Invariants of a quadratic form attached to a tame covering of schemes

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

P

HILIPPE

C

ASSOU

-N

OGUÈS

B

OAS

E

REZ

M

ARTIN

J. T

AYLOR

Invariants of a quadratic form attached to a

tame covering of schemes

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{2 (2000),}

p. 597-660

<http://www.numdam.org/item?id=JTNB_2000__12_2_597_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Invariants of

a

quadratic

form attached

to

a

tame

covering

of schemes

par PHILIPPE

CASSOU-NOGUÈS,

BOAS EREZ et MARTIN J. TAYLOR

À _Jacques Martinet

RÉSUMÉ. Nous étendons des résultats de

_Serre,

Esnault-Kahn-Viehweg

et

_Kahn,

et nous montrons une relation entre des inva-riants dans la

_cohomologie

étale modulo _{2, qui} sont obtenus à

par-tir d’un revêtement modérément ramifié de schémas à ramification

impaire. Le _{premier type} d’invariant est construit à l’aide d’une

forme

_quadratique

naturelle définie _par le revêtement. Dans le cas

d’un revêtement de schémas de Dedekind cette forme est donnée

par la racine carrée de la codifférente avec la forme trace. Dans

le cas d’un revêtement de surfaces de Riemann la forme _provient

de l’existence d’une

_{caractéristique}

théta _canonique. Le deuxième

type d’invariant est défini à l’aide de la

_{représentation}

du groupe

fondamental _{modéré, qui} est attachée au revêtement. Notre for-mule est valable sans restriction sur la dimension. Pour les revê-tements non-ramifiés la formule est due aux auteurs

_précités.

Les deux contributions essentielles de notre travail sont de montrer

₍₁₎

comment ramener la démonstration de la formule au cas non-ramifié en toute dimension et

₍₂₎

comment maîtriser les difficultés _provenant de la

_présence

de points

singuliers

dans

le lieu de ramification du revêtement, en utilisant ce _que nous

appelons

"normalisation le

_long

d’un diviseur" . Notre

_approche

toute entière est basée sur une

_analyse

fine de la structure locale des revêtements modérément ramifiés.

Nous

_présentons

aussi un survol des notions de la théorie des

formes

_quadratiques

sur les schémas et les

_{techniques simpliciales}

de base nécessaires _pour la

_{compréhension}

de notre travail.

ABSTRACT. We build on

preceeding

work of

Serre,

Esnault-Kahn-Viehweg

and Kahn to establish a relation between _invariants, in

modulo 2 étale

_cohomology,

attached to a

tamely

ramified

invariant is constructed

_using

a natural

_quadratic

form obtained from the

covering.

In the case of an extension of Dedekind

do-mains,

mains,

this form is the square root of the inverse different

equipped

with the trace form. In the case of a _covering of Rie-mann

_surfaces,

it arises from a theta characteristic. The second

type of invariant is constructed

_using

the

representation

of the

tame fundamental _group, which

_corresponds

to the

_covering.

Our formula is valid in

_arbitrary

dimension. For unramified

_coverings

the result was

_{proved by}

the above authors.

The two main contributions of our work consist in

(1)

showing

how to eliminate ramification to reduce to the unramified case, in

such a _way that the reduction is

_possible

in

_arbitrary

_dimension,

and;

(2)

getting

around the

_{difficulties,}

caused

_by

the présence of

crossings in the ramification

_{divisor, by introducing}

what we call "normalisation

_along

a divisor" . Our

_approach

relies on a detailed

analysis

of the local structure of tame _coverings.

We include a review of the relevant material from the

_theory

of

quadratic

forms on schemes and of the basic

_{simplicial techniques}

needed for our _purposes.

Introduction

The

_theory

of bilinear and

_quadratic

forms over schemes has

_experienced

a

surge of

activity

in recent years. Even so, it still seems that little is known

in

_general

_(see

however the _very recent work of Balmer

_{[Bal], [Ba2]).}

As an indication that this is the state of

_affairs,

we note that-to the best of our

_{knowledge-Knebusch’s}

_paper of the seventies

[Kne]

remains the

_only

available source on the elements of the

theory.

Of _course, since that time

a lot of _progress has been made on the

theory

of forms on

_rings,

see _e.g.

[Knu],

and some results and definitions on forms over

_rings

are

_sufficiently

functorial to work over

_{schemes; but,}

for

_instance,

the

_theory

of bilinear forms over the rational

_integers,

has not

_yet

been related to the recent work

in arithmetical

_algebraic

_geometry.

_(For

the benefit of the reader we have collected most of the sources on the

_theory

which are known to us in the

bibliography-the

article

_[Pa]

contains a nice

survey.)

One

_explanation

for this

_might

be that there is _not, as

_yet,

a real need to

_develop

a

general

theory,

because few concrete

_examples

and

_problems

have been considered. One of our aims here is to

_study

a _very

special

form,

which arises from the consideration of

_coverings

of

_{connected, regular,}

_proper

Namely,

let

be a

_covering

of such

_schemes,

which is obtained as a

_sub-covering

of a

I N

quotient

1f : X Y =

X/G,

where G is a finite _group

_acting

on the

connected, projective,

regular

_Z[l]-scheme

X. That

_is,

we let X =

X / H

for H a

_subgroup

of G.

_Then,

under suitable

_assumptions,

_among which is

the

_requirement

that the ramification be tame

_along

a divisor b with normal

crossings

and that the ramification indices be

_odd,

we are assured of the existence of a

locally

free sheaf

D-1/2

over

_X,

whose _{square is} the inverse

X/Y

different of

_XIY

_(see

Sect.

_3.d).

When viewed over Y and

_equipped

with the trace form this

_sheaf,

defines a

_{non-degenerate}

_symmetric

bilinear form

and which we shall call the _{square root}

_of

the inverse

_different

_(see

the

Example

before Sect.

_l.c.l).Note

that for unramified

_coverings

gs

/y =

x .

In

_algebraic

number

_theory

this form has been studied

_by

two of us

in

_[E-T]

_(see

also

_[El]).

The aim there was to _compare this form with the natural form on the _group

_algebra,

which is a sum of _squares. For

coverings

of Riemann surfaces it has been studied

_by

Serre in

_{~52~ .}

In

_{[S l] ,}

Serre had

_previously

_given

a formula for the Hasse-Witt invariant of the

trace form for étale

_algebras

over fields of characteristic different from 2

(no

ramification and

_{dim 0).}

The formula in

_[S2]

is an

_analogue

of that formula

in the ramified case of dimension 1

_(and

characteristic

_0).

