The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles

The Lefschetz Trace Formula for the Mo duli Stackof

Principal Bundles

Kai A Behrend

Contents

Intro duction

General Concepts

The Lefschetz Trace Formula

Algebraic Stacks

The Trace Formula for Algebraic Stacks

The Case X X

Gbundles over a curve

The Canonical Parab olic Subgroup

Reductive Group Schemes Over A Curve

Ro ot Systems and Convex Solids

Families of Reductive Group Schemes

DeligneMumford Stacks

Intro duction

Purity

Tractable Stacks

Comparison of the Cohomology with the Coarse Mo duli Space

Gerb es Over a Finite Field

The Trace Formula for DeligneMumford Stacks

Artin Stacks

Intro duction

A Finiteness Theorem

The adic Leray Sp ectral sequence

The Sp ectral Sequence of a Fibration of Algebraic Stacks

ConstantHigherDirectImages

Fibrations With Connected Structure Group

The Innite Dimensional Grassmannian

The Trace Formula for the Action of a Linear Algebraic Group on a Deligne

Mumford Stack

Some Preliminary Remarks on Sp ectral Sequences

The Trace Fromula

The Algebraic Stackof GBundles

Intro duction

Some Elementary NonAb elian Cohomology Theory

The RelativeCase

The Case of Schemes

Algebraicity of the StackofGBundles

Representability of Certain Direct Images

Passing to Covers

Application to Reductive Group Schemes

Smo othness of the StackofGbundles

Ro ot Systems and Convex Solids

Intro duction

RootSystems

Facets and their Corners

Parab olic Sets of Ro ots

Reduction of a Ro ot System to a Facet

Ro ot Systems with Complementary Convex Solids

Complementary Convex Solids

Characterization of F in Terms of Faces

Stability of Ro ot Systems with Complementary Convex Solids

Stability

The Numerical Invariants

Sp ecial Facets

The Canonical Parab olic Subgroup

Intro duction

StabilityofReductiveGroups

The Numerical Invariants of a Parab olic Subgroup

Rationally Trivial Reductive Group Schemes

Alternative Descriptions of the Numerical Invariants

The Ro ot System with Complementary Convex Solid Given by a Generic Maximal

Torus of G

The Main Theorem for Rationally Trivial ReductiveGroups

The Main Theorem

Further Remarks

Families of Reductive Group Schemes over Curves

Intro duction

The Case of a Discrete Valuation Ring

The Fundamental Theorems

The Scheme of Parab olic Subgroups

The Theorems

The Typ e of Instability

Applications to Gbundles

StabilityofGbundles

The Natural Stratication of the StackofGbundles

The Degree of Gbundles

Reduction to a Parab olic Subgroup

The Trace Formula for the StackofGBundles

Intro duction

The Case of Vector Groups

Applications to ParabolicSubgroups

Calculating the Dimension of the StackofGBundles

Algebraic Stacks of Finite Type

Quasi Compact Stacks

Passing to Galois Covers

Borel Subgroups

The Theorem

Level Structures

Applications to the StackofGBundles

Passing to the Limit

Exhausting Topoi

The Condition x

Eigenvalues of Frobenius

Torsors under the Radical of G

Passing to the Limit

Intro duction

General Concepts

First we will give a general intro duction to the ideas of this thesis

The Lefschetz Trace Formula

The classical Lefschetz trace formula reads as follows

tr F j H X Q x

F xx

Here X is a compact oriented C manifold and

dim X

H X Q H X Q

is the singular cohomology of X with values in the rational numbers Q Furthermore F

X X is a C map having only nondegenerate xed p oints The trace of the Qlinear map

F H X Q H X Q induced by F X X on cohomology is by denition

dim X

p p

tr F j H X Q tr F j H X Q

For a xed p oint x ie F x xwe denote by x its index

If X is a complex manifold and F is holomorphic then the index is always p ositive so that

wehave

tr F j H X Q fx X j F x xg

More precisely x sign detDF x where DF xT X T X is the Jacobian of F at x A xed

x x

p oint x is nondegenerate if and only if detDF x

There is also an algebraic version of this formula In the algebraic case X is a complete

smo oth varietyover an algebraically closed eld k and F X X is a morphism of varieties

having nondegenerate xed p oints The formula reads

tr F j H X Q fx X j F x xg

The only dierence from the analytic version is that weusetheetale cohomology of X with

values in the adic rational numbers Q instead of singular cohomology

Avery interesting sp ecial case arises when the variety X is dened over a nite eld Let F

b e the eld with q elements and let k b e an algebraic closure of F That the variety X over k

is dened over F means simply that equations dening X can b e found that have co ecients

in F For example X could b e dened by equations with integral co ecients that we reduce

mo dulo q So in the case that X is dened over F the geometric Frobenius F X X

q q

acts on X and its adic cohomology The geometric Frob enius F acts on X simply by raising

th q

co ordinates to the q power F x x x x Note that the xed p oints of F are

q q n q

precisely those p oints of X with co ordinates in F The geometric Frob enius automatically has

only nondegenerate xed p oints Hence in this case the trace formula reads

tr F j H X Q X F

q et q

Here X F is the set of p oints of X with co ordinates in F

q q

For technical reasons we reformulate this formula in terms of the arithmetic Frobenius

On H X Q acts simply as the inverse of F For the trace formula reads

et q q q

dim X

q tr j H X Q X F

q et q

This follows from Poincare duality One advantage of is that it remains true if X is assumed

to b e a smo oth variety that is not necessarily complete without having to pass to cohomology

with compact supp orts A rigorous denition of the arithmetic Frob enius is given in Denition

A pro of of can b e found as Prop osition

Algebraic Stacks

Algebraic stacks relate to algebraic varieties in the same way group oids relate to sets A group oid

is a category all of whose morphisms are isomorphisms A set X is considered as a group oid

also denoted X by taking for ob jects the elements of X and for morphisms only the identity

morphisms A group G is considered as a group oid denoted BG with one ob ject whose auto

morphism group is GAGset X is considered as a group oid denoted X or XG by taking

as ob jects the elements of X and for the set of morphisms from x to y x y X the transp orter

Hom x y Trans x y fg G j gx y g

X G

For a group oid X wedene

Aut

where the sum is taken over the set of isomorphism classes of X and for an isomorphism class

Aut is the automorphism group of any representativeof If Aut happ ens to b e innite

equal to zero In case X X is a set X is just the numb er of elements of X If we set

Aut

X BG for a group GwehaveBG IfX X foraGset X wehaveX by

G G

the orbit formula at least if one of X or G is nite

and not even necessarily separated

For an algebraic variety X the set of morphisms from another variety S into X denoted

X S HomS X is simply a set For an algebraic stack X XS is always a group oid Recall

that a variety X over k can in fact b e completely describ ed by the collection of sets X S

where S ranges over all k varieties or more generally k schemes Similarly an algebraic stack

X is dened by the collection of group oids XS S ranging over all k schemes This is the way

in which algebraic stacks arise as a generalization of the concept of algebraic varieties The

most basic consequence of this denition is that algebraic stacks form a category instead of an

ordinary category as do algebraic varieties

Clearlyevery algebraic variety X over k is an algebraic k stack If G is an algebraic group

we get a corresp onding algebraic stack denoted BG called the classifying stack of GFor a

k variety S BGS is the group oid of principal Gbundles with base S On the other hand

wehave a morphism fptgBG where fptg Speck is the onep ointvariety is actually

a principal Gbundle in fact the universal principal Gbundle Both of these prop erties are

analogous to the classifying space of a Lie group in homotopy theory hence the notation BG

If X is a variety on which the algebraic group G acts via G X X then there exists an

algebraic stack X characterized by the fact that the following diagram is b oth cartesian and

co cartesian

G X X

X X

So X can b e considered as a quotientofX by GHere X X is a principal Gbundle

G G

The ob ject X always exists as algebraic stack even if the quotientofX by G as varietydoes

not exist at all Moreover X is always smo oth if X and G are see Note that as a sp ecial

case BG is simply the stack asso ciated to the trivial action of G on fptg Speck

Algebraic stacks are the prop er framework for considering mo duli problems Whenever the

ob jects that one wants to classify by a mo duli space have automorphisms one encounters serious

problems if one restricts oneself to algebraic varieties or schemesoreven algebraic spaces For

example let X b e an algebraic curveover the algebraically closed eld k Ifk C the complex

numb ers this just means that X is a Riemann surface Consider the mo duli problem for vector

bundles of rank n and degree d on X Then a mo duli variety exists only for stable vector bundles

Even if one restricts attention to stable vector bundles it is dicult to construct a universal vector

bundle over the mo duli space In fact a universal bundle only exists if n and d are coprime

This comes from the fact that even the stable vector bundles have automorphisms the trivial

automorphisms coming from multiplication by scalars elements of k Using algebraic stacks

all these problems disapp ear There exists an algebraic stack X such that for any k scheme S

XS is the group oid of families of vector bundles of rank n and degree d on X parametrized

by S The algebraic stack X is smo oth and a universal vector bundle exists over X

For the precise denition of algebraic stacks we refer to In accordance with an

algebraic stack is an algebraic stack as dened by M Artin in The stacks dened byP

Deligne and D Mumford in will b e called DeligneMumford stacks The signicance of

a stack b eing a DeligneMumford stack is that the ob jects it classies have no innitesimal

automorphisms

The Trace Formula for Algebraic Stacks

Let us return to the situation where k is an algebraic closure of the nite eld F Assume that

we are given a smo oth algebraic k stack X dened over the nite eld F I prop ose the following

For an Intro duction to categories see

Note that in general the categories BGS andB GS are not equivalentOnlyifH S G do wehave

equivalence In fact BG is obtained by stackafying the analogue of sheafafying the prestack S B GS

trace formula

Conjecture The arithmetic Frobenius satises the fol lowing trace formula

dim X

q tr j H X Q XF

q sm q

Recall that XF is a group oid so that XF is dened as in ab ove For algebraic stacks

q q

wehave replaced the etale cohomology by the smo oth cohomologyFor algebraic varieties the

smo oth cohomology is equal to the etale cohomology but for algebraic stacks the etale top ology

is not ne enough Note that Conjecture is a direct generalization of formula so that

Conjecture is true if X happ ens to b e a smo oth algebraic variety The general problem is to

nd reasonable assumptions on X that imply the truth of Conjecture A very remarkable

feature of this formula is that neither side is necessarily a nite sum Instead the formula

asserts the convergence of two series and the equality of their limits This is the main dierence

of Conjecture from formula In this thesis the truth of Conjecture is established

in some interesting sp ecial cases For example the main result of Section Theorem

establishes the trace formula for the case of smo oth DeligneMumford stacks of nite typ e In

this case the sums on b oth sides of the equation are nite no limiting pro cess is needed

