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An Invariant of Quadratic Forms over Schemes

Marek Szyjewski

Received September

Communicated by Alexander S Merkurjev

0

Abstract A ring homomorphism e W X EX from the Witt ring of

a scheme X into a prop er sub quotient EX of the Grothendieck ring K X

0

is a natural generalization of the dimension index for a Witt ring of a eld

0

In the case of an even dimensional pro jective quadric X the value of e on

the Witt class of a bundle of an endomorphisms E of an indecomp osable

comp onent V of the Swan sheaf U with the trace of a pro duct as a bilinear

0

form is outside of the image of comp osition W F W X E X

Therefore the Witt class of E is not extended

Introduction

An imp ortant role in the theory is played by the rst dimensional

0

cohomological invariant the dimension index e W F ZZ which maps a

Witt class of a symmetric bilinear space V over a eld F onto dim V mo d A

straightforward generalization of this map for symmetric bilinear spaces over rings or

schemes which assigns to a Witt class the rank of its supp orting mo dule or bundle is

0

commonly used We dene a b etter invariant e in Section b elow It is a variant of

0

the construction used in and The map e dened in Section assigns to a Witt

class of a symmetric bilinear space V a class V of V in the group EX attached

functorially to a scheme X The group EX consist of the selfdual ie stable under

dualization elements of the Grothendieck group K X up to the split selfdual ones

0

ie sums of a class and its dual Thus the rank mo d may b e obtained by passing

to the generic stalk The group EX carries much more information on the Witt

0

group W X than ZZ and so do es the map e dened here when compared to the

rank mo d In particular we use it here to show that certain Witt classes are not

extended ie are not of the form V O for a symmetric bilinear space

X

V over a base eld

In the Section basic facts on dualization in the Grothendieck group denition

0

and elementary prop erties of the group EX and map e are given Theorem

describ es EX for a smo oth curve X In the geometric case algebraically closed base

eld the group EX app ears to coincide with the W X of curve X itself

Documenta Mathematica

Marek Szyjewski

Moreover it is shown that Witt classes of line bundles of order two in Picard group

are not extended from the base eld

Section contains a number of examples to show that EX may b e actually

computed the ane space Prop osition the pro jective space over a eld

Prop osition the pro jective space over a scheme Prop osition

The main ob jective of this pap er is to prove that on the pro jective quadric of even

dimension d dened by a hyperb olic form there exist nonextended Witt classes

For this purp ose a close lo ok at the Swan computation of the K theory of a quadric

hypersurface is needed Section contains all needed facts on Cliord algebras and

mo dules the construction of the Swan bundle its b ehavior under dualization and

how to nd a canonical resolution of a regular bundle

In Section we develop a combinatorial metho d for op erations with resolutions

using generating functions Next we use the classical computation of the Chow ring

of a split quadric X to establish the ring structure of K X Theorem gives the

0

description of EX for a split quadric

Thus in Section we show in Theorem that in case of even dimension d

of a quadric the bundle of endomorphisms of each indecomp osable comp onent of the

Swan bundle carries a canonical symmetric whose Witt class is not

0

extended from the base eld since its invariant e has a value outside the image of

the comp osite map W F W X EX

The rst version of this pap er contained only an explicit computation for a

quadric of dimension The referee made several suggestions for simplication of

pro ofs and computations These remarks led author to the present more general re

sults The author would like to thank very much the referee for generous assistance

The author is glad to thank Prof W Scharlau for helpful discussions and Prof K

Szymiczek who suggested several improvements of the exp osition

0

The group EX and the invariant e

Notation

If X is a scheme with the structural sheaf O and M N are coherent lo cally free

X

sheaves of O mo dules vector bundles on X M N is a morphism then we

X

write

M H om M O and N M

O X

X

for the duals

A symmetric bilinear space M consists of a coherent lo cally free sheaf M

which is selfdual ie and a morphism M M

For a subbundle a subsheaf which is lo cally a direct summand N M

dene its N as a kernel of comp osition

^

N KerM M N

MN Thus induces an isomorphism N

There are two imp ortant sp ecial cases the rst when N has trivial intersection

with N or is nonsingular then induces an isomorphism N N the second

when N N and in this case N is said to b e a Lagrangian subbund le

Documenta Mathematica

Quadratic Forms over Schemes

A symmetric bilinear space M is said to b e metabolic if it p ossesses a La

grangian subbundle ie if there exists an exact sequence

^

N M N

for some subbundle N

Direct sum and tensor pro duct are dened in the set B X of isomorphism classes

of symmetric bilinear spaces and in its Grothendieck ring GX the set M X of

dierences of classes of metab olic spaces forms an ideal

The Witt ring W X of X is the factor ring GX M X The Witt class of

a symmetric bilinear space M is its coset in W X Two symmetric bilinear

spaces M and M are Witt equivalent M M i their Witt

1 1 2 2 1 1 2 2

classes are equal or equivalently i M M is metab olic Each

1 2 1 2

Witt class an element of W X contains a symmetric bilinear space and X W X

is a contravariant functor on schemes namely for arbitrary morphism f Y X of

schemes the inverse image functor f induces a ring homomorphism f W X

W Y In fact f M f M and f is an exact functor In the ane case

#

X Sp ec R Y Sp ec S f R S a ring homomorphism f W X

W Y is simply the extension S W R W S Imp ortant sp ecial

R

cases are lo calization or taking a stalk at a p oint x X ie the inverse image for