_Esnault,

Kahn and

_Viehweg

then

_provided

a result valid for all

_tamely

ramified

_coverings

of Dedekind schemes with odd

_{ramification,}

which

_generalizes

both of Serre’s results

_{(see [E-K-V]}

and

_~K~).

In a different direction Lee and Weintraub have

_generalized

Serre’s formula to ramified

_coverings

of

_higher

dimensional

manifolds,

with smooth ramification

_locus,

see

~L-W~ .

It was

puzzling

for us to see that in

_higher

dimension their formula looked

_just

like Serre’s and did not involve more terms. We wondered whether this was due to

the smoothness

_assumption

on the ramification locus. It turns out that the

crossing

points

of the ramification locus do

_play

a

_major

role in our

work,

but

_they

do not contribute to the final result.

We will

_keep

to the

_algebraic

_set-up

and

_provide

a

generalisation

to

_higher

dimensions of the work of

_{Esnault-Kahn-Viehweg.}

We

_give

an

_expression

for invariants attached to the form E. Our

_strategy

consists in

_reducing

to

the case of an étale cover. This case has been treated in full

_generality

in

[E-K-V]

and

_~K~.

_(The

_proof

in this case consists of a

sophisticated

_cocycle

computation

generalizing

that made

_by

Serre to deal with the trace form of étale

_algebras,

see Sect.

_l.h.)

The invariants we consider are

_generalized

Stiefel-Whitney

classes.

_They

live in the étale

_cohomology

groups modulo 2 of Y and are obtained

_{by pulling}

back universal Hasse-Witt classes

_using

a

classifying

_map

corresponding

to E

_(as

in

_[Jl]

and

_[J2],

see Sects. l.d and

_l.e).

_Following

Snaith and

_Jardine,

we shall call the i-th Hasse- Witt

invariant of E the class in

_{Hi (Yet,}

_Z/2Z)

attached to the form E in

this _way.

_Putting

all these classes

_together

we obtain the total Hasse- Witt

invariant of E

which is a class in the

(abelian)

_group

Z/2Z)

of invertible elements

in

_Z/2Z).

To determine this class we shall construct an étale

cover-ing

T ~ Z of

_Y-schemes,

and a _sequence of

locally

free sheaves

9(h)

on

Z,

as follows. Let

G2

denote a

_{2-Sylow subgroup}

of G and

put

Z =

X/G2,

have the

_diagram.

Our first task will be to show that with these definitions 7rz is indeed

étale and that T is

_regular.

We do this

_by

_using

an

_explicit

_description

of the local structure of the cover

X ~Y (see

Sect.

2).

The natural

_thing

to

_try

then is to _compare the

_pull-back

of E

_{along 0}

with the form F =

(7rZ*(OT), TrT/z).

Note that

_by

our choice of Z the

pull-back

_map induced

by 0

in

_cohomology

is

_injective.

The two forms

_0*(E)

and F coincide on the

generic

fiber of Z and we wish _{to compare} their total Hasse-Witt

invariants. For

_this,

_following

_[E-K-V],

we

_try

to show that the

_orthogonal

sum H :=

equipped

with the difference of

forms,

contains an

isotropic

sub-bundle

_I,

whose rank is half of the rank of H. With the

terminology

introduced in Sect.

_l.c.l,

we

_try

to show that H is metabolic with

_lagrangian

I. Note that

_hyperbolic

forms are an

example

of metabolic

forms and the reason for

_trying

to show that H is metabolic is

_that,

as with

hyperbolic

forms,

it is

_expected

that the

_{quadratic~Hasse-Witt}

invariants of

H should then be determined

_by

the

_linear/Chern

invariants of the maximal

isotropic

sub-bundle I. In fact this

_expectation

is

_justified

_(see

the Main Lemma

_1.15).

In dimension

1,

which is the situation considered in

_[E-K-V],

the intersection

_{0* (E)}

f1 F

_gives

the

_required

sub-bundle I. In our

_general

situation, problems

appear due to the fact that the branch locus b of the

covering

X/Y

is not smooth. We show that the form is metabolic

(in

a

generalized

sense),

but to exhibit a maximal

_isotropic

sub-bundle

explicitly

we are led to introduce a _sequence of forms

A~h~

which

interpolate

0*(E)

and F. For this we

decompose

the normalization _map T -~ T’ into a _sequence of

Z-morphisms

numbered

_by

the m

_components

of the branch locus b of the

_covering

X/Y

and which are obtained

_by

"normalisation

_along

a

_component

of b"

(see

Sect.

_3).

The form

A~h~

is the _{square root} of the inverse different for the cover and

A(°)

=

~~ (E),

while = F. For 0 h m - 1 _we

have short exact _sequences of

_locally

free

Oz-modules

where the _map on the

_right

is the difference _map

(see

Prop.

3.12).

This

We need to introduce one last notation before

_stating

our first main

result. For

_{every h}

define an element of

_{H* (Zet, Z/2Z)}

_by

where

_Cï(gCh))

=

Cï(ICh))

denotes the i-th Chern class of

g(h)

and _n denotes

the rank of

_gCh)

_(see

the Main Lemma

_1.15).

Theorem 0.1. Under the above

_assumptions

the

_{following equality}

holds

in

_{H*(Zet, Z/2Z)}

Remark. It should be noted that if we work with another divisor on Y

which differs from b

_by

the addition of a number of irreducible non-ramified

divisors,

then of course the number m of divisors will

_change

and so the

parity

of the first term in the

_right

hand side _may

_change;

_however,

it can

be checked that the

_product

of the two terms on the

right

remains constant

(see

the remark at the end of Sect.

_4.a).

The

_proof

of Thm. 0.1 is

_given

in Sect. 4.a. The invariants of the form

arising

from the étale

_covering

_T/Z,

have been related

to other

_{Stiefel-Whitney}

_type

invariants attached to the

_covering

in

_{[Sl] ,}

[E-K-V]

and

_(K).

These are what we call the Galois theoretic classes

wi(7r),

which can be defined

using

Grothendieck’s

equivariant

cohomology theory.