The Case X X

Let G b e a smo oth algebraic group dened over F and let X b e a smo oth algebraic variety

dened over F endowed with a Gaction also dened over F Then the algebraic stack

q q

X X is dened over F In the case that G is connected by Lemma Conjecture

G q

reduces to

X F

dim X dim G

q tr j H X Q

q G

Q should b e considered as the algebraic analogue of the Gequivariant cohomology H X

of X So can b e thoughtofasaLefschetz trace formula for equivariant cohomology In this

thesis the truth of is established if G is a linear algebraic group See Theorem

The main ingredient in the pro of of Theorem is the Leray sp ectral sequence for a

b ering of algebraic stacks see Theorem Consider for example the case G G the

multiplicative group and X fptg Speck sothat X B G We dene G by the prop erty

m m

that G S S O for any k scheme S In particular G C C We can think of G

m S m m

as the algebraic analogue of the circle group S Wehave

Q with acting trivially for j

Q with acting via multiplication with for j

H G Q

otherwise

The Leray sp ectral sequence for the b ering fptgB G reads as follows

Q if i j

i j ij

H B G Q H G Q H fptg Q

m m et

otherwise

So from this sp ectral sequence together with weget

H B G Q Q with acting trivially

m q

H B G Q

and

i i

H B G Q H G Q H B G Q H G Q

m m m m

for all i So in general

Q for i even

H B G Q

for i odd

This is analogous to the fact that where acts on H B G Q via multiplication with

q m

BS the classifying space of S is the innite dimensional complex pro jective space and has

the same Betti numb ers as those given by Even though B G is of nite typ e and has nite

dimension dim B G it has the cohomology of an innite dimensional space

This exhibits a fundamental dierence b etween algebraic stacks and algebraic varieties

Whereas algebraic varieties just like C manifolds always have nite dimensional cohomology

this is not necessarily the case for algebraic stacks This is the main diculty in proving Conjec

ture One has to make sense of the trace of on the innite dimensional graded algebra

H X Q

In the case of B G we get from

tr j H B G Q

q m

So in this case tr j H X Q converges According to the limit should b e

q sm

dim G

q q q

G F F q

m q

which it is This proves the trace formula for the case of B G

In general wecho ose an emb edding Q C and consider H X C H X Q C

sm sm Q

Thus the eigenvalues of the arithmetic Frob enius b ecome complex numb ers and we are able

to prove the convergence of the series tr jH X C at least in the case that G maybe

q sm

emb edded into GL for some n See Section

Here it also b ecomes apparentwhywe use the arithmetic Frob enius instead of the geometric

Frob enius The trace of the geometric Frob enius do es not converge

Gbundles over a curve

The Canonical Parab olic Subgroup

Let X b e an algebraic curveover the eld k for example a Riemann surface over C Let G be

a reductive algebraic group over k for example one of the classical matrix groups The second

part of this thesis is devoted to the study of the algebraic stack of principal Gbundles on X

We will denote this stackby H X G So for any k scheme S wehave

H X GS group oid of principal Gbundles on X S

If for example G GL the general linear group then principal Gbundles can b e thoughtof

as vector bundles of rank n

Before wegoany further wehave to dene the degree of a principal Gbundle E on X For

a principal Gbundle E the degree dE is the homomorphism

dE XG Z

degE O

G X

where XG is the character group of G and degE O is the degree of the line bundle on

G X

X asso ciated to E via the character LetH X G b e the stackofGbundles of degree d

The nal result of this thesis is that Conjecture holds for X H X G mo dulo a

conjecture whichwe are able to verify in many cases see Theorem Towards that goal

we proveanumb er of results that are interesting in their own right The new dicultyin

dealing with the stack H X G is that it is not of nite typ e This means that not only the left

hand side of the formula of Conjecture is an innite series but the right hand side is also

There are innitely many isomorphism classes of Gbundles of degree d but fortunately their

automorphism groups growvery large so that the sum

H X GF

Aut E F

E H XGF

converges Here H X GF is the set of isomorphism classes of Gbundles of degree d that

are dened over F

The main to ol for studying the stack H X G is a natural stratication on it Every principal

Gbundle E on X has b esides the degree two imp ortantinvariants attached to it The degree

of instability deg E and the type of instability t E The degree of instability is a nonnegative

integer The typ e of instability is more complicated It is also a ner invariant than the degree

of instability and so the typ e if instability induces a ner stratication on H X Gthanthe

degree of instability do es

Let E b e a principal Gbundle on X ThenAut E E G the sheaf of automorphisms

G GAd

of E is a reductive group scheme over X Thetyp e of instabilityof E only dep ends on this

reductive group scheme Aut E For example consider the case G GLn The typ e of

instabilityofa vector bundle E is then the HarderNarasimhan Polygon which is dened in

terms of the canonical ag E E E on E As noted ab ove AutE GLE

is a relative group varietyover X All b ers of Aut E X are isomorphic to GLnas

group varieties The canonical ag denes a subgroup scheme P AutE by taking those

automorphismsof E resp ecting the canonical ag The completeness of the ag varietyof E

over X implies that AutE P is prop er over X ie that P is a parabolic subgroup of Aut E

So the canonical ag can b e thought of as b eing induced by a canonical parab olic subgroup P

of Aut E This motivates the study of reductive group schemes over X and their parab olic

subgroups

Reductive Group Schemes Over A Curve

Let G b e a reductive group scheme over the curve X over the eld k Then the Lie algebra g

of G is a vector bundle of degree zero on X For a parab olic subgroup P G ie a smo oth

subgroup scheme such that GP is prop er over X we dene the degreeof P to b e the degree

of the Lie algebra p of P deg P deg p where p is considered simply as a vector bundle over

X We dene the degree of instability deg G of G to b e the maximal degree of its parab olic

subgroups

deg P deg G max

P G

parab olic

This is easily seen to b e nite If deg Gwe call G semistableFor the case of a vector

bundle the degree of instability turns out to b e twice the area of the HarderNarasimhan p olygon

To G wehave an asso ciated scheme DynG the scheme of Dynkin diagrams of G which

is nite etale over X To x notation let fv v g b e the connected comp onents of Dyn G

Every parab olic subgroup P G denes an op en and closed subscheme of Dyn Gthetype

tP so tP can b e considered as a subset tP fv v gTo v tP we can dene in a

r i

G has now taken the place of Aut E

See Remark

canonical waya vector bundle W P v It can b e obtained as a factor of a suitable ltration of

the unip otent radical of P Its degree nP v deg W P v is called the numerical invariant

i i

of the parabolic subgroup P with respect to the component of its type v

In this thesis weprove that there exists a unique parab olic subgroup P G satisfying

i The numerical invariants of P are p ositive

ii PR P is semistable

See Theorem Here R P is the unip otent radical of P the largest unip otent normal

subgroup of P The quotient PR P is a reductive group scheme over X In the case that

G Aut E foravector bundle E and P is the parab olic subgroup corresp onding to a ag

E E E of E thenPR P is the automorphism group of the asso ciated

r u

graded ob ject E E E EE The parab olic subgroup P of Theorem is called

the canonical parabolic subgroup of GIfP is the canonical parab olic subgroup of G then the

formal sum

nP vv free ab elian group on fv v g

vtP

is called the type of instability of G

We also prove that the canonical parab olic subgroup P of G is the largest element in the

set of parab olics of maximal degree in G In particular the degree of the canonical parab olic

subgroup is the degree of instabilityof G

Ro ot Systems and Convex Solids

Weprove Theorem by reducing the study of parab olic subgroups of G to the study of

parabolic subsets or in another terminology facets of the ro ot system of G First we reduce the

question of unique existence of a canonical parab olic subgroup to the case that G is rationally

trivial This means that one and hence any generic maximal torus of G is split Let K b e the

function eld the eld of meromorphic functions of X A generic maximal torus is a maximal

torus of the generic b er G of GFor a generic maximal torus T G to b e split means that

K K

n n

G T for some n K

So assume that G is rationally trivial and let T G b e a generic maximal torus Then we

get an asso ciated ro ot system G T The set is a subset of the character group XT

of T and can b e characterized as follows

g g g

K K K

where g fA g j AdtA tA for all t T g and Ad G GLg is the adjoint

K K

representation of G Let V b e the Rvector subspace of XT R generated by Then is a

root system in V

Nowsince G is the generic b er of the reductive group scheme G over X G T

K K

has an additional structure on it I have called this structure a complementary convex solidIt

comes ab out as follows The Borel subgroups of G containing T are in bijection with the Weyl

chambers of For any Borel subgroup B of G containing T letdB b e the elementofV

the dual space of V such that

h dB i deg LB

Borel subgroups are parab olic subgroups that are solvable They are minimal parab olic subgroups

The Weyl chamb ers of a ro ot system are the facets of maximal dimension The facets of are the

equivalence classes of the equivalence relation on V dened bythesetofhyp erplanes H whereH is the

hyp erplane orthogonal to The facets of are in bijection with the parab olic subsets of and the parab olic

subgroups of G containing T

for all Here LB is a line bundle on X asso ciated to B and in a canonical way The

comp onents of the typ e of B v v are in bijection with the elements of the basis

r r

of dened by B and in fact LB W B v for i rFor p ositiveroots

i i

LB is constructed as a factor in a suitable ltration of the unip otent radical R B ofB

To stay within the context of ro ot systems we index the vectors dB by the Weyl chamb ers of

So let C b e the set of Weyl chamb ers of A complementary convex solid for is a collection

of vectors dc of V satisfying the following two axioms see Denition

i If is a fundamental dominantweight with resp ect to b oth the Weyl chambers c and d

then

h dci h ddi

ii If c is a Weyl chamber and a simple ro ot with resp ect to c and d cistheWeyl

chamb er obtained from c by reection ab out the hyp erplane orthogonal to then

h dci h ddi

Most of Section is devoted to proving that in the ab ove situation these two axioms are indeed

satised

The convex solid giving rise to the name of this structure is actually the convex hull F of

the vectors dc c C The family dc can b e reconstructed from F We call the ro ot

system with complementary convex solid d semistable if F We also dene the degree

and the numerical invariants for any facet of d Our main result on a ro ot system with

complementary convex solid d is that d has a unique special facet P It is the largest

element in the set of facets of maximal degree See Corollaries and

For the facet P to b e special means that

i all numerical invariants of P are p ositive

ii the reduced ro ot system with complementary convex solid d obtained from d

P P

by reduction to P is semistable

The convexityofF plays a crucial role in the pro of of this result since the sp ecial facet turns

out to b e the facet containing the unique p ointofF closest to the origin of V

If d is the ro ot system with complementary convex solid asso ciated to the generic maximal

torus T of G Corollary gives via the corresp ondence b etween facets of and parab olic

subgroups of G containing T a parab olic subgroup P of G sp ecial or canonical with resp ect

to T The existence of the canonical parab olic subgroup follows easily from this One of the

main observations that makes this pro of work is that anytwo parab olic subgroups of G contain

in their intersection generical ly a maximal torus of G

Iwould liketopoint out Conjecture to the eect that H X gp where p is the

Lie algebra of the canonical parab olic subgroup P of G The signicance of this conjecture is

essentially that the canonical parab olic subgroup is rigid The truth of this conjecture would

improve many of the results of Sections and

Families of Reductive Group Schemes

The following results lay the foundation for the stratication of H X Gmentioned ab ove

i Let G b e a reductive group scheme over the curve X over the scheme S For example G

could b e the automorphism group of a family of vector bundles parametrized by S Then

the degree of instability deg sdeg G isupper semicontinuous on S

i i

ii Let GXS b e as b efore and assume that the degree of instabilityof G is constanton