Sp ec O X and the extension ie taking the inverse image for the structure

X x

map f X Sp ec F for a variety X over a eld F In the latter case a Witt class

of the form f M f M O for genuine bilinear space M over

F X

F is said to b e extended or induced from the base eld F

Rank mo d

In the ane case X Sp ec R we write as usual W R instead of W Sp ec R The

classical situation is if R F is a eld of characteristic dierent from two In this

case there is a ring homomorphism

0 0

e W F ZZ e M dim M mo d

0

known as dimension index One may put the denition of e into a K theoretical

framework as follows

The map e M M induces a ring homomorphism

e

=

GF K F Z

0

which is surjective since each over F carries a symmetric bilinear form

Any metab olic form M is hyperbolic ie the sequence splits and

M N N

0

Since each vector space is selfdual eM F K F Z so e is the induced

0

ring homomorphism

0

e

W F ZZ K F K F

0 0

Documenta Mathematica

Marek Szyjewski

In general the forgetful functor M M induces a ring homomorphism

which in general neither is surjective nor maps M X into K X We shall show

0

b elow how to handle this using a prop er sub quotient of K X

0

The involution and the group E X

Denote by PX the category of lo cally free coherent O mo dules The dualization

X

op

functor is an exact additive functor PX PX which preserves tensor

pro ducts and commutes with inverse image functors Since

op

K PX K PX K X

the functor induces a homomorphism on K groups known also as the Adams op

1

eration We shall denote it by

Definition K X K X is the homomorphism induced by the exact

op

functor PX PX

Proposition The homomorphism K X K X enjoys the following

prop erties

i is an involution

ii is a graded ring automorphism of K X

iii if f Y X is a morphism of schemes then f f

iv if i Z X is a closed immersion and X is regular of nite dimension

then i K Z i K Z

0 0

Pro of iv Consider a nite resolution of i M by vector bundles for a bundle M on

Z The stalk of this resolution at any p oint outside Z is exact so its dual is exact

Hence the class of the alternating sum of the members of the resolution vanishes

outside Z

We fo cus our attention on the Grothendieck group K X The main ob ject of

0

this pap er are the homology groups of the following complex

^ ^ ^ ^

1+ 1 1+ 1

K X K X K X K X

0 0 0 0

Definition

EX Ker Im

E X Ker Im

0

We shall dene a natural homomorphism e W X EX The group

E X will play only a technical role here although one may consider a natural map

L X E X The EX is the group of symmetric or selfdual elements in

2k +1

K X mo dulo split selfdual elements ie elements of the form M M The

0

following observations are obvious

Documenta Mathematica

Quadratic Forms over Schemes

Proposition i Ker is a subring of K X and the groups Im

0

Ker Im are Ker mo dules

ii EX is a ring and E X is an EX mo dule

iii an arbitrary morphism f Y X of schemes induces a ring homomorphism

f EX EY and an EX mo dule homomorphism f E X E Y

iv for a regular No etherian X EX and E X carry a natural ltration induced

by the top ological ltration of K X K X

0

0

v EX and E X

Note that the forgetful functor M M induces a ring homomorphism

GX K X which admits values in Ker and maps M X onto Im

0

since for a metab olic space M there is exact sequence ie the equality

M N N holds in K X

0

0

Definition e W X EX is the ring homomorphism induced by the

forgetful functor M M

This notion enjoys nice functorial prop erties

Proposition Let f X Y b e a morphism of schemes Then the following

diagram commutes

0

e

W X EX

x x

 

f f

0

e

W Y EY

Example Let X b e an irreducible scheme with the function eld F X and

let j Sp ec F X X b e the embedding of the generic p oint Then there is a

commutative diagram

0

e

W F X E F X ZZ

x x

 

j j

0

e

W X EX

0 0 0

and the comp osition j e e j is rank mo d usually used instead of e Since

O carries the standard symmetric bilinear form the surjection j EX

X

ZZ splits canonically The kernel of the map j EX ZZ has b een used in

It is easy to see that this kernel is a nilp otent ideal of a ring EX for a regular

No etherian X of nite dimension

Documenta Mathematica

Marek Szyjewski

Example Retain the notation of example and assume in addition that

X is a variety over a eld F char F Let f X Spec F b e the structure map

Thus we have a commutative diagram

0

e

W F X ZZ

I

0

e

W X

EX

id

I

0

e

ZZ W F

0

The values of e f are inside the direct summand ZZO of EX If we

X

pro duce a variety X with nontrivial ie having more than two elements EX and

0

a symmetric bilinear space with a nontrivial value of e then the Witt class of this

space must b e nonextended

Curves

The case dim X is exceptional for several reasons so we treat it here as an

illustration The following theorem covers the classical case of sp ectra of Dedekind

rings

Theorem Let X b e an irreducible regular No etherian scheme of dimension one

Then

i EX ZZO I where I I and I is canonically isomorphic to the

X

group PicX of the elements of order in the Picard group

2

ii E X is canonically isomorphic to PicX PicX

0

iii the map e W X EX is surjective

Pro of The rank map ie the restriction to the generic p oint yields the splitting

1

K X Z O F K X

0 X 0

1

where F K X K X is the top ological ltration on K X The map

0 0 0

maps each direct summand onto itself

Documenta Mathematica

Quadratic Forms over Schemes

V

1

Under assumptions on X the map F K X PicX induced by taking

0

the highest exterior p ower of a bundle is an isomorphism An arbitrary element of