Then,

with some further

work,

in low

degree

Thm. 0.1 can be reformulated

to

_give

a

_{simple expression}

for the difference between the second Hasse-Witt and Galois theoretic invariants. For this let us introduce an element

p(X/Y)

in

_{H (%t,}

_Z/2Z)

which

_{only depends}

on the ramification data of the

_covering

_X/Y.

We shall denote

_by

the same

_symbol

_(the

class

_of)

a

line bundle and its

_image

in

_Z/2Z)

under the

_composition

where the second map is the

coboundary

induced

by

the Kummer sequence

(note

that under our

_assumptions

we _may

_identify

_p2 with

Z/2Z).

We define

_p(X/Y)

as a divisor class.

Let ~h

denote the

_{generic point}

of the

irreducible

_{component bh}

of the branch locus b and consider the divisor

where the sum runs over the

_{points g§}

on X of codimension 1 above the

p(X/Y)

on Y is defined as the direct

image

where

f (~h)

denotes the residue class extension

_degree.

Our second main

result can then be formulated as follows. Theorem 0.2.

a)

Let

_dXjY

denote the

_{function field}

discrirrcinant viewed as an element

in

_Z/2Z),

then

b)

The

_{following equality}

holds in

_H2(Yet,

_Z/2Z):

The main effort in

_proving

this

_{theorem, given}

the

_{preceding work,}

_goes

into a determinantal

computation,

which

again

uses in an essential _{way the}

local

_description

of the tame cover

X / Y .

More

precisely

in Thm. 4.2 we show that in

_Z/2Z):

The

_origin

of the coefficient

_{(e2 -}

_1)/8

is in the combinatorial Lemma 4.3. We do not have a

conceptual

definition of the ramification term

_p(X/Y).

A word about invariants of forms over schemes

_might

_help

_clarify

the

significance

of our results. Not all invariants of forms over a field extend to invariants of forms over

schemes,

_except

in

degree

2 and smaller. Global obstructions _appear. So for instance the Arason

_invariant,

in

_degree

_3,

does

not extend to

_schemes,

see

[E-K-L-V].

The Hasse-Witt invariants we

study

exist in

_arbitrary

_degree,

but one should

_keep

in mind that

_they

are not

independent

one of the other. So for

_instance,

as

_might

be

_expected

from the

_analogies

with

_{algebraic topology,}

if

_{wi (E)}

and

_{w2 (E)}

are both _zero, then

_{w3 (E)}

is zero as well. The formulae in Thm. 0.2 can be viewed as

expressing

the difference between the

_{"topological"}

Hasse-Witt invariant

and the "discrete" Galois theoretic invariant

_(see

_~J1~,

_p.

_84,

_[J3]

Sect. 3 or

[J4]

Intro.).

They

should also be viewed as

_"twisting

formulae"

_following

Frôhlich’s work

_~F~.

Yet another

_{interpretation}

is

_given

_by

Kahn in terms

of equivariant cohomology

(see

[K]).

The invariants in

_degree

smaller than 2 have been refined to

_give

invari-ants in

_K-theory

_(see

_{[G1], [0-P-S], [Sz], [G2]}

and

_[B-0];

the last two also deal with invariants in

_higher

_degree).

These invariants are related to the

invariants in étale

_cohomology

via Chern class _maps; we do not know if our formula can be lifted to

_K-theory.

The _{paper is} structured as follows. In Sect.l we fix notation and recall the _necessary definitions and

_background

material used in the rest of the

paper. We offer a rather detailed review of the basic material on forms

over schemes and their invariants as the literature on the

subject

is rather scarce

and/or

quite

technical. In Sect. l.h we state and sketch a

_proof

of the result of

_[E-K-V]

which we use in the

_proof

of Thm. 0.2. Sect.

_2,

contains the material on tame covers on which our contribution is based. There we describe the normalization T

_locally

_(see

_Prop.

_2.6).

_Next,

we

analyze

the normalisation

_procedure

and

_decompose

it into

_steps,

which

we call "normalisation

_along

a divisor"

(see

Sect.

3).

Sect. 4 contains the

proofs

of Thm. 0.1 and Thm. 0.2. The first will follow

_easily

from the

preceding

work. As

_explained

_above,

to obtain Thm.

_0.2,

we have to make more

_explicit

the

_right

hand side in the

_equality

of Thm. 0.1 as well as

bring

in the Galois theoretic invariants. We have collected in an

Appendix,

some material on

_homotopical

_algebra

which we could not find

_presented

in

_compact

form

_anywhere

in the literature.

To conclude this Introduction let us mention another related line of re-search which consists in

_studying

forms on

_cohomology

of

_{(complexes of)}

sheaves. This has been

_pursued

_by

Saito in

_[Sa]

and

Chinburg-Pappas-Taylor

in

_C-P-T1.

Saito’s work also has as its

_{starting point}

_Sl,

whereas

[C-P- Tl]

deals with forms

_coming

from Arakelov metrics. This work sheds new

light

on the role that the trace form

_plays

in Galois module

_theory.

We

would also like to indicate that in the future we

_hope

to relate the material

in this _{paper to the programme}

_developed

in

_[CEPT2]

and other papers of

that consortium of authors

_(see

_[E2]

and

_[C-P-T2]

for a

_general

overview of that

_programme).

Other

_developments

_arising

from Serre’s

_original

formula for étale

_algebras

can be found in

[Mo]

and the references therein.

We would like to

_acknowledge

initial discussions with G.

_Pappas

and

T.

_Chinburg

and _{to express} our

gratitude

to

_Q.

Liu for his

_help

and

ad-vice. The second author would also like to thank P.

_Balmer,

J.F.

_Jardine,

B.

_Kahn,

M-A. Knus and M.

_Ojanguren

for

_helpful

remarks

_concerning

their work and the literature on the

subject.

1. Framework and notation

l.a. Notation and conventions.

Induced

_algebras.

In our local

description

of the structure of tame

_coverings

we will need the

following

construction. Let A be a

_ring

and let B be an

A-algebra equipped

with an action of the finite _group

H,

which _preserves the

algebra

structure. For a _group G and a

_subgroup

H of G let

MapH(G, B)

denote the

_B-algebra

of all _maps

_f

from G to B such that for

_{all g}

in

G and all h in

_H,

_{f (gh)}

=

f (g)h.