S Then if we passtoacover of S that is universally homeomorphic to S there exists

a parab olic subgroup P G such that for every s S P is the canonical parab olic

subgroup of G

Nowlet G b e a reductive group scheme over the curve X over the eld k From nowonwe will

assume that G and all its twists G for Gtorsors E satisfy Conjecture In other words

we assume that G satises Condition x see Denition

Let H X G b e the algebraic stack of principal Gbundles of degree d and degree of

instability less than or equal to m By i H X G is an op en substackofH X G We

d d

also provethat H X G is of nite typ e Theorem and even of the form Y for some

m H

DeligneMumford stack Y and a linear algebraic group H acting on Y Hence the trace formula

Conjecture holds for H X G To prove the trace formula for all of H X G weuse

d d

the exhaustion bythevarious H X G Tomake this pro cess work we need information on

H X G the stack of principal Gbundles on X of degree of instability equal to m By ii

X G decomp oses as follows H

H X G H X G

m v

d d

Here H X G is the stack of principal Gbundles of typ e of instability v the disjoint sum b eing

taken over all typ es of intability that give rise to the degree of instability mWe also prove that

H X G has the same cohomology as H X G for a suitable reductive group scheme G

d d

on X and degree d This group G can b e obtained by taking G PR P where P is the

canonical parab olic of Aut E for any Gbundle E of typ e of instability v The groups G that

o ccur in this manner are nite in numb er so that we get go o d control over H X G making

it p ossible to prove the absolute convergence of tr jH H X G C We obtain a pro of of

q sm

the trace formula which reads in this case

g dim G

X G C q tr j H H

sm q

AutE F

E H XGF

where g is the genus of the curve X Note that the cases in which G satises Condition x

include the following

X G is the stackofvector bundles of rank n and degree d i G GL so that H

ii G is rationally trivial and the genus of X is zero or one

iii Dyn G is connected

DeligneMumford Stacks

Intro duction

In this section weprove the Lefschetz trace formula for a smo oth DeligneMumford stack X

of nite typ e over a nite eld F Theorem The pro of contains several steps First

we consider the case of a certain sub class of algebraic F stacks These are the stacks we call

tractable see Denition This terminology is not standard

The reason for intro ducing these tractable stacks is the following If X is a tractable stack

then the coarse mo duli space X of X is very nice ie a smo oth variety and the pro jection X X

is also very nice ie etale So it is p ossible to deduce the trace formula for tractable stacks

from the trace formula for smo oth varieties The trace formula for smo oth varieties is reviewed

in Prop osition The trace formula for tractable stacks is Lemma The remarkable

fact is that for a tractable stack X the structure morphism X X X the coarse mo duli space

induces an isomorphism on the level of adic cohomology see Prop osition On the other

hand wehaveX F a result which is due to Serre This is the contents

Aut

of Section So b oth sides of the trace formula yield the same value for X and X

To deduce the trace formula for general smo oth DeligneMumford stacks from that for

tractable stacks weneedtwo results First every nonempty reduced DeligneMumford stack

contains a nonempty tractable op en substack Secondlywe need the Gysin sequence Corollary

for smo oth pairs of algebraic k stacks The existence of the Gysin sequence is the contents

of Section

X Q Note that in the case of a DeligneMumford stack X the adic cohomology algebra H

is nite dimensional over Q Wehave H X Q for p dimX So in this case unlike

for the general case there is no limiting pro cess needed to make sense of the trace formua

Let S beascheme An algebraic S stack is an algebraic S stack in the sense of Only

concepts not explained in will b e reviewed here Note in particular that all algebraic stacks

and hence all schemes and all algebraic spaces are assumed to b e quasiseparated

Purity

Let k b e a separably closed eld

Denition Let Z and X be smooth k stacks and i Z X a closed immersion Wecall

the pair i Z X a smooth pair of algebraic k stacks of codimension c if for every connected

comp onent Z of ZlettingX b e the connected comp onentof X suchthatiZ X wehave

dim Z c dim X

Let i Z X b e a smo oth pair of k stacks and let U X Z Let F b e an ab elian sheaf on

X F X We denote the p cohomology of X with values in F and support in Z by H

sm sm sm

The group H X Fisthe p derived functor of

F kerX F U F

sm sm

evaluated at F Wehave a long exact sequence

p p

H X F H X F H U F

sm sm sm

Let j U X b e the inclusion morphism Then we denote the right derived functors of

sm sm

F kerF j j F

p p

by H X F The sheaf H X F is the sheaf on X asso ciated to the presheaf

sm sm sm

Z Z

U H U FjU

where U ranges over the smo oth Xschemes There is an E sp ectral sequence

pq q

X F X F H H X H

sm sm sm

Z Z

Let Zn where n is prime to chark

Prop osition Cohomological Purity Let i Z X beasmooth pair of k stacks of

codimension cLet F bealocal ly constant sheaf of modules on X Then

for p c

X F H

i i F c for i c

Proof Since the isomorphisms whose existence we claim are also claimed to b e canonical the

question of their existence is lo cal on X Sowemay pass to a smo oth presentation X X of

X and thus assume that i Z X is a smo oth pair of k schemes of co dimension c Then our

prop osition follows from VI Theorem Remark and Corollary

Corollary Gysin Sequence Let i Z X beasmooth pair of k stacks of codimension

cLet F bealocal ly constant sheaf of modules on X Then we have a long exact sequence

pc p p

H Z i F c H X F H U F

sm sm sm

Proof From the sp ectral sequence and purityweget

H X FH Z i F c

sm sm

So the sequence gives us the Gysin sequence

Tractable Stacks

For an S stack X we dene AutXtobetheS stack given by the following cartesian diagram

AutX X

X X X

Note that the two morphisms AutX X are isomorphic but not canonically so Therefore

it is unambiguous to discuss the prop erties of AutX X without sp ecifying which of the two

morphisms we mean

Remark Let k b e a eld and let X b e an algebraic k stack lo cally of nite typ e If

Aut X X is at and X X is the coarse mo duli space of XthenX is an algebraic space

lo cally of nite typ e and X X is faithfully at and lo cally of nite typ e see Corollaire

Let k b e a eld and let X b e an algebraic k stack lo cally of nite typ e Assuming that Aut X X

is at wedene B Aut X to b e the image of the morphism

X X X

So wehave a faithfully at lo cally of nite typ e presentation

X B Aut X

of B Aut X making B Aut X into an algebraic k stack lo cally of nite typ e We clearly havea

diagram

AutX X

X B Aut X

which is b oth cartesian and co cartesian justifying the notation B Aut X Letting X b e the

coarse mo duli space of X wehave a diagram

B AutX X

X X

which is also b oth cartesian and co cartesian

Denition An algebraic S stack X is called a DeligneMumford stack if the diagonal

X X X

is unramied

Denition Let k be anyeldThenwe call X tractable if it satises the following condi

tions

i X is an algebraic DeligneMumford stack of nite typ e over k

ii AutX X is nite etale

iii The coarse mo duli space of X is a smo oth k variety

Note If X is a tractable k stack then X is smo oth integral of nite typ e over k IfX is

the coarse mo duli space of X then the natural morphism X X is etale

Proof Since Aut X X is nite etale the same holds for X B Aut X That in turn implies

that B Aut X X is at least etale Again by descent X X is etale Since X is smo oth X is

so to o It remains to prove that X is connected But this is clear b ecause X is a variety

Lemma Let X be an integral normal k scheme of nite type Let f Y X bea

separatedetale morphism of nite type Then there exists a nonempty open subset U X such

that the induced morphism f f U U is nite etale

Proof Without loss of generalitywemay assume that Y is connected Then Y is integral Let

L b e the function eld of Y and let Y b e the normalization of X in L Then Y is naturally an

op en subscheme of Y see I Theorem

Y Y

f f

Let Z Y iY Then f b eing nite implies that f Z is closed in X LetU X f Z

Then U is nonempty b ecause it contains the generic p ointofX Since f U iY wehave

that f f U U is a base change of f and hence nite

Prop osition Let X be a nonempty reduced DeligneMumford stack local ly of nite type

over the eld k Then there exists a nonempty open substack X X which is tractable

Proof Let Y X b e a smo oth presentation where Y is a k scheme lo cally of nite typ e

Since X is reduced Y is also So Y contains a nonempty op en subscheme Y that is a smo oth

k variety Let X X b e the image of Y under The stack X is then an op en substackofX

that is smo oth and of nite typ e over k So up on replacing X by X wemay assume that X is

of nite typ e and has a presentation Y X where Y is a smo oth k variety Consider the

morphism Aut Y induced via the following base change

Aut Y

Aut X X

The morphism Aut Y is representable of nite typ e and separated Hence Aut isa

separated algebraic k space of nite typ e By generic atness Theoreme there exists

an op en subset Y Y such that the induced morphism Aut Y is at Here denotes

the restriction of to Y So up on replacing Y by Y and X by the image X of wemay

assume that Aut Y is at Since X is a DeligneMumford stack X X X and hence

Aut Y is unramied Therefore Aut Y is etale and separated By lemma

there exists an op en subset Y Y such that Aut Y is nite etale So without loss of

generalitywemay assume that Aut Y is nite etale The same holds then for Aut X X

Wehavenow satised conditions i and ii To satisfy condition iii consider the coarse mo duli

space X of X According to Remark X is an algebraic k space of nite typ e Since

Aut X X is nite etale X X is etale In particular X is smo oth over k By I I

Prop osition there is a nonempty op en subspace X X that is a scheme Restricting X

even further wemay assume that X is a smo oth k variety Let X b e the base change

X X

Clearly X is an op en substackofX that satises even condition iii

Comparison of the Cohomology with the Coarse Mo duli Space

Let k b e a separably closed eld

Lemma Let X beatractable k stack with coarse moduli variety X Then there exists a

nite group G and an etale epimorphism Y X of k schemes yielding a cartesian diagram

BG Y

X X

Proof Let Y X be an etale presentation of XLetG b e the pullbackofAut X to Y

G Y

Aut X X

Then G is a nite etale group scheme over Y So after passing to an etale cover of Y wemay

assume that G is constant say equal to G for a nite group G considered as a nite etale

group scheme over k Then the following diagram is also cartesian

BG Y

B Aut X X

This implies the lemma

Prop osition Let X beatractable k stack with coarse moduli variety X Then we have