1

the group F K X may b e expressed as a dierence of the classes of two bundles of

0

the same rank r

M N

V

The isomorphism maps onto the class of a line bundle L

r r

N M L

V

in PicX The isomorphism maps L O onto the class L in PicX to o So

X

1

any element a of F K X may b e expressed as a dierence of a line bundle and the

0

trivial line bundle

L O

X

Moreover for arbitrary line bundles L L

1 2

L O L O L L O

1 X 2 X 1 2 X

1

Hence the involution acts on F K X as taking the opp osite and it acts

0

trivially on Z O Therefore

X

1

Ker Z O F K X Im Z O

X 0 X

1 1

Ker F K X Im F K X

0 0

and assertions i ii follow

To prove iii note that a line bundle L which has order two in PicX is isomorphic

to its inverse L so is automatically endowed with a nonsingular bilinear form

L L This form must b e symmetric lo cally at any p oint hence is symmetric

0

globally Finally e maps the Witt class of L O onto the class of L

X

V

in PicX via

Remark If R is a Dedekind ring X Sp ec R then PicX PicR is sim

1

ply the ideal class group H R the claim on the form of element of F K X is

0

a consequence of the structural theorem for pro jective mo dules if rankP r

then there exist fractional ideals I I such that P I I moreover

1 r 1 r

V

r

r 1 1 r 1

R P In this case L X PicX PicX and P R I I

1 r

1 0

L X is isomorphic to E X via obvious generalization of e

Remark If PicX is nontrivial PicX then there exist nonextended

2 2

Witt classes on X

Corollary If X is a smo oth pro jective curve of genus g over an algebraically

closed eld F then

1+2g

i if char F then EX ZZ

ii the degree map induces isomorphism E X ZZ

Documenta Mathematica

Marek Szyjewski

Remark The result in Corollary i has b een p ointed out to author by W

Scharlau

Remark The prop osition of states that for F C the Witt group W X

1+2g

of a smo oth pro jective curve X is itself isomorphic to ZZ but the pro of

remains valid for an arbitrary algebraically closed eld F provided char F So

0

under assumptions of Corollary i the map e W X EX is an isomorphism

0

The map e W X EX for certain quasiprojective X

We shall show now that the group EX may b e actually computed and compare the

result with known Witt rings The simplest case is following

n

Proposition If R is a regular ring and X A the ane space then the

R

inverse image functor f for the structure map f X Sp ec R induces isomorphisms

W R W X ER EX E R E X

Pro of By the homotopy prop erty of K theory the map f K R K X

0 0

is an isomorphism and commutes with so the assertion on E and E follows

The assertion on W X is a consequence of the Karoubi theorem see Ch VI

Corollary

Now let X b e a quasipro jective variety over a eld F char F with the

structure map f X Sp ec F Consider the commutative diagram

0

e

W X EX

x x

 

f f

W F EF

0

e

We shall refer to left f and right f in for various X

n

Next x a pro jective embedding i X P and denote

F

O the unit element in K X

X 0

n

O i O

X P

F

H O the class of hyperplane section in K X

X 0

We summarize some technicalities as follows

Lemma If d dim X then

d+1

i H

d

X

i 0

1

H in K X here H ii O H

0 X

i=0

d

X

H

i

iii H H

H

i=1

dk

k

X

k i H

i

k k k

H H iv H

H i

i=0

d d d

v H H

Documenta Mathematica

Quadratic Forms over Schemes

1

Pro of H O so O H O H H b eing

X X X

1

nilp otent Thus H O O O H H and

X X X

k k k

H H H

d d

In the case i id X P the family H H forms a of a free

F

Ab elian group K X which allows us to compute E X E X

0

d

the pro jective space then Proposition If X P

F

i b oth vertical arrows in the diagram are isomorphisms

d

ii E X ZZ H for o dd d and E X for even d

Pro of The left f in the diagram is an isomorphism by Arasons theorem

Note that the statements on EX E X are valid for d and by Theorem

d1

ab ove for d Consider Y P and a closed embedding k Y X of Y as a

F

hyperplane in X There is an exact sequence



k

d

Z H K X K Y

0 0

since k O i O i Thus we have a short exact sequence of complexes

X Y

^ ^ ^

1 1+ 1

K Y K Y

0 0

x x

 

k k

^ ^ ^

1 1+ 1

K X K X

0 0

x x

d d d

1(1) 1+(1) 1(1)

d d

Z H Z H

and an induced exact sequence in homology For even d this lo oks like

d

E X E Y ZZ H EX EY

and if by induction the prop osition holds for Y then maps the generator of

d1 d d

E Y ZZ H onto H mo d Z H so the prop osition holds for X E X

k EX EY is an isomorphism In case of an o dd d we have an exact sequence

d

EX EY ZZ H E X E Y

in homology By induction EY ZZ O so k EX EY is an

X

d

isomorphism Thus ZZ H E X is an isomorphism since E Y

Remark The idea of this pro of is due to the referee

d

Proposition For an arbitrary variety Y let X P Y and let

F

d

p X P p X Y b e the pro jections Then

1 2

F

d d

E Y p E P EY p p E P EX p

2 F 1 2 F 1

d d

E X p E P p E Y p E P p EY

1 F 2 1 F 2

Documenta Mathematica

Marek Szyjewski

Pro of By the pro jective bundle theorem p p yield the identication K X

0

1 2

d

K Y Denote K P

0 0

F

^

1

d d d

A KerK P K P B K P

0 0 0

F F F

d

The complex for X P Y may b e included into the short exact sequence

F

of complexes

^ ^ ^

1 1+ 1

B K Y B K Y

0 0

x x

^ ^

(1 )1 (1 )1

^ ^ ^

1 1+ 1

K X K X

0 0

x x

^ ^ ^

1 1+ 1

A K Y A K Y

0 0

Note that restricted to A K Y coincides with and induces

0

on B K Y Therefore the exact hexagon in homology

0

EX

R

A EY B E Y

x

2 1

y

B EY A E Y

I

E X

breaks into short split exact sequences

d d

E P E Y E X E P EY

F F

d d

E P EY EX E P E Y

F F

1 1 1

ie X P P Then Example Put d Y P

F F F

EX ZZ O ZZ H  H

X

E X ZZ H  ZZ  H

where  is induced by op eration F  G p F p G Since Witt ring is an invariant

1 2

of birational equivalence in the class of smo oth pro jective surfaces over a eld F