The _group G acts on this

_{algebra by}

f9(9~~ _

Normalisation. For a

_ring

A which

_injects

into its total

_ring

of fractions

Fr(A)

we write A for its

integral

closure. We will also use a tilde

_superscript

to denote certain

_objects

related to the cover of

schemes X/Y.

This should

not cause _any confusion.

Coef,ficients.

We will work with mod 2 étale

_cohomology

of schemes on which 2 is

_invertible,

so we will

always

identify

_p2 with

Z/2Z.

l.b. Tame

_coverings

with odd ramification.

In what

_follows

all schemes will be

_defined

over Let

X

be

a

_{connected, projective,}

_regular

scheme which is either defined over the

spectrum

Spec(Fp)

of the

_prime

field of characteristic

_{p ~}

2 or is flat over

Spec(Z[~]).

Tameness. Assume X is

_equipped

with a tame action of

_G,

in the sense of Grothendieck-Murre. In

_particular

the

_quotient

exists and is a G-torsor outside a divisor b with normal

crossings

(see

[Gr-M], [CEPT1], [C-E]

Appendix

and

_[CEPT2]

1.2 and

_Appendix).

Once and for all fix a

subgroup

H of

G,

and let

denote the

_quotient

_map. Let

be the induced _map, so that 7r = 1r o A. These _maps are all finite.

Regularity

and

_flatness,.

If à is

_flat,

then Y is

_regular

_(see

_[EGAIV]

IV

6.5.2).

Conversely

a

morphism

between

equidimensional regular

schemes

of the same dimension is flat

_(see

_[Ka-M],

Notes to Ch. 4 or use

[EGAIV]

IV

_6.1.5).

We will assume that Y and X are

regular,

or

equivalently

that

ir and .~ are flat. Note that

_being

_quotients

of a normal scheme

_by

a finite

group y and X are

certainly

normal,

however it is

only

under the above

regularity

assumptions

that we will be able to

_give

a

_precise

_description

of

the local structure of the

_quotient

maps

(see

Lemma

2.3).

It follows from

these

_assumptions

that ir is flat as well.

Different;

branch and

_ramification

loci. The different is the annihi-lator of

_(see

_[Mi]

Rem.

_1.3.7).

The reduced closed subscheme of X

x

B =

B(X/Y).

Then b =

b(X/Y)

is the reduced sub-scheme of Y defined

by

the

_image

of B under if. There are

_{decompositions}

where

_{the bh}

are the irreducible

_components

of b and where for _any fixed

integer h

between 1 and _m, the

_Bh,k

are the irreducible

_components

of B such that =

oh.

The tameness

_assumption

on the ramification

_implies

that the

ramifi-cation locus

_{B(X /Y)}

on X

is a divisor with normal

_crossings.

For each irreducible

_component

_bh

of

_b,

let

_Çh

denote the

_{generic point}

of

_bh,

and let

g£

denote a

_{generic point}

of the branch locus on X above

_Çh.

Put

g§ = ~(~)*

Let

_e(g§)

_(resp.

_{f (~h))}

denote the ramification index

_(resp.

residue class extension

_degree)

of

_g§

over

_~h.

Parity.

We assume that the inertia group of each closed

_point

of X has odd

order,

so _every

_point

has odd inertia and in

_particular

the

_integers

are odd.

The base

_change.

Choose a

2-Sylow subgroup

G2

of G and consider the

quotient

Z :=

X/G2.

We will show in Sect.

_2,

that

_by

the

_parity

assump-tion X is étale over

_Z,

so Z is

regular

and flat over Y. We

_give

a local

description

of

_Z/Y

_analogous

to that of

_X/Y,

without

_using

the

regular-ity

of Z in

_(2.4).

As mentioned in the Introduction our main effort will go into the construction of Z-schemes which will allow us to reduce our

computation

of the invariants to the case of an étale cover.

Mernotechniques.

For future reference and to

_help

the reader with

book-keeping,

we

_anticipate

some of the notation to come.

Components

of the branch locus b will be numbered

_by

h

_(from

1 to

_m).

We fix

points

y in Y and x in X over _y. The n

_components

of b which are in the

_image

of a

component

of the ramification divisor

_{through x}

will be numbered

_{by ~}

_(see

(2.2);

of course there

_might

be none of

these).

Let

I(x)

denote the inertia group of x for the

covering

X /Y,

this is a _group which is the direct

_product

of n

cyclic

_groups indexed

by

the

_components

of b

through

_~. Let _eî denote

the order of the ~-th

_component.

In

_describing

the local structure of our

coverings

we will consider the

_spectrum

S =

Spec(Ay)

of a

ring containing

a

regular

_sequence of

_parameters

_{al, ... , an,}

_providing

a

_sufficiently

small étale

_{neighbourhood of y}

in Y

_(see

Lemma

_2.3).

_Then,

we will construct

corresponding

to inclusions of

_algebras

where for a certain divisor of e.l and where

te,

= _a£

(see

e Ê

also the

_{diagram just}

before Sect.

_2.a).

The index i arises from the double

coset

_{decomposition}

of G with

_respect

to H and

_{I (x) .}

_Analogously,

the

index j (i)

arises from a coset

_{decomposition}

f1

_HBH

_{(see (2.b)).}

Fixing

i and

_going

to the fields of fractions we obtain a _sequence of abelian Galois extensions of the field

_Ky

=

Fr (Ay ) .

Namely

let

and

_Hi

=

I(x)/I(x,i),

then _we have the

diagram

of fields

There is an

analogous

description

of S -

Zs

and

quite

naturally,

in the

local

_description

of the normalisation T of T’ = Z _{x y}

X,

_we will consider

algebras

indexed

_by

_pairs

_{(j, i) (see}

_Prop.

_2.6).

Moreover the

_algebras

appearing

in the local

_descriptions

will be

_{decomposed according}

to the characters of the group

Hi,

which will be indexed

by

a set

_A(i),

whose elements will be a’s

_(see

Lemma

_2.5).

l.c.

_(auadratic

forms on schemes.

Quadratic

forms on schemes are defined

_by

_generalizing

the familiar

defi-nitions

_given

for such forms over fields or

_rings

_(see

e.g.

[Kne]

and

[Knu]

Chapt.

VIII).

The

_only

definition which is not

_{straightforward}

is that of a metabolic form

_(see

below Sect.

_l.c.l).