H X Q H X Q

Proof Let X X denote the structure morphism It suces to prove that

Q for i

R Q

for i

where wework with Q sheaves This question is lo cal in X soby Lemma wemay

assume that X BG for a nite group G But then R Q is clearly constant equal to

i i

H BG Q But H BG Q is nothing but the cohomology of the nite group G with

sm sm

values in the eld Q This proves the prop osition

Gerb es Over a Finite Field

Prop osition Let G be an algebraic F gerbe etale and of nite type over F ThenG is

q q

neuter

Proof We will use the following cartesian diagram

S Sp ec F

i q

Sp ec F Sp ec F

q q

where for every i n the scheme S is just a copyofSpecF The morphism is dened

i q

by jS id for all i n The morphism is dened by jS Spec where

i Sp ec F i

n n n

F F is the Frob enius automorphism dened by for F

q q q

Since G is etale over F there exists an etale presentation X GsuchthatX is a

nite etale F scheme Let X b e dened by the following cartesian diagram

X X

X G

The algebraic space X is then also a nite etale F scheme Hence there exists an n such

that for every p oint of the underlying top ological space of X there exists an x X F

factoring through Nowcho ose an arbitrary F valued p oint x of X We will construct a

S X yielding two commutative diagrams here written as one diagram morphism z

Sp ec F S

q i

z x

X X

Consider the morphism p p ofF schemes

X X X

which comes via base change from

G G G

Since G is a gerb e and hence is an epimorphism By our choice of n this implies that

n n n

X F X F X F

q q q

is surjective So for every i n wemaycho ose z X F such that z x x

i q i

Sp ec Dening the restriction of z to S to b e equal to z wehave constructed z as required

i i

Using z it is easy to verify that x x Thus x desceds to the required

section of G F

Prop osition Let G be a nite etale group scheme over F Then

Aut

BGF

Proof The group oid BGF is the category of nite etale F schemes X endowed with an action

q q

X G X such that the diagram

X G X

X F

is cartesian Let C b e the category dened as follows Ob jects of C are pairs X where X is

a nite set and X X is a p ermutation Morphisms in C are dened by

Hom X Y f X Y j g

Then wehave an equivalence of categories

nite etale F schemes C

X X

X Y X Y

Here X X F where F is some xed algebraic closure of F and X X is the Frob enius

q q q

Using this equivalence of categories we can reinterpret BGF as the group oid B G of ob jects

X ofC endowed with an action X G X such that the underlying action

G on X is simply transitive Now the nite group G acts on its underlying set via of

G G G

g gg

e e

G Denote the group oid dened by this action by GFor an ob ject of G ie an element of

we dene an ob ject X ofC as follows The set X is just the set G and X X

is dened by g g for g GNowwe dene a right action of G onX

G Let G be two ob jects of GFor a morphism via right translations in

ie an element Trans which is just an elementof G satisfying we dene

a morphism X X by left multiplication with inside G This morphism

resp ects the Gactions on X and X So wehave dened a functor

G B G

This functor is easily seen to b e fully faithful Toprove essential surjectivity let X bean

ob ject of C with a right G action making X a simply transitive Gset Cho ose an element

x X and G suchthatx x Then g x g denes an isomorphism of X

with X Thus our functor is an equivalence of group oids and we are reduced to proving that

Aut

Or in other words that

Stab

where the sum is taken over all orbits under the action of G on its underlying set But this

is simply the orbit equation

Corollary Let G be an algebraic F gerbe etale and of nite type over F Then

q q

Aut

Proof According to Prop osition G is neuter So let E b e an ob ject of the category GF

Let G b e its sheaf of automorphisms which can b e dened by the following cartesian diagram

G F

G G G

Since is representable and etale of nite typ e weseethatG is a nite etale group scheme

over F Now G is isomorphic to BG and the claim follows from Prop osition

Corollary Serre Let X be an algebraic DeligneMumford stack of nite type such that

Aut X X is etale with coarse moduli space X Then we have

X F

Aut

Proof Let X X b e the structure morphism Then is etale of nite typ e Let x X F

beapoint and let G b e the F gerb e dened by the pullback

x q

G F

x q

X X

Clearly G is an etale F gerb e of nite typ e Wehave a cartesian diagram of set theoretic

x q

group oids

G F fg

x q

XF X F

q q

Hence wehave

X X X

Aut Aut

xX F G F XF

q x q q

X F

which nishes the pro of

The Trace Formula for DeligneMumford Stacks

Let F b e the eld with q elements Let F b e an algebraic closure of F LetX b e an algebraic F

q q q q

stackofnitetyp e X X F Wewant to examine the action of the arithmetic Frob enius

F q

q i

X Q So let F F a a be the Frob enius Weget on the cohomology H

sm q q q

induced morphisms Sp ec SpecF Sp ec F and id Sp ec X X Let us abbreviate

q q q X q

id Sp ec by b X X The morphism b induces a morphism of top oi whichweagain

X q q q

denote by b X X LetF b e a sheaf on X By functoriality of cohomology b induces a

q sm sm sm q

i i

X F H X b F Since F is dened over X wehave b F F homomorphism b H

sm sm sm

q q q

i i

and an induced homomorphism b H X F H X F

sm sm

Denition The homomorphism thus constructed is called the arithmetic Frobenius act

ing on H X F and is denoted by

i i

H X F H X F

q sm sm

n i

Taking F Z Z passing to the limit and tensoring with Q we obtain H X Q

q sm

n n n

X Q Taking F and X F we get which is given by H

sm q q

Passing to the limit and tensoring with Q we see that acts on Q by

q q

multiplication with q OnQ c the endomorphism acts bymultiplication with q

Lemma Let X be an algebraic stack over the nite eld F Then for any integer c we

have

p p

X Q c H X Q Q c H

sm sm

as Q vector spaces with action In particular

p c p

tr jH X Q c q tr jH X Q

q sm q sm

Proof Clear

Prop osition Let X beasmooth variety over the nite eld F Then we have

dim X

X Q X F q tr jH

sm q q

Proof Let n b e the dimension of X The usual trace formula for the geometric Frob enius F

reads as follows

i i

tr F jH X Q X F

q et q

Consider the cup pro duct pairing

ni i

X Q H X Q n Q H

et et

which is p erfect byPoincare duality The cup pro duct satises F F F

q q q

Hence wehaveF F The arithmetic Frob enius acts as the inverse of F on

q q q q

H X Q m for any j m Hence is the transp ose of F with resp ect to the pairing

et q q

Since the trace of the transp ose of a matrix is the same as the trace of the original matrix we

get from

i ni

tr jH X Q nX F

q et q

which is equivalentto

i i n

tr jH q X Q X F

q et q

Since smo oth and etale cohomology coincide for X the prop osition follows

Lemma Let X beatractable F stack Then we have

dim X

q tr jH X Q

q sm

Aut

Proof By Prop osition wehave for any p

p p

X Q H X Q H

sm sm

where X is the coarse mo duli space of X Since X is a smo oth F varietywehaveby Prop osition

the following

dim X

X Q X F q tr jH

sm q q

By Corollary wehave

X F

Aut

So together this implies the trace formula for X

Theorem Let X bea smooth DeligneMumford stack of nite typeandconstant dimension

over the nite eld F Then we have

dim X

q tr jH X Q

q sm

Aut

Proof Let jXj b e the set of p oints of X The Zariski top ology on jXj is dened by calling U jXj

op en if there exists an op en substack U X suchthatU jUjWe then have a bijection b etween

op en subsets of jXj and op en substacks of X The set jXj with the Zariski top ology is a no etherian

top ological space b ecause X is of nite typ e over a eld Let

M fU jXjjThe theorem holds

for the op en substack U corresp onding to U g

Then M has a maximal element say U Let U b e the corresp onding op en substackofXWe

will prove that U X

So assume that U X and let Z b e the complementof U endowed with the reduced substack

structure By Prop osition there exists a nonempty op en substack Z of Z which is tractable

Let X U Z whichisanopensubstackofXIfwecanprove that the theorem holds for X

then wehaveacontradiction to the maximalityof Uthus proving U X and our theorem So

wemay assume that X X and hence Z Z

So wehave a smo oth pair i Z X of algebraic F stacks of co dimension c dim X dim Z

such that Z is tractable and the theorem holds for U X Z By Lemma the theorem also

holds for ZNowwe consider the Gysin sequence Corollary

pc p p

H Z Q c H X Q H U Q

sm sm

It follows that

dim X

q tr jH X Q

q sm

dim Zc dim U

q tr jH Z Q c q tr jH U Q

q q sm

dim Z dim U

q tr jH Z Q q tr jH U Q

q q sm

X X

Aut Aut

UF ZF

q q

Aut

Done

Artin Stacks

Intro duction

In this section we prove the Lefschetz trace formula for a smo oth algebraic F stack of nite typ e

X provided there exists a DeligneMumford stack X and a smo oth ane algebraic group G over

F acting on X suchthatX XG Letting BG b e the classifying stackofG the principal

Gbundle X X induces a morphism X BG This morphism is the b ering with b er X

asso ciated to the universal principal Gbundle over BG The main ingredientintheproofofthe

trace formula is the Leray sp ectral sequence of this b ering X BG

The general form of the Leray sp ectral sequence for morphisms of algebraic stacks is Theorem

It rests heavily on the niteness theorem Theorem proved in Section In Section

we examine the sp ecial case that the higher direct images of our morphism of stacks are

constant The result is Theorem One case in which the higher direct images are constant

is that of a bration with connected structure group This result is given in Theorem

which is the nal form of the Leray sp ectral sequence we will use

Toprove the trace formula for X XG wemay easily reduce to the case that G GL

bycho osing an emb edding G GL and noting that XG X GL GL

n G n n

So our pro of relies on the sp ectral sequence of the frib ering X BGL which motivates the

study of BGL in Section Even though the dimension of BGL is n wecallBGL the

n n n

innite dimensional Grassmannian b ecause of the universal mapping prop ertyofBGL and

the analogue with homotopy theory

The pro of of the trace formula is then carried out in Section Wehavetoworry ab out

convergence of the trace in this case Thus wecho ose an emb edding Q C andconsiderthe

X C The result is Theorem trace of the arithmetic Frob enius on H

sm q

A Finiteness Theorem

Throughout this discussion wexaninteger N If A is a no etherian ring wesay that a

sheaf of Amo dules F on a top os X is noetherian if it is a no etherian ob ject in the category of

sheaves of Amo dules on X IfX is a no etherian scheme then a sheaf of Amo dules on X is

no etherian if and only if it is constructible

Denition A morphism of top oi Y X is said to satisfy the niteness theorem

with respect to N if for any no etherian ring A suchthatNA and any no etherian sheaf of