1 1 2

char F Theorem and X P P is birationally equivalent to P the

F F F

left f in the diagram is an isomorphism while the right f is not This example

0

shows that e W X EX need not b e surjective in general

Documenta Mathematica

Quadratic Forms over Schemes

Remark Probably there exists a skew symmetric bilinear space M on X

1 1

such that M H  H in EX P P

F F

1 1 3

Remark X P P may b e embedded into P by Segre immersion as a

F F F

quadric surface x x x x In fact in the preliminary version of this pap er this

0 1 2 3

example was given using Swans description of the K theory of a quadric The idea

to use inverse images for pro jections was p ointed out to author by the referee

Remark Note that we know W X and EX for three quadrics of maximal

index

X equation W X EX E X

2 2

two p oints z z W F W F ZZ ZZ

0 1

1 2 2 2

P z z z W F ZZ ZZ

0 1 2

F

1 1

P P x x x x W F ZZ ZZ ZZ ZZ

0 1 2 3

F F

We shall compute EX and E X for all pro jective quadrics of maximal index

To do this some preparational work is required

The Swan K theory of a split projective quadric

To compute EX and E X we need some facts on dualization of vector bundles on

quadrics All needed information is known in fact since indecomp osable comp onents

of a Swan sheaf corresp ond to spinor representations Nevertheless we give here

complete pro ofs of the needed facts

We shall apply the results of in the simplest p ossible case of a split quadric

X is a pro jective quadric hypersurface over a eld F char F dened by the

quadratic form of maximal index

Notation

Consider a vector space V with basis v v v over a eld F char F

0 1 d+1

Denote z z z the dual basis of V Let q b e the quadratic form

0 1 d+1

d+1

X

i 2

z q

i

i=0

1 1

v v f v v for all p ossible values of i Moreover let e

2i 2i+1 i 2i 2i+1 i

2 2

Thus if d is even d m then e f e f e f form a basis of V with

0 0 1 1 m m

the dual basis x y x y x y and

0 0 1 1 m m

m

X

x y q

i i

i=0

If d is o dd d m then f e f e f v form a basis of V

0 1 1 m m d+1

with the dual basis x y x y x y z and

0 0 1 1 m m d+1

m

X

2

x y z q

i i

d+1

i=0

Documenta Mathematica

Marek Szyjewski

We shall compute EX and E X for a ddimensional pro jective quadric X dened

d+1

by equation q in P ie for

F

X Pro j S V q Pro j F z z z q

0 1 d+1

The

In case of an o dd d m the even part C C q of the Cliord algebra

0 0

m+1

C q is isomorphic to the matrix algebra M F where N In particular

N

K C K F

p 0 p

In case of an even d m the algebra C has the center F F where

0

1 1

2

are orthogonal central v v v and Thus

0 1 d+1

2 2

idemp otents of C so

0

C C C

0 0 0

m

F For even where each direct summand is isomorphic to the matrix algebra M

2

d m consider the principal antiautomorphism C C

0 0

k

w w w w w w for w w w V

1 2 k k k 1 1 1 2 k

Note that

m+1

1

Moreover for every anisotropic vector w V the reection w w in V

induces an automorphism of C which interchanges with its opp osite

w 0

w

Regarding subscripts i mo d denote

i

P C for even d

i 0

Lemma In case of an even d m

i the involution of the algebra C provides an identication of the left C

0 0

mo dule P Hom P F with the right C mo dule P

i F i 0 i+m+1

ii for any anisotropic vector w V the reection interchanges P s P

w i w i

P

i+1

Swan K theory of a quadric

Recall some basic facts and notation of Denote by C the o dd part of the Cliord

1

algebra C q We shall use mo d subscripts in C Recall the denition of the Swan

i

bundle U Put

d+1

X

z v X O V

i i X

i=0

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Quadratic Forms over Schemes

The complex

O n C O n C

X n+d X n+d+1

O n C

X n+d1

is exact and lo cally splits Prop a

Definition

O n C U CokerO n C

X n+d+2 n X n+d+3

U U

d1

Since the complex is up to a twist p erio dical with p erio d two we have

U U

n+2 n

Consider the exact sequences

O n C U O n C

X n+d+2 n X n+d+3

for two consecutive values n twist the rst one by For any anisotropic vector w V

the isomorphism given by right multiplication by w ts into the commutative

diagram

O n C U O n C

X n+d+3 n+1 X n+d+4

1w 1w

= =

y y

O n C U O n C

X n+d+2 n X n+d+3

Thus we have proved the following lemma

Lemma

U n U and U U

0 n n n+1

for arbitrary integer n

There is an exact sequence

O C U U

X 0 1 0

where an isomorphism w was used to replace O C by O C for even d

X 1 X 0

Lemma End U C acts on U from the right

X n 0 n

Pro of Lemma

Documenta Mathematica

Marek Szyjewski

We are now ready to compute U

n

U d U n in particular U Lemma U

n n

Pro of We have chosen a basis v v v of V in ab ove The set of naturally

0 1 d+1

ordered pro ducts of several v s in an even number forms a basis of C Dene a

i 0

quadratic form Q on C as follows let the distinct basis pro ducts b e orthogonal to

0

each other and

q v q v v q v v Qv

i i i i i i

2 1 2 1

k k

The form Q is nonsingular and denes by scalar extension a nonsingular

2

symmetric bilinear form on O C Since q v a direct computation

X 0 i

U is a totally isotropic subspace O C U shows that Im O C

0 X 0 0 X 1

of O C Therefore

X 0

U O C U U U U

1 X 0 0 0 0 0

follows quickly from sect ab ove and the exactness of Thus

U U U

0 1 0

and in general

U n U n U n U n U

n 0 0 0 n

Remark This argument was p ointed out to the author by the referee

Corollary i U U d and U d U d d

in K X

0

1

d

ii rank U dim C

0

2

In case of an even d m the algebra End U C splits into the direct

X 0

pro duct of subalgebras dened in ab ove C P P

0 0 1

Definition In case of an even d

P U P U U U

1 n C 0 n C

0 0

n n

P P U U U U

1 0 C C

0 0

Note that U U U U U U U and U corresp ond to spinor repre

n

n n 0 0

sentation and we shall copy here the standard argument on dualization compare

sect

In case of an even d m another prop erty of and the quadratic form Q

introduced in the pro of of Lemma may b e veried by direct computation

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Quadratic Forms over Schemes