_However,

for the definition of

the

_invariants,

it is also

_good

to have an

algebraic

description

of the set of

isometry

classes of forms of

_given

rank over a scheme Y. We will recall the

description

in terms of non-abelian first

_cohomology

_sets,

and the

descrip-tion in terms of

_homotopy

classes of maps from Y to a certain

_classifying

space. For

completeness,

we

_briefly

_go over the basic material on forms over schemes. As

_usual,

since 2 is invertible over

_Y,

then the

_theory

of

symmetric

bilinear forms is

_equivalent

to that of

_quadratic

forms. Here we

will concentrate on the former.

Let Y be a scheme. A vector bundles E on Y is a

locally

free

Oy-module

of finite rank. The dual of a vector bundle E is the vector bundle E’ such

that,

for _{any open} subscheme Z of Y

There is a natural evaluation

_pairing

_, &#x3E; between E and

E’

and one can

identify

E with the double dual

E~"

by

such that

_a,K,(u)

&#x3E;= _{u, a} &#x3E;. A

_symmetric

bilinear

_form

on X is a

vector bundle E on Y

equipped

with a _map of sheaves

which on sections over an _open subscheme restricts to a

_symmetric

bilinear form. This defines an

_adjoint

_map

which because of the

_symmetry

_{assumption equals}

its

_transpose:

We shall _say that

_{(E, B)}

is

_{non-degenerate}

_{(or unimodular)}

if the

_adjoint

cp is an

isomorphism.

(Note

that some authors call

non-degenerate

forms

spaces.)

One can show that a form on a

locally

free module of rank n over a

ring R

in which 2 is invertible is

non-degenerate

if and

only

if

locally

in the étale

_topology

it is

_isomorphic

to

see

[Sw]

Cor.

_1.2,

[Knu]

IV

(2.2.1),

IV

(3.2.1)

or

[M-R]

1.11.5

(In

[M-H]

1

(3.4)

it is shown how to

_diagonalize

a

_{non-degenerate}

form over a local

ring

in which 2 is

_invertible.)

_Morphisms

among

forms,

orthogonal

sums and

tensor

_products

are defined as

_expected.

Example

1.1. Let 1r : X ~ Y be an

oddly, tamely

ramified cover as in

Sect. l.b. We obtain a

non-degenerate

symmetric

bilinear form on Y

by

letting

1 1

and

_by

_considering

the bilinear form B : E _Xy E ~

_{Oy given}

_by

_{B(x, y) =}

tracex/y(x.y)

(see

[EGAIV]

IV

_18.2.1).

Note that twice the divisor

_defining

E is

_isomorphic

to the

_(relative)

canonical divisor of

_X/Y

and that if 7r is

There is a further useful _way to view a form over a scheme. For E a vector

bundle over

_Y,

let

_{P(E) -&#x3E;}

&#x3E; Y denote the associated

_projective

scheme.

Then, giving

a

_symmetric

bilinear form on E is

_equivalent

to

_giving

a

section of the invertible sheaf

_O(2)p(E)

over

P(E)

(a

section of

S2(E))

(see

[Sw]

Lemma 2.1 and

_[Del]).

l.c.l. Metabolic

_forms.

The notion of a metabolic form was introduced

by

Knebusch for his definition of the Witt _group of a scheme in

_[Kne].

Hyperbolic

forms are metabolic and the

_greater

_generality

of the latter

notion is the

_right

one in the

_global

situation of non-affine schemes

(see

below).

More

_recently

in

_{~Bal~, [Ba2]}

Balmer has extended the notion of a metabolic form in the context of

_triangulated

_categories

with

_duality

and he has shown its usefulness. In fact the

_{interpolation}

_procedure

which we will

_set-up

in Sect. 3 shows that even in our _very

_special

situation it is

quite

natural to consider forms as self-dual

complexes

(see

the remark after

Prop.

3.12).

Let E be a vector bundle over Y. A

_{sub-Oy-module}

V of E is a sub-bundle of E if it is

_locally

a direct

_summand,

i. e. for _{any y in}

_Y,

there is an

open Z

containing y

such that

VIZ

has a direct summand in

Elz.

If a V is

a sub-bundle of

E,

then V and the

_quotient

E/V

are both vector bundles.

Let now

(E, B)

be a form. For a sub-module i : V C E one defines an

orthogonal

complement,

which is the

_{sub-Oy-module}

_V,

whose sections

over the _open Z consist of those sections of E which are

orthogonal

to all

sections of V over _{any open} subset of Z.

Alternatively:

If furthermore V is a sub-bundle of

_E,

then i’

is an

_epimorphism,

and if

(E, B)

is

_{non-degenerate,}

then _{cpB is} an

isomorphism

(by definition).

So under these

_assumptions

we have an

_isomorphism

is

_locally

free and

V

is also a sub-bundle. There also is an

isomor-phism

, . ,

(The

form

_(E/V)

can be identified with the

sub-Oy-module

of E whose

sections over Z C Y are the linear forms ~ :

Oz

which vanish on

_V,

so

V

= The

maps a

and 0 correspond

to bilinear _maps which are

_{perfect dualities,}

obtained

_by

"restriction" from B:

which

_gives

V =

V .

We have the

following

commutative

_diagrams

and

A sub-bundle V of a

_{non-degenerate}

bilinear form

(E,

B)

is a

totally

isotro-pic

sub-bundle

_(also

called a

_{sub-lagrangian)}

if V C

V 1

. If V is a

sub-lagrangian

of

_{(E, B),}

then

_{V /V}

is a sub-bundle of

_E/V

and the form on

V /V

obtained

_{by reducing}

B modulo V is

_{non-degenerate.}

A sub-bundle

V of a

non-degenerate

form

(E, B)

is called a

Lagrangian

if it is such that

V =

V-L.

The form

(E, B)

is called metabolic if it contains a

lagrangian.

If

V is a

_lagrangian

in

(E,

B),

then

rank(E)

= 2 ~

rank(V)

and V is in _{a sense}

a maximal

totally

_isotropic

sub-bundle.

Using

the

above,

we see that V is a

_lagrangian

in E if and

_only

if one has a commutative

_diagram

That is a metabolic form is

_given

by

a

symmetric/self-dual

short exact sequence

(see

[Kne]

Chapt.

3).