Amo dules F on Y R F is a no etherian sheaf of Amo dules for all q

A top os X satises the niteness theorem with resp ect to N if the morphism X pt do es

where pt is the punctual top os

Prop osition If Y X and X Z are morphisms of topoi satisfying the niteness

theorem with respect to N then Y Z satises the niteness theorem with respect to N

also

Proof Let A b e a ring annihilated by N For a no etherian sheaf of Amo dules F on Y consider

the Leray sp ectral sequence

p q pq

R R F R F

The claim follows from the following fact If

F F F

is an exact sequence in an ab elian category in this case the category of sheaves of Amo dules

on Z then F is no etherian if and only if F and F are no etherian

Prop osition Let Y X be a morphism of topoi where X and Y are quasiseparated

and noetherian Then the property satises the niteness theorem with respect to N islocal

on the base X

Proof Let A b e a no etherian ring annihilated by N Then the prop osition follows immediately

from the fact that the prop erty of b eing no etherian is lo cal for a sheaf of Amo dules on a

quasiseparated and no etherian top os

Lemma Let Y X be a morphism of topoi and let Y be an object of Y covering

Y Assume that for al l p the induced morphism of topoi Y X satises the niteness

p p

theorem with respect to N where Y Y Y Then Y X satises the niteness

p Y Y

theorem with respect to N also

Proof Let A b e a no etherian ring annihilated by N LetF b e a no etherian sheaf of Amo dules

on Y For anyobject U of X Y U is a oneelementcover of U Consider the E sp ectral

sequence of this covering

q pq

H Y U F H U F E

p Y

If we let U vary over all ob jects of X thenwe can consider this as a sp ectral sequence in the

category of presheaves on X Applying the asso ciated sheaf functor we get a sp ectral sequence

q pq

E R F jY R F

p p

Using the fact that the restriction of a no etherian sheaf of Amo dules is again no etherian the

claim follows

Lemma Let Y X be a morphism of schemes Then Y X satises the

et et et

niteness theorem with respect to N if and only if Y X does

sm sm sm

Proof There is a natural morphism of top oi

j X X

sm et

j is the restriction of sheaves from the smo oth to the etale site j extends the emb edding of

the etale site into the smo oth site j is exact in fact j even has a right adjoint j b ecause j

is not only continuous but also co continuous

Let A b e a no etherian ring suchthatNA Since j j id and j is faithful wehave

for any sheaf of Amo dules F on X F is no etherian if and only if j F is no etherian This

implies for any sheaf of Amo dules G on X G is no etherian if and only if j G is no etherian

The same is of course also true for j Y Y

sm et

Let F b e a sheaf of Amo dules on Y For anyscheme U etale over X wehave

q q

H U j F H U F

et sm

q q

Hence R j F j R F using the fact that j commutes with the asso ciated sheaf

et sm

functor which follows from the co continuityof j Sowehave

q q

R j j R

et sm

which implies also

q q

j j R R

sm et

It follows immediately that satises the niteness theorem with resp ect to N of do es

et sm

F For the converse let F b e a no etherian sheaf of Amo dules on Y Tocheck whether R

sm sm

q q

is no etherian it suces to check whether j R F or equivalently R j F is no etherian

sm et

whichisnow clear

Theorem Let k be a eld and let Y X be a morphism of algebraic k stacks of nite

type Then Y X satises the niteness theorem with respect to N if the characteristic

sm sm

of k does not divide N

Proof First note that if X is an algebraic k stack of nite typ e then X is a quasiseparated

no etherian top os So by prop osition we reduce to the case that X X is a scheme of

nite typ e over k Nowcho ose a presentation Y Y suchthatY is a k scheme of nite typ e

Then Y Y Y is of nite typ e for all p By lemma it suces to showthat

p Y Y

Y X satises the niteness theorem with resp ect to N for all p From a niteness

p sm sm

theorem of Deligne th nitude weknowthatY X satises the niteness theorem

p et et

with resp ect to N Our theorem nowfollows from lemma

The adic Leray Sp ectral sequence

In this section we review the construction of the higher direct images for no etherian adic

sheaves and morphisms of top oi satisfying the niteness theorem As an application we prove

the existence of the Leray sp ectral sequence for morphisms of algebraic stacks of nite typ e

First recall the denition of adic and ARadic sheaves Exp V

Denition Let E b e a top os and a prime numb er A pro jective system F F

n n

of ab elian sheaves on E is called ARnul l if there exists an integer r suchthatF F

nr n

is the zero map for all n

F is called an adic sheaf or a Zsheaf if the following two conditions are satised

i For all n F

ii For all m n the canonical map F F F is an isomorphism

m m n

FiscalledanARadic sheaf if condition i ab ove is satised and there exists an adic sheaf G

and a morphism G F that is an ARisomorphism ie a morphism whose kernel and cokernel

are ARnull

An adic resp ARadic sheaf F is called noetherian if for all n F is a no etherian

ab elian sheaf on E The category of no etherian Zsheaves on E is denoted by ZfnE If

E X is the etale top os of a scheme X we also say constructible instead of no etherian

The following result is essentially proved in Exp V

Prop osition Let Y X be a morphism of topoi satisfying the niteness theorem

with respect to for al l n Let F beanoetherian adic sheaf on Y Then for any q

R F is a noetherian ARadic sheaf In particular there exists a unique adic sheaf

n n

q q q

G on X with an ARisomorphism G R F

n n

Denition Let Y X and F b e as in the prop osition Then we call the adic sheaf

q th q

G from ab ove the q higher direct image of F under and denote it by R F IfX pt is

q q

the punctual top os then R F is denoted H Y F

q q q q

Note that in general R F R F and H Y F H Y F Wehave dened an

n n n n

exact functor

R ZfnY ZfnX

Prop osition Let Y X and X Z be morphisms of topoi satisfying the

niteness theorem with respect to for al l n Then for any noetherian adic sheaf F on

Y thereisaLeray spectral sequence functorial in F

p q pq

R R F R F

Proof The pro of is completely analogous to the pro of of prop osition in Exp VI

Theorem Let k beaseparably closedeld char k Then for any morphism Y X

of algebraic k stacks of nite type and any noetherian Zsheaf F on Y we have a spectral

sequence

p q pq

H X R F H Y F

sm sm

in the category of projective systems of nite abelian groups

Proof By theorem Y X and X Sp ec k satisfy the niteness theorem with

sm sm sm

resp ect to for all n So by prop osition wehave a sp ectral sequence

p q pq

H X R F H Y F

sm sm

q q

using the fact that k is separably closed when replacing R by H

The Sp ectral Sequence of a Fibration of Algebraic Stacks

Constant Higher Direct Images

To prove the next result wehavetopasstoQ sheaves Recall that in the category of adic

sheaves morphisms are simply morphisms of pro jective systems of ab elian sheaves The category

of no etherian adic sheaves is a Zcategory which essentially means that all Homspaces are

Zmo dules and all Hompairings are Zbilinear This justies the term Zsheaves

Denition Let E b e a top os The category of noetherian Q sheaves on E is the category

whose ob jects are no etherian adic sheaves ie Zsheaves and whose morphisms are dened

Hom F G Q Hom F G

Q Z Z

We denote this category by Q fnE

Note that no etherian Zsheaves on the punctual top os are just adic systems of nite ab elian

groups where the denition of adic system is as in denition

Denition If M is an adic system of nite ab elian groups let

M fx M jr m n y M y x and y g

tors n n m

The adic system M M is called the torsionsubsystem of M IfM M M

tors tors n n tors

is called torsion IfM M is called torsionfree

tors

The following are some basic facts ab out adic systems of nite ab elian groups

Prop osition Let M beanadic system of nite abelian groups

i If M is torsionfreeitisfreeofniterank

ii MM is torsionfree

tor s

iii If M is torsion then there exists an r such that M

iv there exist integers r r such that

r r

M Z Z Z Z Z Z

where we have written Z for the canonical adic system given by Z Z Z for

al l n

Proof Straightforward

Prop osition Let M bean adic system of nite abelian groups Let E be a topos satisfy

ing the niteness theorem with respect to for al l n Consider M as a constant Zsheaf

on E Then for any p the canonical homomorphism

p p

H E Z M H E M

is an isomorphism in the category Q fnpt of nitely generated Q abelian groups In

we abuse notation in a similar fashion as in proposition with regardto Z

Proof For any n wehave a canonical homomorphism

p n p

H E Z Z M H E M

n n

x H E x

where we consider x M as a homomorphism x Z Z M Putting these homomor

n n

phisms together gives a homomorphism of pro jective systems

p n p

H E Z Z M H E M

n n

p p

Nowwehave adic systems H E ZM and H E M together with natural ARisomorphisms

p p n

H E Z M H E Z Z M

and

p p

H E M H E M

Comp osing we get an ARmorphism Exp V

p p

H E Z M H E M

p p

which is actually a homomorphism of pro jective systems since H E Z M and H E M

are adic So wehave established the existence of the homomorphism

Nowwehave a short exact sequence of adic systems of nite ab elian groups

M M MM

tors tors

where MM is free of nite rank say r Hence splits and M M Z Sowehave

tors tors

p p p r

H E Z M H E Z M H E Z

tors

and

p r p

H E M Z H E M

tors

By similar arguments as those proving the existence of weget

p p p

H E M N H E M H E N

for anytwo adic sheaves M N on E Using this our homomorphism is given by

p p r p p r

H E Z M H E Z H E M H E Z

tors tors

So it suces to show that the canonical homomorphism

p p

H E Z M H E M

tors tors

s s

is killed by for some integer s But this is clear b ecause M is killed by for some

tors

Corollary Let X be a topos having suciently many points and satisfying the niteness

theorem with respect to for al l n Let Y X be a morphism of topoi satisfying

the niteness theorem with respect to for al l n Let F bea noetherian Zsheaf on Y

such that R F is a constant projective system of abelian sheaves on X Thenthereisa

n n

spectral sequence

p q pq

H X Z R F H Y F

in the category Q fnpt of nite Q abelian groups

Proof If X is the empty top os there is nothing to prove So wemay assume that X has p oints

Let pt X b e one R F is a no etherian ARadic sheaf on X So there exists a

n n

no etherian adic sheaf H andan ARisomorphism H R F Pulling backtopt via

n n

the exact functor we get an ARisomorphism H R F Sincept X pt

n n

is isomorphic to the identitywehaveproved that there exists an ARadic system of nite

e e

ab elian groups N N such that N R F where we denote by N the sheaf

n n n n

on X asso ciated to N SinceN is AR adic there exists en adic system of nite ab elian

f e

groups M together with an ARisomorphism M N ThenM N is an ARisomorphism of

q q

pro jective systems of ab elian sheaves on X SoM R F and wehaveproved that R F

is constant So we can apply prop osition to conclude that

p q p q

H X R F H X Z R F

in the category Q fnpt From prop osition wenow get the sp ectral sequence

p q pq

H X Z R F H Y F

in the category Q fnpt

Denition For a pro jective system M M of ab elian groups suchthat M

n n n

for all nwe dene

M Q Q lim M

Z n

For a top os X and a pro jective system F of ab elian sheaves on X suchthat F forall

nwedene

p p

H X F H X F Q Q lim

n Z

Note If X is a top os satisfying the niteness theorem with resp ect to and F a no etherian