Lemma In case of an even d m

i if m is even then P C are orthogonal to each other hence selfdual

i 0

ii if m is o dd then P C are totally isotropic hence dual to each other

i 0

iii

Corollary In case of an even d m

U d for even m U d and U i U

U d for o dd m U d and U ii U

m

iii End U End U F M

X X 2

iv the exact sequence splits into two exact parts

U O P U

X 0

0 0

U O P U

X 1

0 0

The standard way to determine indecomp osable comp onents is tensoring with

the simple left mo dule over an appropriate endomorphism algebra We will use from

here onwards sup erscript for the direct sum of identical ob jects

Definition

m+1

2

F i in case of an o dd d m V U

C

0

m

2

ii in case of an even d m V U F

0

m

M (F )

2

m

2

V U F

1

m

M (F )

2

n n

For convenience we will use mo d subscripts in V Since M F F

i n

as a left M F mo dule indecomp osable comp onents inherit prop erties of the Swan

n

bundle we have

Proposition a In case of an o dd d m

m+1

2

i U V

ii V V d

m

F and rank V iii End V

X

m

iv V d V d in K X

0

b In case of an even d m

m m

2 2

i U V and U V

0 1

ii V V d

i i+m

m1

iii End V F and rank V

X i i

m

iv V d V d in K X

i i+1 0

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In particular there is no global morphism V V

i i+1

Corollary In case of an even d m following identities hold in K X

0

i V V H

0 1

ii V V O n V V

0 1 X 0 1

m

iii V V V V

0 1 0 1

Pro of Prop osition b iv yields

V d V d V d V d

0 1 1 0

Tensoring with O d one obtains

X

V V V V O

0 1 0 1 X

hence i and ii Thus iii results from b ii

Proposition K X is a free K F mo dule of the rank m moreover

i in case of an o dd d m the classes O O O d V

X X X

form a basis of K X

ii in case of an even d m the classes O O O d V

X X X 0

V form a basis of K X

1

Pro of Apply Theorem of

We have expressed the action of on K X in terms of a twist We need a plain

0

expression in order to determine EX and E X

We recall here several facts known from section of needed for establishing plain

formulas for the action of

Every regular sheaf F on X has a canonical resolution innite in general

k

k k

0

1 p

F O O O p

X X X

where sup erscript k means a direct sum of k copies One may compute the co e

p p

cients k and the dierentials recursively as follows put Z F Since a regular

p 1

sheaf is generated by its global sections put k dim X Z p and dene Z

p p1 p

as the twisted kernel in

k

p

Z p Z p O

p1 p

X

Then Z p is a regular sheaf Therefore the sequence

p

k

k k

p

0 1

F p O p O p Z p O

X X p X

is exact Twisting it by one obtains an exact sequence of regular sheaves

k k k

0 1 p

F p O p O p Z p O

X X p X

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Since the functor of global sections is exact on regular sheaves there is following

recurrence for k dim X Z p

p+1 p

dim X F p k dim X O p

0 X

p1 p

k dim X O k

p X p+1

In case of a regular F to obtain an expression for F K X in terms of

0

the basis from Prop osition one truncates the canonical resolution of F

k

k k

0

1

d1

F O Z O d O

X d1 X X

Z Then in K X Hom U Z and replaces Z by U

d1 0 X d1 d1 C

0

d1

X

Hom U Z O i U F

X d1 X C

0

i=1

Dep ending on the parity of d we have there

Hom U Z aV U

X d1 C

0

or

Hom U Z aV bV U

X d1 0 1 C

0

where the integers a b in turn dep end on the decomp osition of Hom U Z into

X d1

a direct sum of simple left C mo dules Conversely for a given F the equality

0

d1

X

F O i W

X

i=1

holds where W is either aV or aV bV then k is the Euler characteris

0 1 0

P

i i

tic dim H X F of F So if F is regular then k dim X F Next

0

k

0

F is regular and iterating yields that for a regular F Z KerO

0 X

the congruence

F O i mo d ImK C K X

X 0 0 0

holds if and only if integers k satisfy In case of an o dd d m in order

i

to express a class F of a regular sheaf F in terms of the basis of Prop osition

it is enough to know the dimensions of X F i for i d to

determine the k s and the rank F to determine the co ecient a of V An analogous

i

statement remains valid for an arbitrary sheaf F with Euler characteristic of F i in

place of dim X F i In case of an even d m in view of Corollary ii and

Prop osition ii the bundles V and V have the same Euler characteristic rank

0 1

and even the highest exterior p ower Thus without sp ecial considerations one can

express a class F in terms of basis of the Prop osition only up to a multiple

of V V

0 1

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The group EX for a split projective quadric

A Poincare series of a sheaf

We introduce here a metho d for the determination of the co ecients of the canonical

resolution of a large enough class of regular sheaves A Poincare series t of a

F

sheaf F is the formal p ower series

X

def

i

dim X F i t Zt t

F

i=0

The Poincare series t of a variety S is the Poincare series of its structural

S

sheaf

def

t t

S O

S

In particular if S Pro j A for a graded algebra A then t is the usual

S

Poincare series of A

n

Example If S is the pro jective space S P then dim S O i

S

F

n i

so

i

X

n i

def

i n1

t t P t t

n S

i

i=0

Example Let f b e a homogeneous p olynomial of degree k in homogeneous

d+1

co ordinates in P Pro j B B F x x x A B f S Pro j A

0 1 d+1

F

d+1

Since the exact sequence a hypersurface f in P

F

f

B A B

n+k n

splits for every n the following equality holds

k

t P t t P t

S d+1 d+1

k k 1

t t t

Thus t

S

d+2 d+1

t t

Lemma For a pro jective quadric X of dimension d

t

def

Q t t

d X

d+1

t

Proposition If F F F is an exact sequence of O mo dules

X

and either F F are regular or F F are regular then

00 0

t t t

F F F

Pro of By Sect Lemma either F F F are regular or F F F are

regular Hence each exact sequence of sheaves

F i F i F i

induces an exact sequence of global sections

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The generating function for the canonical resolution