Example

1.2. A

_special

case of the notion of a metabolic form is that

of a so-called

split

metabolic

form,

which

corresponds

to the case of a

split

exact sequence. Let

_{(U, C)}

be a form with

_adjoint

_cp. The

split

metabolic space associated with

(U, C)

is the form

M(U, C),

which has as

_underlying

bundle U EB

UV

and form B the form with

_adjoint

1 - - 1

The form

_H(U)

:=

M(U, 0)

is called

hyperbolic.

For a form

(E,

B),

the

form

_(ELE,

B1 -

_B)

is metabolic. It is shown in

_[Kne]

_Prop.

3.1 and

_3.2,

that

_{M(U, 2C) = H(U),}

and that

Since 2 is

_invertible,

a

_split

metabolic form is

_isomorphic

to a

_hyperbolic

form:

_just

consider the

_equality

/ _ "1" / - , / - - -" / - - ,

Moreover,

for an affine

scheme Y,

one shows that _any metabolic _space

form

_{(E, B)}

over

Y,

there is a

splitting analogous

to the classical Witt

decomposition. Namely

(E, B)

is isometric to the

_orthogonal

sum of a form which does not _{contain any}

_{sub-lagrangian}

_(i.e.

is

_anisotropic)

and a form which is

_split

metabolic

_(loc.

cit.

_Prop.

_3.3).

_(One

should be aware that such a

decomposition

does not determine either summand. One obtains a better

_analogue

of the Witt

decomposition by

_working

inside the

Grothendieck-Witt

ring

of Y

_[Kne]

Thm.

_4.3.)

Over a

_{complex elliptic}

curve there exist

_infinitely

_{many non-isometric} metabolic forms which are

not

_split

metabolic

_(see

_[Knu]

VIII

_(1.1.1)).

Example

1.3. In our

_set-up,

metabolic forms will be useful in

comparing

two forms over an irreducible scheme Z which are isometric when restricted

to the

_generic

fiber of Z.

_Namely

if

_{(E, BE)}

and

_{(F, BF)}

are defined over

Z and _agree on the

_generic

fiber

then under suitable

_assumptions

_(E1F,

BE1 -

BF)

is metabolic

_(see

_e.g.

Prop.

3.12).

A natural

_approach

is to consider the

sub-sheaf 9

of

_{EI,7_,}

=

Fll1Z

defined as

and the exact sequence

where the _maps are obtained

by

_restricting

to E1F the _maps defined at the level of the

_generic

fibers

_given

_by

the

_diagonal

_map and the _map

_sending

(x,

y)

to

_{(x -}

_~)/2.

_Then,

_{if 9}

is

_{locally free,}

it follows that E fl F is a

sulrbundle of

_ELF,

which is a

_lagrangian

and

The

_point

is

_{that 9}

_might

not be

_locallg

_free

in

_general;

on the other

hand 9

is

_certainly

_locally

free in the case of Dedekind

_schemes,

which is the case which was considered in

[E-K-V].

l.d. Classifications.

As with forms over

_{fields, non-degenerate, symmetric,}

bilinear forms of

a

_given

rank n over a scheme Y can be considered as torsors under an

orthogonal

group and hence are classified

by

the non-abelian

cohomology

set

Jardine shows in

_[Jl]

how to

_identify

this set with the set of

_morphisms

from Y to the

_"classifying

_space"

_BO(n)

inside the

_homotopy

_category

of

of forms on schemes which is _very close to the

_approach

to characteristic classes

_following

Borel

_{(see [St]}

_App.

10 and Rem. l .d. 2

_below).

We think it is worthwhile to _go over Jardine’s

description

in some detail. The

powerful

techniques

and ideas he

_exploits

are seldom

_presented

in a form suitable

to the

_{non-specialists. Going}

over this material also allows us to see how

simplicial

schemes arise

_quite

_naturally

in our

_set-up

(see

the

Appendix).

1.d.l.

_Cohorrcology

sets. We start

_by

_{giving precise}

references for the clas-sification in terms of first

_cohomology

sets. The

_orthogonal

_group of a

_given

form is a _group scheme. It

_{is, however,}

most

_easily

defined in terms of its

associated functor of

_points.

Let R be a

_ring

in which 2 is invertible. Models over R are

_R-algebras.

The functor we are interested in is a functor from models to sets which satisfies some

special properties,

making

it a sheaf for the

_{(Grothendieck)}

_{(fppf)-topology.}

Let

_{(M, BM),}

_(N,

_BN)

be two

qua-dratic forms over R. Let

_{Isom(BM, BN)}

denote the functor from models over R to

_sets,

which sends an extension R’ of R to the set of isometries

between the forms

_BM

and

BN

extended to

R’.

If

_BM

=

BN,

this defines

a sheaf

which is the

_orthogonal

_group

_(functor)

of

_{(M, BM)}

_([D-G]

_III,

Sect.

_5,

_n.2).

This sheaf is

_{representable by}

an affine

scheme,

but this

_property

does not

make the discussion _{any easier.} Let us fix a

_quadratic

form

_{(M, BM)}

over R. We say that the

quadratic

form

(N, BN)

is a twisted

form

of

(M,

BM)

if there is a

(fppf)-covering

_R-algebra

R’ such that

(N, BN)

becomes

iso-morphic

to

_{(M, BM)}

over R’. Recall that a torsor X under a

(sheaf of)

group(s)

G over a sheaf Y

(e.g.

a

scheme)

is a sheaf X

_{covering Y,}

which

is

_equipped

with an action of G such that

under the _map

_given

on sets of sections

_by

_{(x, g)}

~

_{(x, xg). (We}

_are

essentially

saying

that the action of G on X is

_simply

_transitive.)

In loc.

cit. one finds a

proof

that the _map

is a

_bijection

between the set of

_isometry

classes of twisted forms of

_(M,

and the set

_{ÎI- 1 (R,}

of

_isomorphism

classes of torsors under

_0(BM)

over R. This is a

pointed

_set,

pointed by

the class of the trivial torsor. All

goes

through

for a scheme Y in

_place

of the

_ring

R and in our

_set-up

the

torsors X are schemes.

Example

1.4. In view of what we have recalled about

_{non-degenerate}

forms,

we see that

_{non-degenerate}

forms of rank n, over a scheme Y over

Example

1.5. One can

_give

a

_{cohomological description}

of metabolic forms over a

_ring

R

(see

[K-02]

and

[Knu]

IV

(2.4.2)).