Zsheaf on X then

p p

H X F H X F Q Q lim

n Z

p p

This is b ecause H X F and H X F have the same limit b eing ARisomorphic

n n

Corollary Under the same hypotheses as in corol lary we have a spectral sequence

of nite dimensional Q vector spaces

p q pq

H X Q R F Q H Y F Q

Proof Simply take the limit of and tensor with Q

Theorem Let k beaseparably closedeld char k Let Y X be a morphism

of algebraic k stacks of nite typeand F anoetherian Zsheaf on Y such that R F is

sm sm n

constant for al l n Then we have a spectral sequence of nite dimensional Q vector spaces

p q pq

H X Q R F Q H Y F Q

sm Q sm sm

Proof X has suciently manypoints b ecause it is quasiseparated and no etherian It satises

the niteness theorem with resp ect to for all nby theorem and b ecause k is

separably closed Y X satises the niteness theorem with resp ect to for all

sm sm sm

n also by theorem So corollary applies

Fibrations With Connected Structure Group

Lemma let E be a topos Let X ob E Let G beagroup object in E acting on X on

the left Let P be a right Gtorsor and let Y P X Let j E E and j E E

G X Y

X Y

be the localization morphisms Then for any abelian object A of E we have

q q

R j AjY P R j AjX

Y G X

for al l q

Proof R j AjY is the sheaf asso ciated to the presheaf

U H Y U A

P R j AjX is the sheaf asso ciated to the presheaf

G X

U P U H X U A

by exactness of a the functor that maps a presheaf to the asso ciated sheaf Note that g GU

induces

g X U X U

x u g ux u

The op eration of g on H X U Aisthengiven by g g for a cohomology class

H X U A Note also that s P U induces an isomorphism

s X U Y U

x u suxu

We denote the isomorphism induced bys on the cohomology level by s s s for

a cohomology class H X U A The following facts are easy to check

g g

sge x u s gx u

sg g s

where and are used to prove Wecannow dene

q q

U P U H X U A H X U A

s s

That U iswelldened follows from We get an induced map of sheaves

q q

P R j AjX R j AjY

G X Y

is obviously an isomorphism

Prop osition Let G bea connected algebraic group over the separably closed eld k

acting from the left on an algebraic k stack X of nite type Then for every j n the induced

j n

action of G on H X Z Z is trivial

j n

Proof If we give the nite set H X Z Z the obvious k scheme structure then the action

j n j n

of G on H X Z Zis morphic Hence the stabilizer of any element H X Z Z

sm sm

is a nonempty closed and op en subset of GieG itself

Theorem Let k be a separably closed eld let X be an algebraic k stack of nite type

and let G beaconnected algebraic group variety over k acting from the left on X asmooth

algebraic k stack of nite type Let P be a principal Gbund le over X and let Y P X be

the associated bund le with ber X Thenif char k we may write the Leray spectral sequence

of the morphism Y X as the fol lowing spectral sequence of nite dimensional Q vector spaces

i j ij

E H X Q H X Q H Y Q

sm Q sm sm

e e e e

Proof Denote by G X P and Y the sheaves on X induced by the stacks G X P and Y

sm X X

that are all smo oth over XLet and b e the corresp onding lo calization morphisms

e e e e

G X P Y

in X

e e e e e

The sheaf P is a principal Gbundle and Y P X Letting Y X denote the natural

morphism and Y X the induced map of smo oth top oi wehave

sm sm sm

j n j n

R Z Z R Z Z

By Lemma wehave

n j n j

Z Z R Z Z P R

e e e

X G Y

j n

P H X Z Z

From Prop osition we get

j n j n

P H X Z Z H X Z Z

sm sm

which proves that

j n j n

R Z Z H X Z Z

sm sm

Then our theorem follows from Theorem

The Innite Dimensional Grassmannian

Let k b e a separably closed eld Consider BGL the classifying stackofGL over k Since

n n

BGL classies GL bundles or equivalently vector bundles of rank nwemay consider BGL

n n n

as an anolog of the innite dimensional Grassmannian which is used in homotopytheory

Let Grass h b e the Grassmannian over k of lo cally free quotients of rank n of O The uni

versal quotient on Grass h induces a natural morphism Grass h BGL The connection

n n n

between the cohomology of Grass h and BGL is given by the following lemma

n n

Lemma The natural morphism Grass h BGL induces an isomorphism

n n

i i

H BGL Q H Grass h Q

n n sm

for every i h n

Proof Wemay consider Grass h as the quotientofM n h by the natural action of

GL Here we denote by M n h the subvarietyofM n h of nonsingular n hmatrices

Hence we get a natural op en immersion of algebraic k stacks Grass h G nM n h Since

n n

M n h is cohomologically trivial we get from Theorem that G nM n h has the same

cohomology as BGL Now our lemma follows from purity Corollary up on estimating

the co dimension of the singular part of M n h

Let V b e the vector bundle on BGL asso ciated to the universal GL bundle via the standard

n n

representation of GL Letc H BGL Q i for i n b e the Chern classes of V

n i n

We will call c c the universal Chern classes

Theorem We have

Q c c H BGL Q

n n

The arithmetic Frobenius acts as the identity on H BGL Q

q n

Proof This may b e deduced from the corresp onding results on nite dimensional Grassmannians

using Lemma

Cho ose an isomorphism Q Q For any algebraic stack X over k we get an induced

isomorphism

i i

H X Q H X Q

sm sm

for anypairofintegers i In particular we get an isomorphism

Q c c H BGL Q

n n

of Q algebras and the action of the arithmetic Frob enius on the generators c c is

given by

c q c

q i i

for all i n

The Trace Formula for the Action of a Linear Algebraic Group on

a DeligneMumford Stack

Some Preliminary Remarks on Sp ectral Sequences

pq pq

Denition Let E E b e a family of C vec tor spaces E with

pq N

pq pq

E E a C linear endomorphism for every p q N Wesay that E satises condition

with resp ect to N a nonnegativeinteger and the four functions

a fNg R

b fNg N

c N R

d N N

if the following conditions are satised

i For every qN wehave E

ii If is an eigenvalue of on E then jj aq cp

iii dim E bq dp for every p q N

Lemma Let E E be a family of C vector spaces satisfying condition

pq N

P P

pq pq

with respect to N a b c d Assume that cpdp Then tr jE

p pq

is absolutely convergent

Proof Wehave

X X X

j tr jE j aq cpbq dp

pq p

X X

aq bq cpdp

which proves the lemma

Denition Let E E b e a family of C vec tor spaces satisfying condi

pq N

tion with resp ect to N a b c dsuchthat cpdp Thenwe dene the trace of

on E to b e

pq pq

tr jE tr jE

Lemma Let E bea spectral sequenceofC vector spaces with a compatible endomor

phism Let n be an integer n Assume that E satises condition with respect to

cpdp ThenE also satises condition with respect N a b c d where

to N a b c dMoreover we have tr jE trjE

n n

Proof It is clear that E satises condition So tr jE is welldened We consider

n n

E as an innite family C of nite complexes of nite dimensional C vect or spaces The

next level E is then the family of graded C vector spaces H C Sowemaycompute

tr jE tr jH C

tr jC

tr jE

where wehavetokeep track of the signs of course

Lemma Let E beanE spectral sequenceofC vector spaces with a compatible en

domorphism Assume that E satises condition with respect to N a b c d where

cpdp Let E abut to F

F E

Then

n n

tr jF tr jF

is absolutely convergent and we have

tr jF trjE

Proof First note that our sp ectral sequence degenerates at E sothatwehave E E

N N

By Lemma and induction E satises and wehavetrjE trjE Sowemay

estimate as follows

X X X

n pnp

j tr jF j j tr jE j

n n p

X X

an pbn pcpdp

n p

X X

aq bq cpdp

q p

This proves the absolute convergence of tr jF Toprove the formula wemay calculate

n n

tr jF tr jF

X X

n pnp

tr jE

n p

X X

n pnp

tr jE

n p

pq pq

tr jE

nishing the pro of

The Trace Fromula

Cho ose an emb edding Q C Ifwe are given a top os X we write H X C for H X Q C

Let X b e a smo oth equidimensional DeligneMumford stack of nite typ e over the separably

closed eld k together with an action of GL on X Let X b e the quotientstack for this

action Wehave a cartesian and co cartesian diagram of k stacks

GL X X

X X

The stack X is a smo oth algebraic k stack of nite typ e The structure morphism X X

is a left principal GL bundle and wehave another cartesian and co cartesian diagram of

k stacks

X X

Sp ec k BGL

We will consider the Leray sp ectral sequence of f the morphism of asso ciated smo oth top oi

induced by f Since f is just the bundle with b er X asso ciatied to the universal GL bundle

we get from Theorem the following E sp ectral sequence of nite dimensional Q vector

spaces

i j ij

H BGL Q H X Q H X Q

n Q sm sm

By the remarks following Theorem the E term of this sp ectral sequence maybewritten

as E H X Q c c where c c are the universal Chern classes and

sm n n

denotes an isomorphism Q Q Sowemay write this sp ectral sequence byslight

abuse of notation as

E H X Q c c H X Q

sm n sm

Now assume that X and are dened over the nite eld F Then the arithmetic Frob enius

acts on the sp ectral sequence and up on tensoring with C we get a sp ectral sequence of

C ve ctor spaces with compatible endomorphism

X C c c H X C H

sm n sm

Nowdene N dim X and

aj max jj for j N

bj dim H X Q for j N

ci q for i N

di dimQ c c for i N

n i

In the denition of aj the index ranges over the eigenvaluesof on H X Q In the

q sm

denition of di the subscript i denotes the homogeneous part of degree iWe dene the degree

cidi Moreover on Q c c by declaring the degree of c to b e Note that

the family H X C c c satises condition with resp ect to N a b c d

sm n q

Hence by Lemma we get that tr jH X C converges absolutely and wehave

q sm

tr jH X C tr jH X C c c

q sm q sm n

tr jH X C tr jC c c

q sm q n

tr jH X Q

q sm

Now dim X dim X dim GL so wehave

dim X dim GL dim X

q X C q tr jH X Q q q tr jH

sm q sm q

n n dim X

X Q q q q tr jH

sm q

GL F

n q

Aut

X F

by Theorem

Lemma Let X beasmooth DeligneMumford stack of nite type over F and let G bea

connected algebraic group over F acting on X Let X be the associated algebraic stack Then