The recursive metho d of nding a canonical resolution

k

k k

0

1 p

F O O O p

X X X

of a regular sheaf F describ ed in ab ove namely the identity yields following

X

def

i

k t identities for the generating function G t

i F

i=0

t

F

t G t t and G t

F F X F

Q t

d

Example The generating function for the canonical resolution of the sheaf

O

X

t

O

X

t

O (1)

X

t

so

1t

d+1

d+1

t t t Q t

(1+t)

O d

X

G t

O (1)

X

1t

Q t tQ t t t

t

d d

d+1

(1+t)

l l

Example For a linear section H O of co dimension l in X

X

l

G l t

H

Example Continue the notation of Since X splits it contains linear

subvarieties S Pro j F x x given by the following equations

k 0 k

a in case of an even d m

y y x x for k m and

0 m k +1 m

y y for k m

0 m

b in case of an o dd d m

y y z x x for k m and

0 m d k +1 m

y y z for k m

0 m d

k

in particular its structural sheaf L is regular Therefore S is isomorphic to P

k k

F

dk k 1

P t t t

k

G t

L

k

1t

Q t t

d

d+1

(1+t)

Lemma

dk

G G t

L L

k k 1

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The generating function for a truncated canonical resolution

Truncating a generating function G one obtains a p olynomial T For l d the

F F

l

canonical resolution for H is itself truncated

l

T l t for l d

H

The sequence c of co ecients of the canonical resolution of the sheaf L sta

i k

bilizes from the degree d k onwards

dk

X X

t

i i dk

c t t G t

i L

k

t

i=0 i=0

so

dk

c c

dk dk +1

Thus

dk d

t t

T t

L

k

t

Proposition If for xed k L is a structural sheaf of a linear subvariety S

k k

of dimension k in X then in K X

0

a in case of an o dd d m

i d1

X X

d k

i mk

O i V L

X k

p

p=0

i=0

b in case of an even d m for a suitable integer a

i d1

X X

d k

i mk

O i aV aV L

X 0 1 k

p

p=0

i=0

Pro of Substituting t O into the expansion for T t yields dep ending

X L

k

on the parity of d the expressions

i d1

X X

d k

i

O i aV L

X k

p

p=0

i=0

i d1

X X

d k

i

O i aV bV L

X 0 1 k

p

p=0

i=0

for suitable integers a b Thus

d m

T a

L

k

d m dk 1

a for d m

rankL

k

d m1

T a b

L

k

d m1 dk 1

a b for d m

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The topological filtration

Now we shall nd a basis of K X which is convenient for computations Since the

0

quadric X is regular K X K X and one may transfer the top ological ltration

0

0

p

F K X subgroup generated by

0

the stalk F for all generic p oints

x

F

x of subvarieties of co dimension p

of K X to K X We omit the standard pro of of following

0

0

p

Proposition For a split pro jective quadric X the Chow groups A X are

isomorphic to the corresp onding factors of the top ological ltration

p p+1

p

A X F K X F K X

0 0

Continue the notation of Recall the classical computation of the Chow ring

of a split pro jective quadric

Proposition For a split pro jective quadric X of dimension d

a in case of an even d m

m p

Z Z Z for p m p m and A X A X

b in case of an o dd d m

p

A X Z for al l p p m

Explicit generators are given as follows

Case d m

i for p m a class of any linear subvariety of dimension d p eg

S y y x x

dp 0 m dp+1 m

p

ii for p m a class H of a linear section of co dimension p

m

iii for p m A X is generated by two classes of linear subvarieties

S x x and S y x x

0 m 0 1 m

m m

m

the classes in A X remain unchanged if an even number of x y are

i i

exchanged in these equations

Case d m

i for p m a class of any linear subvariety of dimension d p eg

S y y z x x

dp 0 m d+1 dp+1 m

p

ii for p m a class H of a linear section of co dimension p

For a sketch of pro of and references see Thm

Now we can give an explicit description of the ring structure and the action of

the involution on K X To do this denote L L the class of the structural

0 p p

sheaf of the linear subvariety S of dimension p Moreover in case of an even d m

p

resp ectively and S the class of the structural sheaf of S and L denote by L

m m m m

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Theorem Let X b e a split pro jective quadric of dimension d Then

2 m

i in case of an o dd d m classes H H H L L form a

m 0

basis of the free Abelian group K X

0

2 m1

ii in case of an even d m classes H H H L L L

m1

m m

L form a basis of the free Ab elian group K X

0 0

iii in case of an even d m classes may b e chosen as follows

i d1

X X

m

i

O i V L

X 0

m

p

p=0

i=0

i d1

X X

m

i

O i V L

X 1

m

p

p=0

i=0

and for dimensions k m

i d1

X X

d k

i mk 1

O i V V L

X 0 1 k

p

p=0

i=0

m

iv if d m then H L L L

m1

m m

L H L v H L L H L

m1 p p1

m m

d

dk d d+1

vi H L L for k H L H

k k 1 0

vii L L L L L L

p q p p

m m

2 2

if d m and L L L L viii if d m and m is even then L

0

m m m m

2 2

m is o dd then L L L L L

0

m m m m

d1

k

Pro of First of all note that the classes H L for k g L and the pair fL

k

m m

2

are determined uniquely by the conditions of irreducibility of the underlying subva

p

riety and to form a basis of some appropriate A X In fact by Prop osition

d

d

this is clear for F K X A X Thus the general case results by induction

0

Statements i and ii follow from Prop osition and To verify iii note

xes v v v and changes v into the opp osite that the reection

0 2 d+1 1 v

1

ab ove Thus this reection induces an automorphism of the symmetric algebra

S V which interchanges x with y and xes other co ordinates and q Therefore

0 0

it induces an automorphism of S V q X Pro j S V q a semilinear auto

morphism of O n for all n and an automorphism of K X By Lemma ii

X 0

interchanges the P s with each other So the induced automor the reection

i v

1

phism of U interchanges direct summands U U P and U U P of U and

0 1

their indecomp osable comp onents V V Therefore the induced automorphism of

0 1

K X xes the basic elements O O O d and interchanges

0 X X X

V with V By the uniqueness statement this automorphism xes L L

0 1 0 m1

The explicit description given in Prop osition ii shows that this automorphism

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interchanges L with L Hence by the explicit formula of Prop osition ii for