If

O(2n, n)

denotes the group functor which sends an extension

R’ / R

to the group of isome-tries of the

_hyperbolic

_space

_{H(R n)}

=

R n 1 R ,nv

which stabilize

then metabolic _spaces

_(with

a

distinguished

lagrangian)

are classified

by

Hl(R,

O(2n,

n)).

By

the Theorem of Torsor Descent recalled

_below,

one can

"compute"

the

cohomology

set

Hl

in terms of a set

_{(of orbits)}

of

_cocycles,

and more

precisely

as the

_cohomology

of a

_cosimplicial

_group, called the Amitsur

complex.

In fact one first identifies the set

_Hl(Y,

_O(BM))

with the direct limit of sets of torsors which are trivialized over a

_{given covering}

Y’ - Y

(i.e.

a sheaf

epimorphism).

More

_generally

for _any

(sheaf

of) group(s)

G one has

where the limit is taken over the

_coverings

Y’/Y.

The sets on the

_right

are defined

_by

(Here

ker

consists of elements in the inverse

_image

of the

_{distinguished}

point.)

We recall the definition of the Amitsur

_complex

and of the first

cohomology

set

_G)

in the

_Appendix,

see

(5.4).

The Theorem of

Torsor Descent is the

_following

statement

_(see

_[D-G]

_III,

Sect.

_4,

n.

6.5).

Theorem 1.6. Let Y’ ~ Y be a

sheaf epimorphism,

and let G be a

sheaf

of

groups. There is a canonical

_bijection

l.d.2.

_Morphisms

in the

_homotopy

_category.

Let now

[Y, BG]

denote the

set of

_morphisms

between the

_simplicial

sheaves Y and BG in the

_homotopy

category.

(see

Appendix

5.c for

_{definitions).}

The next result is due to Jardine

[Jl]

Cor. 1.4. We sketch a

proof

of this in

Appendix

5.d.

Proposition

1.7. For _any

_{sheaf of}

_groups G on

Y,

there is a

bijection

Remark 1.8. The

_proposition

is an

algebraic/combinatorial

version of a result which is

_quite

_typical

in

_geometry,

where

_principal

G-bundles on a

space Y are classified

by

maps into a

_classifying

_space

_BG,

so that the bundles are obtained

_{by pull-back}

from a universal bundle EG on BG

(see

[St]

Sect.

_19).

For _{certain groups}

_G,

one has a _very

_{explicit description}

of the space BG. For instance the

classifying

space for the real

orthogonal

group

O(n, R)

is

given

by

the Grassmann manifold

Gn

(see

[M-S]

5.6).

In the first two sections of

_[Mac-Mo]

_Chapt.

_VII,

the reader will find how

to

_identify

_principal

G-bundles over a _space Y with G-torsors over

Y,

in the case of a discrete _group G. For the

_simplicial

version of this see

_[Cu]

Sect. 6. A version of the

_proposition

for

_simplicial

sets can be found in

[Go-Ja]

Thm. 3.9.

l.e.

_{Cohomological}

invariants I: Hasse-Witt.

Let

_{(E, B)}

denote a

_{non-degenerate}

form over a scheme Y on which 2 is invertible. We recall the definition of the Hasse-Witt invariants

at-tached to

_{(E, B)}

in

_Hi(Yet,

_Z/2Z),

_following

Jardine’s work and

_[E-K-V].

We

_{begin by defining}

the invariants in low

_degree.

_Then,

we

_give

two

defini-tions of the

_{higher degree}

Hasse-Witt invariants: the first is

_by

_pulling

back universal

_classes,

the second is via the

_{decomposition}

of the

_cohomology

of the

_complement

of the

_quadric

defined

_by

B inside the

_projective

bundle associated to E.

l.e.l. Low

_degree.

To a form

(E, B)

of rank 1 over

Y,

there

_corresponds

a class

Let

_det(-)

denote the

_top

exterior _power

_operation.

Then the

_first

Hasse-Witt invariant of a form

(E, B)

of

_arbitrary

_rank,

is

where the

_right

hand term is the Hasse-Witt invariant of the rank one

form

_{(det(E), det(B))}

as defined above.

Thus,

under the _map induced in

cohomology by

the _map

_Z/2Z

=

p2 -

Gm ,

the first Hasse-Witt invariant

is sent to the first Chern class

_ci(E):

(see [Knu]

111.4.2).

Below we shall view ci as

taking

values in

Z/2Z),

via the

_coboundary

_map

_coming

from the Kummer _sequence. In

_degree

2 one Let 7r : X -~ Y can define an invariant

by

a

construc-tion,

which

_generalizes

that of the Clifford

_{algebra. Namely,}

to a form

(E, B)

over

_Y,

one associates a sheaf

C(E)

of

_algebras

over

_Y,

called the

Clifford algebra

of

_{(E, B)}

_(see

_{(K-01~, [Knu]}

IV

_(2.2.3);

in odd rank one

only

considers the even

_part

of the full Clifford

algebra).

The

_algebra

_C(E)

is an

Azumaya algebra,

that is an

Oy-algebra

which

locally

in the étale

topology

is

_isomorphic

to the

_Oy-algebra

_{Endo, (W)}

of

_{endomorphisms}

of some

_Oy-bundle

W. The class of this

_algebra

in the Brauer group of Y is 2-torsion and is called the

_Clifford

invariant of

_{(E, B)}

where n’ =

n/2,

if the rank n of E is

even, and n’ _

(n -1)/2,

if the rank is

odd. The Clifford invariant can be viewed in

Gm) (see

_e.g.

[Mi]

Chapt.

4,

Thm.

_2.5).

The invariant we will be interested in is a lift of this

invariant under the map

In

_[F],

Frôhlich has shown how to construct an extension

which,

in the case B is the standard

form, gives

rise to a _map

The second Hasse- Witt invariant of

_{(E, B)}

is

see also

_[E-K-V]

Sect.

_1.9,

and

_[J2]

_Appendix.

_(Note

that the Pinor

exten-sion constructed in

_[A-B-S]

_gives

_w2+wI.)

Further

_approaches

to invariants in

_degree

2 are

given

in

[Pat]

and

[Pa-S].

l.e.2. Universal classes. To obtain invariants in

_higher

_degree

one has to

work harder. The

_approach

outlined here is that of Jardine

_~J1~.