X X

Aut Aut

X F XF

q q

Proof From a theorem of S Lang wehave H F G This implies that X F XF is

q q q

essentially surjective Hence wemay calculate

X X X

Aut Aut

X F

XF X F

q q

Aut

proving the lemma

From Lemma we get nally

dim X

X C q tr jH

sm q

Aut

Theorem Let G be a nonsingular linear algebraic group over F acting on a smooth

equidimensional DeligneMumford stack X of nite typeoverF Let X X XG be

q G

the corresponding smooth algebraic F stack of nite type Then tr jH X C is absolutely

q q sm

convergent and we have

dim X

X C q tr jH

sm q

Aut

Proof Chooseanemb edding of G into GL for some nLetY X GL ThenX Y

n G n GL

Y GL and so the theorem follows from

The Algebraic StackofGBundles

Intro duction

This section is of a technical nature We dene the stackofGtorsors H XS G where G

is an algebraic group scheme over the curve X over the base scheme S See section In

section we show that H XS G is an algebraic stack lo cally of nite typ e for the case we

are interested in This is the case that S is the sp ectrum of a eld and G is a reductive group

scheme over X See Prop osition In section weshow that H Xk G is smo oth See

Corollary The calculation of the dimension of H Xk G will b e deferred to section

Some Elementary NonAb elian Cohomology Theory

Let X be a topos Wewant to prove some basic facts ab out groups in X

Denition Let G a group in X and U ob X Thenwe denote the group oid BGU of

right G torsors byU G so that H U G is the set of isomorphism classes of U G If

no confusion seems likely to arise we write G for X G Note that X has the same

relation to H X asX has to H X

E Prop osition Let G bea group on X and E a Gtorsor Let G E G Aut

GAd

be the inner form of G obtainedfrom G by twisting with E Then we have a natural equivalence

of groupoids

G G

F Hom E F

with quasiinverse

G G

F F E E

Proof Straightforward

Remark This lemma do esnt necessarily apply to an arbitrary inner form G of G

Let G G b e a homomorphism of groups in X Then there is a natural morphism of group oids

G G

E E G

Prop osition If G G is a monomorphism and E ob G then there is a cartesian

diagram of groupoids

EG fg

G G

If E ob G thenwehave

GG fg

E G

G G

Proof The rst diagram expresses the fact that the category of reductions of E to G is a set

since G G is injective and that this set is equal to the set of global sections of EG The

second diagram follows from the rst since E GG E GG GG

G G

Prop osition If G G G is a short exact sequenceofgroups in X andif

E ob G thenwehaveacartesian diagram of groupoids

G fg

E G

G G

Proof Wehavetoshow that G is equivalent to the category of reductions of the structure

E E

group of E G to GLetF ob G ie F is an G torsor Under the vertical map

F is mapp ed to F E E The action of G on E is given bye g e eg Sothe

induced action on E G is given bye g e g eg g e g ie it is trivial So

F E E G E G Thus F F denes a reduction of structure group of

G G

E G to G

e e

ConverselyletE be a Gtorsor and E G E G an isomorphism of G torsors

G G

Construct F as the following b ered pro duct in the top os X

E E F Hom

E G E G X Hom

G G

Then F is an G torsor and F E E E

Prop osition Let G G G be a short exact sequenceofgroups in X with

G abelian If H X G for every E ob G thenG G is essential ly

surjective

Proof Let E be a G torsor Let K E b e the gerb e of liftings of E to GThelien of the

gerb e K E is the lien asso ciated to the group G The gerb e K E is trivial or neuter if

and only if E admits a global lifting to G see Chap IV Prop osition Now K E

denes a class in H G in the notation of Chap IV Denition By Chap

E E E

IV H G so that K E is isomorphic to B G the trivial G gerb e Hence

K E has a section

The ab ove prop ositions imply the standard facts ab out the long exact cohomology sequence in

nonab elian cohomology theory Just apply the following lemma to the ab ove diagrams

Lemma Let

A fg

B C

bea cartesian diagram of groupoids Then we get an induced exact sequence

A B C

Here denotes the associated set of equivalence classes and C is made into a pointed set using

fgC

Proof Clear

The RelativeCase

Let X S b e a morphism of top oi Wewant to generalize the results of the previous

section to this situation For a sheaf of groups G on X we denote the direct image BG of the

classifying X stack BG by H XS G

Prop osition Change of Origin Let G beasheaf of groups on X Then for any

Gtorsor E there is an isomorphism of S stacks

H XS G H XS G

Proof This follows immediately from Prop osition

Lemma Let G beasheaf of groups on X Then the natural morphism of S stacks

H S G H XS G is a monomorphism So we may consider H S G as an S

substack of H XS GThenH S G is the image S substack of the trivial morphism of

S stacks S H XS G

Proof First let us showthatforevery U ob S the functor H S GU H XS GU is

fully faithful In other words we need to show that the functor U U G U G

is fully faithful Without loss of generalitywemay assume that U S so that we need only

show that S G X G is fully faithful

Let E be a Gtorsor Let E E E GNotethatwehave a natural

G G

E Since lo cally this is obviously an isomorphism it follows morphism of Gsheaves E

E E

that E E We also have a natural morphism of S group sheaves G G Since

E E

this is also lo cally an isomorphism it follows that G G Therefore the following

diagram is commutative

S G X G

E E

S G X G

Here the vertical arrows are the change of origin isomorphisms from prop osition whereas

the horizontal arrows are the functors and of the lemma so bychanging the origin in

the indicated waywe reduce to proving that

i S G X G

ii X E S E

E aswe already did Now i is clear and ii likewise noting that E

This nishes the pro of that H S G H XS G is a monomorphism Now S

H XS Gobviously factors through H S G Also S H S Gisby denition an

epimorphism So wehaveproved the lemma

If G G is a homomorphism of groups in X then there is an induced morphism of stacks

over S

H XS G H XS G

Prop osition Let G G be a monomorphism of groups in X Let E ob H XS GU

where U ob S Then we have a cartesian diagram of S stacks

EG U

H XS G H XS G

Let E ob H XS G U Then we have

GG U U

E G

H XS G H XS G

Proof Follows immediately from Prop osition

Prop osition Let G G G be a short exact sequenceofgroups in X Let

Y be dened by the diagram below being cartesian Then we have the fol lowing diagram of

S stacks

H XS G Y H S G

H XS G H XS G

Here H XS G Y is a principal G bund le is a monomorphism and and give

the decomposition of as an epimorphism fol lowed by a monomorphism

If E ob H XS GU then we have a cartesian diagram of S stacks

U H X U G

E G

H XS G H XS G

Proof Clear

Prop osition Let G G G be a short exact sequenceofgroups in X and

G for every E ob H XS G U and for al l assume that G is abelian If R

U ob S then H XS G H XS G is an epimorphism

Proof Let E be a G torsor We need to show that we can lo cally lift E to GLetK E be

G gerb e of lifts of E to G The gerb e K E denes as in the pro of of Prop osition the

an element dE H X G It suces to show that lo cally over S this elementvanishes

But this follows from the assumption on R G

The Case of Schemes

Let X be a scheme Endow the category of schemes over X with the fppftop ology and pass to

the asso ciated top os This top os is the at topos associated to the scheme X We will denote

it by X A morphism of schemes X S induces a morphism of the asso ciated top oi and

a group scheme G over X induces a sheaf of groups on the top os X

If G is a group scheme over X thenaGtorsor is a torsor under the asso ciated sheaf of

groups on X IfX is an S scheme and G an X group ie a sheaf of groups on X then we

fl fl fl

denote by H XS Gthe S stack of families of Gtorsors In the notation of section the

stack H XS G is equal to H X S G

fl fl

Sometimes we will also have to consider the etale top os asso ciated to the scheme X This is

the top os asso ciated to the category of etale X schemes with the etale top ology and is denoted

by X

Lemma Let X be scheme and GX agroup scheme Then any Gtorsor is an algebraic

X space

Proof Let E be a Gtorsor Then there exists a faithfully at X scheme U of nite presentation

and a cartesian diagram of X spaces

U G U

E X

induced by a section s of E over U Sos p U G E is a presentation of E This proves

that E is an algebraic X space Another wayofsaying this is that the prop erty of b eing an

algebraic space is lo cal with resp ect to the fppftop ology

Lemma Let S be a scheme and E X a smooth surjective morphism of algebraic

S spaces Assume that X is quasicompact Then thereexistsanetale morphism of nite type

U X such that has a section over U ie such that there exists a morphism of algebraic

S spaces s U E such that s id

Proof This follows immediately from the corresp onding result for schemes whichiswell

known

Corollary Let X be a scheme and GX a smooth group scheme Then any Gtorsor is

local ly trivial with respect to the etale topology on X

Proof Without loss of generalitywemay assume that X is quasicompact Let E byaGtorsor

By lemma E is an algebraic X space Of course E is smo oth over X Soby lemma

E X has lo cal sections with resp ect to the etale top ology

Lemma Let Y X be a nite morphism of schemes Let G beagroup scheme over

i If is at then G G is a monomorphism

ii If is a nilpotent closed immersion and G is smooth over X then G G is an

epimorphism even with respect to the Zariski topology on X

Proof First let T be any X scheme Wehave GT GY T GY T So we

X X

need to check whether the restriction map GT GY T is a monomorphism But this

follows immediately from the faithful atness of Y T T

For the second part note that if T is an ane X scheme then the restriction map GT

GT is GY T is an epimorphism since G is formally smo oth over X SoGT

surjectivewhenever T is an ane X scheme This clearly implies the result

Algebraicity of the StackofGBundles

Representability of Certain Direct Images

Prop osition Let X beaprojective scheme over the eld k Let X Sp ec k be

the structure morphism Let S beak scheme and Y X an ane X scheme of nite

S S

presentation Then Y is an ane S scheme of nite presentation

Proof Since the prop osition is lo cal in S wemay assume that S is ane Then using

Prop osition we reduce to the case that S is the sp ectrum of a no etherian ring Cho ose

avery ample invertible sheaf O on X Wehave Y SpecAwhereA is a quasicoherent

submo dule algebras of nite typ e Then by there exists a coherent O sheaf of O

X X

S S

O n E Agenerating A Then there exists an O mo dule E of the typ e E

X i X

S S

together with an epimorphism EEThus we get a surjective morphism of O algebras

SE A Let I be the kernel Again by there exists a coherent submo dule F I

generating I LetF b e a sheaf of the typ e F O m mapping onto F Thenweget

X i

a morphism of O algebras SF SE suchthatF generates I Let E SpecSE and

F SpecSF b e the corresp onding geometric vector bundles over X Wehave a cartesian

diagram of X schemes

Y X

E F

This follows easily from the exactness of the following sequence of ab elian groups

Hom A O Hom E O Hom F O

O alg U U O mo d U U O mo d U U

U U U

where U is any X scheme

we get a cartesian diagram of X sheaves By the formal prop erties of the functor

Y S

F E

S S

O n But then we are also reduced to So wehave reduced to the case that Y V

X i

the case that S Speck forwhich it is easily seen that VE V E for any lo cally free

coherent O mo dule E

Corollary Let X beaprojective scheme over the eld k with structure morphism X

Sp ec k Let G be an ane group scheme of nite type over X Let S beak scheme and E F