m m

k m

i d1

X X

d k

i mk

O i aV aV L L

X 0 1 k k

p

p=0

i=0

mk 1

the integer a must b e equal to This same argument for k m yields

i d1

X X

d k

i

O i aV aV L

X 0 1

m

p

p=0

i=0

and

i d1

X X

d k

i

O i aV aV L

X 0 1

m

p

p=0

i=0

Since the statement ii of the theorem holds the integer a must b e or this

to o Statements and S follows from the regularity of the structural sheaves of S

m m

iv vii are obvious consequences of the uniqueness and the explicit equations of

Prop osition For to prove viii assume without loss of generality that L is

m

the class of S and L is the class of S Consider the class L of the subvariety

m

m m m

S y y In case of an even m the classes L and L coincide and

m 0 m m

m

Moreover L transversally at the empty set of p oints so L S meets S

m

m m m

2

S meets S transversally at the rational p oint S so L L Analogously

m 0 0

m m

2

L L

0

m

2 2

In case of an o dd m L L so L L L L L

m 0

m m m m m

Theorem For a split pro jective quadric X of dimension d the involution acts

as follows

k

X

d d k i

dk

L for k i L

k i k

i

i=0

ii in case of an even d m

m

X

m i

m

L L L

mi

m m

i

i=1

m

X

m i

m

L L L

mi

m m

i

i=1

m1

X

j m

k k j

A

H H L L

m m

k k

j =k

d

X

j j

k

A

L

dj

k k

j =m+1

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iii in case of an o dd d m

m1

X

j m

k k j

A

H H L

m

k k

j =k

d

X

j j

k

A

L

dj

k k

j =m+1

dk

Pro of i Since H L L L H L by the Theorem iv vi

k k 1 k k

dk

H H

and H L

k

H H

by Lemma iii

dk

H

dk

H

H

dk

L L

k k

dk 1 dk 1

H

H H H

H

d d

X X

d k i d k i

i dk dk

L H L

k i k

i i

i=0 i=0

k dk m

ii To obtain the formula for H substitute H L L and H L

k k 1

m

L L into the formula of Lemma iv Analogously one proves iii In case

m1

m

d m

m m+1

H H

m m

L L H L H

m1

m m

H H

so

m

m

H H

m

L L

m m

m1

H

H H H

H

d

X

m i

i m m

H L L L L

m m m m

m1

i H

i=0

and thus

m

X

m i

m

L L L L L

mi

m m m m

i

i=1

On the other hand by Theorem iii

V V L L

0 1

m m

and by Corollary iii

m m

L V V L L L

0 1

m m m m

The formula for L and L follows directly since we know their sum and dierence

m m

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Consider the matrix A of the involution in the free Ab elian group K X with

0

resp ect to the basis given in Theorem i ii We shall write it in a slightly unusual

way

A a i j m

ij

In case of an even d m we regard A as a blo ck matrix B arranging two central

rows and two central columns into separate blo cks

a a

mm mm+1

2 2

HomZ Z b

mm

a a

m+1m m+1m+1

a

mi

2

HomZ Z b

mi

a

m+1i

2

b a a HomZ Z for i m

im im im+1

a for i j m

ij

a for i m j m

ij +1

b

ij

a for i m j m

i+1j

a for i j m

i+1j +1

As one may exp ect in view of Prop osition iv the matrix A is triangular in the

o dd dimensional case and the matrix B is triangular in the even dimensional case

We summarize the most imp ortant results of Theorem as follows

Corollary a In case of an o dd d m the matrix A is triangular with

i

a for i m

ii

a for i

i0

i

i for i m

m

a

m for i m

i+1i

i

i for i m m

b In case of an even d m the matrix B is blo ck triangular with

i

b for i m

ii

m

b

mm

b for i

i0

i

i for i m

b

i+1i

i

i for i m m

m1

b m

mm1

m

b m

m+1m

m

m

b

2mm

m

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Theorem Let X b e a split pro jective quadric of dimension d

a If d m is o dd then

EX ZZ O

X

ZZ L for even m

m

E X

m

ZZ H for o dd m

b If d m is even then

EX ZZ O ZZ L

X 0

for even m

E X

ZZ L ZZ L for o dd m

m m

Pro of Consider the complex

^ ^ ^

1+ 1 1+

K X K X K X K X

0 0 0 0

with the top ological ltration

0 1 d d+1

K X F K X F K X F K X F K X

0 0 0 0 0

and the corresp onding sp ectral sequence

p+q

pq

p+q p+q (1)

E Ker Im E X

1

1 1

where E X EX E X E X The E term has p erio d with resp ect to q

1

a In case of an o dd d m the term E lo oks like

1

q =1

d1

1 0

q =0

ZZ ZZ ZZ ZZ

The dierential is induced by the multiplication by the entry a of the matrix

i i+1i

q

A of Thus for each even q we have complex E

1

0 1 2

ZZ ZZ ZZ

(m1)

2m m

ZZ ZZ

(2m1)

ZZ ZZ

m+1q iq

Therefore for even m we have E q E ZZ and E for other values

2

2 2

of i Since the left the zeroth column of A has zero entries except a all the

00

0q

are trivial So EX ZZ O E X ZZ L dierentials starting from E

X m

r

0q mq iq

Analogously for an o dd m we have E E ZZ and E for other values

2 2 2

m

of i so EX ZZ O E X ZZ H

X

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b In case of an even d m the term E lo oks like

1

m1

d1

m 0

2

ZZ ZZ ZZ ZZ ZZ

The dierential is induced by the corresp onding blo ck of the matrix B of For

i

q

each even q we have a complex E

1

0 1 2

ZZ ZZ ZZ

2 3

4 5

(m1)

(m2) (m1)[1 1]

m

2

ZZ ZZ ZZ

(2m2)