It relies on his

_computation

of the

_cohomology

algebra

of the

classifying

simplicial

scheme

_BO(n)

_(over

the

_big

étale site of

_Spec(Z[2]).

The

_computation

shows that this

_algebra

is a

polynomial algebra

over a finite number of

elements. These elements are the universal Hasse-Witt invariants. We will use these universal invariants to

_give

one definition of the Hasse-Witt

invariants of

_forms,

but

_also,

in Sect. l.f

_below,

to define invariants of

orthogonal

representations.

As before let Y be a scheme over

Spec(Z [)] ) .

Theorem 1.9. The

_cohomology

_ring

_{H*(BO(n)et, Z/2Z)}

is a

_polynomial

algebra

over the

_algebra

_Z/2Z),

with n

_{distinguished}

gen-erators

_HWi,

where

_for

1 i _n,

_deg(HWi)

= i.

This allows one to define the Hasse-Witt invariants of a form

(E, B)

over Y as follows. Recall from

_Prop.

1.7 that if

_{(E, B)}

has rank n, then it

corresponds

to an element

[E]

in the set

_~Y,

_BO(n)~

of

_homotopy

classes of

maps from Y to

_BO(n).

One sets

It can be shown that in low

degree

these classes _{agree with} those defined

l.e.3.

_Projective

bundles. A

_{further, equivalent,}

_way of

_defining

_higher

degree

invariants

_proceeds

_along

the lines indicated

_by

Grothendieck’s

defi-nition of Chern classes in

_[Gr].

_Following

_Delzant,

Laborde has shown how

to use this

approach

to define invariants in étale

cohomology

for

quadratic

forms over schemes

_{(see [Dz] [La]).}

His work has been detailed in

_[E-K-V]

and

_proceeds

as follows.

Given a

_{non-degenerate}

form

_{(E, B)}

over Y of rank _n, one considers the

_complement

U of the

_quadric

associated to B in the

_projective

bundle

p :

P(E) ~

Y.

By identifying

the form with a section s :

Oy ~

S2(E) (as

in the end of Sect.

_l.c),

one obtains a

_{non-degenerate}

form on the invertible sheaf

₀₍₁₎

:= This form defines a class _{ri in}

Z/2Z).

The

_following

result is Claim 5.2 in

_[E-K-V].

Theorem 1.10. Let

_p’

denote the restriction to U

_of

the bundle _{map p.}

For any

integer

i &#x3E;

_0,

one has the

decomposition

Corollary

1.11. There exist

_unique

elements

_{wl(E), w2(E), ... , wn(E) in}

H*(Yet,

Z/2Z)

such that

The

_equivalence

of this definition of the Hasse-Witt invariants

_wi(E)

with the

_previous

one is shown in

[E-K-V]

Prop.

1.4.

l.e.4. Axiorraatic

_approach.

We

_give

a list of

_properties

_{characterizing}

the

Hasse-Witt invariants

_wi(E)

=

wi(E, B)

in

Hi (Yet,

Z/2Z).

(HWO)

Normalization.

_wo(E)

=

1,

wi(E) =

~(det(E), det(BE))~

and

wi(E)

= 0 for i &#x3E;

rank(E)

_{+ 1.}

(HW1)

Functoriality.

For any

morphism

f :

Z -

_Y,

(HW2)

Whitney

formuda.

Let

_{wt (E) =}

_Zi

_w2(E)t2

be viewed as a formal

power series. Then for

non-degenerate

forms

(E,

BE)

and

(F,

BF)

one has the

_identity

Remark 1.12. The

_{topological Stiefel-Whitney}

classes

_satisfy

a list of

properties

analogous

to the ones above

_(see

_[M-S]

_Chapt.

_4).

The two

kinds of invariants can be related as follows

(see

[Ca]).

Let Y be a _compact,

connected,

locally

connected

_topological

_space and write

_R(Y)

for the

_ring

of continuous real valued functions on Y. One can _compare the Hasse-Witt

correspondence

between bundles over Y and modules over the

_ring

R(Y).

There is a

homomorphism

A

_{non-degenerate}

form

_(E,

_B)

over

R(Y)

determines an

orthogonal

bundle E over Y. This defines a

_homomorphism

from the Grothendieck-Witt _group

L(R(Y))

to the Grothendieck group

KO(Y)

of

orthogonal

bundles over Y. Let

_{G* (-, Z/2Z)}

denote the _group of invertible elements in the

_cohomology

ring

H* (-,

Z/2Z).

Then,

as shown in

_[Ca]

Thm.

_5.18,

the Hasse-Witt invariants w2 and the

topological Stiefel-Whitney

classes sw2

give

rise to a

commutative

_diagram

l.f.

_{Cohomological}

invariants II: Galois theoretic.

Let 7r : X ~ Y be a

tamely

and

oddly

ramified cover of

degree n

as in

Sect.

_l.b,

and assume Y is connected. The classes we are

_going

to

define in this section will

_depend

on the natural

representation

of the tame

fundamental _group of Y defined

_by

7r. In fact we will

_only

need the class w2, but since it does not cost more effort we

_give

the

_general

definition.

The cover 7r is determined

by

a

homomorphism

e1r from the tame-odd

fundamental _group 7rl

(Y)’,’

into the

symmetric

group

Sn.

For étale covers

this is well known: in this case

_equals

the fundamental _group of

Y and the

_homomorphism

is described

_in-say-[Mu]

Remark at the end of Sect. 12. For

_general

tame covers see

[Gr-M]

2.4.4. In

principle

we should indicate the

_dependence

of 1fl

(Y)’,’

upon the choice of a base

_point

and the

fixed branch

_locus,

but we do not need to be so

_precise.

Let K denote the

_algebraic

closure of the residue field of some

_point

on Y.

By

embedding

sn

into the

orthogonal

_group

O(n) (K)

by

permuta-tion matrices we obtain an

_{orthogonal representation}

of the discrete _group

which we denote

by

_p~.. We want to use this

representation

to

pull

back universal classes to the

_cohomology

of

_7ri(y)~.

Consider the

composition

c of the _map

induced

_by

the

_change

of sites

_given

_by

_{Spec(K) -~}

Y and of the _map

which is induced

_by

a _{map E} on

_simplicial

sheaves defined as follows. View