a ( V^x) = V p A

w i t h A t h e " d i a g o n a l " o f t h e s u b b u n d l e G * V^x/ V o f P. C l e a r l y G ^ Vx/ V « A * Vx/V.

Thus V ^ A i s a d i r e c t summand o f

P = (V P U) x (A P Vx/ V ) ,

and we have v e r i f i e d t h a t a ( Vx) i s a s u b b u n d l e o f P. We have r k a ( V^x) = r k ( V^x) = r k V + r k V^x/V.

On t h e o t h e r hand

r k P « 2 r k V + r k G +. r k V^x/V = 2 ( r k V + r k V^x/ V ) . Thus a ( Vx) i s i n d e e d a L a g r a n g i a n o f ( F , B!) .

S i n c e now X i s a g a i n an a r b i t r a r y scheme. The vector- b u n d l e a ( V^x) i s i s o m o r p h i c t o V¹. Thus we o b t a i n i n L ( X ) t h e r e l a t i o n

(*) [ E , B ] + [ V^A/ V , - B ] = [ H ( V^X) ] .

By t h e p r e c e d i n g P r o p o s i t i o n 2

[ H ( V^X) ] = [ H ( V ) ] + [ H ( V^X/ V ) ] . On t h e o t h e r hand

[v^x/v,-E]

+

[vVv,E] = [ K( v V v, 5) l =

rii(vVv)].

Thus a d d i n g [vVv,IT] on b o t h s i d e s i n (*) we a r r i v e a t t h e d e s i r e d r e l a t i o n

[ E , B ] . [ H ( V ) ] + [ V V V . S ] .