ZZ ZZ

0q 2mq

Thus for even m and even q only E E ZZ are nonzero By the dimension

2 2

argument the sequence degenerates from E onwards Hence

2

EX ZZ O ZZ L E X for even m

X 0

0q 2mq mq

2

For o dd m and even q only E E ZZ and E ZZ are nonzero

2 2 2

0q

since the entries of the left zeroth There is no nonzero dierential starting from E

r

2mq m+1 mq

E column of B are except b All the dierentials but E

00

m m

must b e zero This exceptional one is zero to o since it is induced by b

2mm

m m

m

and is even for o dd m Therefore the sp ectral

m m

sequence degenerates and nally

EX ZZ O ZZ L E X ZZ L ZZ L for o dd m

X 0

m m

The theorem is proved

Nonextended Witt classes on certain split projective quadrics

We shall show here that if the dimension d of a split pro jective quadric X is even and

0 2

greater than two then the invariant e W X EX ZZ is surjective

For an arbitrary lo cally free coherent sheaf M the sheaf E E nd M

O

X

M M is selfdual and supp orts a canonical symmetric bilinear form which

reduces to the trace of a pro duct on stalks

tr for E x X

x

E E is the multiplication map then E E is adjoint of or if E

O

X

tr E E O

O X

X

Theorem If X is a split pro jective quadric of dimension d m m then

for an indecomp osable comp onent V of the Swan sheaf U

0

0

V L e E nd

0 0 O

X

V represents a nonextended Witt class in W X Thus E nd

0 O

X

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Pro of The case m is sp ecial so assume m We shall compute the class of

V V V d V d in EX We know from Prop osition b iv that

0 0 0 0

for d m

m

V d V d

0 1

On the other hand by Corollary ii and Theorem iii

V d V d V V L L

0 1 0 1

m m

Thus

m

V d O V d V d L L

0 X 0 1

m m

or

m

V d H L L

0

m m

The rules of multiplication in K X given in Theorem and Lemma yield

0

that multiplying b oth sides of this equality by

m m1 d

X X X

mj m+j di i m di i

H H L L L H

m1

m m

j =1 i=0 i=0

m m1

X X

mj di i m

L L H L L L

mj mj 1 m1

m m

j =1 i=0

m1

X

di i m

H L L

m m

i=0

we obtain

d+1 d+1 d+1

V d V d H

0 0

m1

X

di i m

H L L L L L

m1

m m m m

i=0

m1

X

mi1 i m m+1

H L L L L L

m1

m m m m

i=0

m1

X

d+1 mi1 i d d

H L L L L

m m m m

i=0

m1

X

mi1 i d+1 d+1

H L

m

i=0

m1

X

mi1 i

Thus H L Since K X is torsion free V d

0 0

m

i=0

m1

X

mi1 i

H L V d

0

m

i=0

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Note that mo d Im since

is a member of Im Also mo d Im Therefore

E V V V d V d

0 0 0 0

m1

X

2(mi1) i i

H H L L mo d Im

m m

i=0

If m then the rst summand equals while the second is For m

m1

X

d2i2 i i

H H L L E

m m

i=0

m2

X

d2i2 i i m1 m1

H H H H L L

m m

i=0

m1 m1

H H L L since

m m

m m

2 m1 m1

L L by Lemma H H H H

m m

mm

d2 d1 d

H m H H L L

m m

L L m L m L mm L

2 1 1 0 0

L L by Theorem vi

m m

2

L d L m L L L

2 1 0

m m

L L m L

2 2 0

m

L L terms of higher co dim by Theorem i

m m

mL L for even m

0 0

by Theorem viii

mL for o dd m

0

L mo d Im

0

0

Anyway e V L for m

0 0

In the particular case d there exists another symmetric bilinear form on

E V V the tensor pro duct of exterior multiplications

0 O 0

X

2 2

O V V O and V V V V

X 0 0 X 0 O 0 0 0 O

X X

On the stalks the asso ciated quadratic form is the determinant map Since the

0

value of e dep ends only on supp orting bundle E is nonextended as well as E

The symmetric bilinear space E has the following interesting prop erty it is

0

not metab olic since it has a nontrivial e but is hyperb olic on stalks ie locally

hyperbolic In fact any stalk V at x X is a free rank two O mo dule so

0x X x

any stalk of E is M O det which is hyperb olic Thus there is no lo cal

2 X x

invariant to detect the symmetric bilinear space E and such a global invariant as

0

e is useful If is a sum of two squares in F then E is lo cally hyperb olic to o

Documenta Mathematica

Marek Szyjewski

Note that the case d is of particular interest since the split fourdimensional

quadric X is the smallest nontrivial Gramann variety G Thus on general

2

Gramann varieties there may exist nonextended Witt classes contrary to the case

of pro jective spaces ie Gramann varieties G n

1

References

J Kr Arason Der Wittring projektiver Raume Math Ann

F FernandezCarmena On the injectivity of the map of the Witt group of a

scheme into the Witt group of its function eld Math Ann

Corr Math Ann

F FernandezCarmena The Witt group of a smooth complex surface Math Ann

M M Kapranov On the derived categories of coherent sheaves on some homo

geneous spaces Invent Math

M Knebusch Symmetric bilinear forms over algebraic varieties Conference on

quadratic forms ed G Orzech Queens pap ers in pure and applied math

T Y Lam Serres conjecture Lect Notes in Math Springer Berlin

D Quillen Higher algebraic K theory I Lect Notes in Math Springer

Berlin

A Ranicki Algebraic Ltheory Pro c London Math So c III ser

A Ranicki The algebraic theory of surgery Pro c London Math So c III ser

R G Swan Vector bund les projective modules and the K theory of spheres ed

W Bowder Algebraic top ology and algebraic K theory Princeton Ann

Math Studies

R G Swan K theory of quadric hypersurfaces Ann Math Studies

Marek Szyjewski

Instytut Matematyki

Uniwersytet Sl aski

PL Katowice ul Bankowa

Poland

szyjewskigatemathusedupl

Documenta Mathematica