E F is an ane S scheme of nite presentation Isom two G torsors over X Then

S S S

E E

Proof The sheaf IsomE F isan G torsor Since G is a form of G and G is ane

S S S S

of nite presentation G is by faithfully at descent again an ane group scheme of nite

presentation over X By the same reasoning IsomE F itself is an ane X scheme of nite

S S

presentation Then the lemma follows from Prop osition

Passing to Covers

Theorem Let X beaprojective scheme over the eld k Let f Y X beaprojective

at covering of X Let G be an ane group scheme of nite typeoverX Then the natural

morphism of k stacks

H Xk G H Ykf G

is representable ane of nite presentation

Proof Let U be a k scheme and E ob H Ykf GU ie E is an f G torsor over Y Let

U U

V b e dened so as to make the following diagram cartesian

V U

H Xk G H Ykf G

We need to checkthatV is an ane scheme of nite presentation over U

Consider the following pro jections

p Y Y Y

i X

y y y for i

q Y Y Y Y

i X X

y y y y for i

Y Y Y Y Y

ij X X X

y y y y y for i j

i j

For any U scheme T V T is the set of descent data for E A descent datum is an isomorphism

p E p E

T T T

satisfying the co cycle condition

T T T

In other words a descent datum is a preimage of the trivial element under the map

E E Autq E p Isomp

T T T T

T T T

This proves that wehave a cartesian diagram of sheaves over U

V U

p q E p E E Aut Isom

U U

Sp ec k is the structure map for i So it suces to prove where Y Y

i X X

that is an ane morphism of nite presentation But this follows from Lemma which

says that is a morphism b etween ane U schemes of nite presentation

Application to Reductive Group Schemes

Prop osition Let X beaprojective scheme over the eld k Let V bea vector bund le of

rank n over X and G GLV a closedsubgroup such that the quotient GLV G is an ane

scheme over X Then the natural morphism of k stacks

H Xk G H Xk GLV

is representable ane of nite presentation In particular H Xk G is an algebraic k stack

local ly of nite type

Proof By Prop osition for any GLV torsor E where S is a k scheme wehavea

cartesian diagram of k stacks

EG S

S S

H Xk G H Xk GLV

where X Sp ec k is the structure morphism By Prop osition to prove the rst part

all weneedtodoisexhibitEG as an ane X scheme of nite presentation This question is

S S

lo cal with resp ect to the fppftop ology on X sowemay assume that E GLV But then

S S

EG GLV G GLV G which is ane of nite presentation by assumption

S S S S

For the second part we will note that H Xk GLV is an algebraic k stack lo cally of nite

typ e By Change of Origin wehave H Xk GLV H Xk GL But H Xk GL is

n n

an op en substackofCoh the k stack of coherent O mo dules which is algebraic and lo cally

of nite typ e byTheoreme Note that even though Theoreme states

that Coh is of nite typ e all that is true and proved is that Coh is local ly of nite typ e

Xk Xk

Prop osition Let X S beaprojective at morphism of schemes Let V bea vector

bund le of rank n over X and G GLV a closedsubgroup Then the natural morphism of

S stacks

H XS G H XS GLV

is representable local ly of nite presentation In particular H XS G is an algebraic S stack

local ly of nite presentation

Proof By Prop osition for any GLV torsor E where T is an S scheme wehavea

cartesian diagram of S stacks

EG T

T T

H XS G H XS GLV

Now since EG is an algebraic X space of nite presentation wehaveby that

T T

EG isanalgebraicT space of nite presentation So we are reduced to proving that

T T

H XS GLV is an algebraic S stack lo cally of nite presentation This follows as in the

pro of of Prop osition from Theoreme

Prop osition Let X be a curve over the eld k and let G beareductive group scheme

over X ThenH Xk G is an algebraic k stack local ly of nite type

Proof Let f Y X b e a nite cover suchthatf G is rationally trivial Suchanf exists by

the b eginning of the pro of of Theorem Then by Note f G is an inner form So by

Theorem wemay replace X by Y and thus assume that G is an inner form Let G b e the

constant group of the same typ e as G The group scheme G is dened over k So there exists

an emb edding G GL making G into a closed subgroup of GL for some n The fact that

n n

G is an inner form means that there exists an Ad G torsor E suchthatG E G So

Ad G

wemay assume that G E G Via the induced map Ad G Ad GL PGL weget

Ad G n n

an asso ciated PGL torsor E PGL To this PGL torsor wehave the asso ciated inner

n Ad G n n

form E PGL GL E GL of GL We also have a homomorphism G

Ad G n PGL n Ad G n n

E G E GL which identies G as a closed subgroup scheme of E GL

Ad G Ad G n Ad G n

since it do es so lo cally

Now consider the exact sequence

G GL PGL

m n n

k b e an algebraic closure of k Thenwehave H X G of group schemes over X Let

et m

by Arcata Prop osition Hence over k any PGL torsor can b e lifted to a GL torsor

n n

In particular our PGL torsor E X The PGL can b e lifted to a GL torsor F over

n Ad G n n

torsor F is dened over a nite extension K of k Now it is easily seen using Theoreme

that to prove the prop osition wemay replace k by K and thus wemay assume that F is dened

over k

So wehaveaGL torsor F such that F PGL E PGL Nowwehave

n GL n Ad G n

GL F GL F PGL GL E PGL GL

n GL Ad n GL n PGL n Ad G n PGL n

n n n n

E GL Hence wehave exhibited G as a closed subgroup scheme of GL GLV

Ad G n n

where V is a vector bundle over X suchthatF is the bundle of frames of V

G GLV

This homomorphism is by construction lo cally isomorphic to G GL Butby a theorem

of Kostant the quotient GL G over k is an ane k scheme since G is reductive Hence

GLV G is an ane X scheme So by Prop osition H Xk G is an algebraic k stack

lo cally of nite typ e qed

Smo othness of the StackofGbundles

Prop osition Let S beanoetherian ane local scheme X S a curve and T S a

closed subscheme denedbyanideal of squarezero Let G beasmooth group scheme over X

Then the natural morphism

X G X G

T T

is essential ly surjective

Proof Let X X b e the natural morphism which is a closed immersion dened byan

ideal sheaf I of square zero Since I wemay consider I as an O mo dule By Lemma

wehave a natural epimorphism of sheaves of groups on X

Zar

G G

Let g b e the Lie algebra of Gwhich is a lo cally free coherent O mo dule If e X G denotes

the identity section and C is the conormal sheaf of e then wehave g C Nowfrom

eG eG

general principles wehave for any ane op en subset U X an exact sequence

Hom C jU IjU GU GU

O T

This gives rise to an exact sequence of sheaves of groups on X

Zar

Hom C I G G

O T

Or in other notation

g I G G

Now apply Prop osition to this sequence and the morphism of top oi X S Wehave

Zar Zar

R g I for every S S Zariskiop en and any G torsor E since b eing a

S S S

coherent mo dule is lo cal for the fppftop ology and by Chap I I I Corollary Hence S

b eing a lo cal scheme implies that X G X G is essentially surjective

T T

Corollary Let G beareductive group scheme over the curve X over the eld k Then

H Xk G is a smooth algebraic k stack

Proof By Prop osition H Xk G is an algebraic k stack lo cally of nite typ e So it

suces to provethatH Xk G is formally smo oth which is precisely Prop osition

Ro ot Systems and Convex Solids

Intro duction

We will dene the notion of a ro ot system with complementary convex solid A ro ot system

with complementary convex solid consists of a ro ot system and an additional structure d the

complementary convex solid If is a ro ot system in the real vector space V then d is the family

of vertices of a solid F V where V is the dual of V This solid F gives rise to the name of

the structure d d is a family of p oints in V parametrized by C the set of Weyl chamb ers of

This family d is sub ject to two axioms see denition The convex hull of d is F the

complementary convex solid Alternatively F can b e describ ed by its faces The faces of F are

parametrized by the set of fundamental weights of see the all imp ortant Lemma

The terminology complementary is suggested by the fact that there is a bijection b etween

the facets of and the solids b ounding F To a facet P there corresp onds a solid F P F

the dual solid of P This corresp ondence has the prop ertythatdimF P n dim P where

n dim V is the rank of at least if F P is not degenerate

The terminology convex is used b ecause F is by denition convex and this convexityis

the key p oint used in the pro of of our main result on ro ot systems with complementary convex

solids

The convexityofF is used as follows Cho ose a metric on V identifying V with V Then

since F is compact nonempty and convex there exists a unique p oint y F closest to the origin of

V This p oint y lies in a unique facet P of This facet is special Prop osition Our main

results Corollaries and can b e understo o d as giving dierentcharacterizations of

this sp ecial facet P See also Denition

Ro ot Systems

Here we will assemble some elementary facts ab out ro ot systems that do not usually seem to

emphasized in the literature

We will always assume all ro ot systems to b e reduced For convenience we briey review

the denition of a reduced ro ot system Let V b e a real vector space A root system in V is

a subset V satisfying

i is nite and generates V

ii For every there exists V such that h i and such that the map V V

dened by h i maps into

iii For every wehave Z

iv If then

Because of i the element whose existence is guaranteed by ii is unique Hence iii makes

sense

For the basic facts ab out ro ot systems see Chap VI

Facets and their Corners

Let V b e a nite dimensional Rvector space of dimension n Let b e a ro ot system in V Let

C denote the set of Weyl chamb ers of

For letL be the hyp erplane in V dened by

L f V jh i g

L is a nite set of hyp erplanes in V Hence L determines a partition of V into

facets We call these the facets of By denition V are in the same facet if and only

if for every either L and L or and are strictly on the same side of L ie

h ih i The facets of dimension n are precisely the Weyl chamb ers of The facet of

dimension zero is the origin There is a canonical bijection b etween the facets of and those of

For a facet P of weletP denote the corresp onding facet of We partially order the set

of facets of by

P Q P Q

Denition For a facet P of let chamP fc C j P cg

Let b e the set of fundamental weights of ie

f V jc C is a fundamental dominantweight wrt c g

Recall that if is the basis of dened by the Weyl chamber c then the dual basis

of is the set of fundamental dominantweights with resp ect to c

n n

For any let denote the corresp onding fundamental weightof

P The set Denition Let P b e a facet of Then is called a corner of P if

of corners of P will b e denoted byvert P We also dene

vert P f jc chamP vertcg

Wehavevert P vert P and vertP vert P if and only if P isachamb er This follows

for example from prop osition

Lemma Let c be a Weyl chamber of Let be the fundamental dominant

weights of with respect to c Then vert c f g

Proof is the dual basis of where is the basis of dened by c

n n n

It is wellknown that

c f V jh i for all i ng

f V j with for all i ng

i i i

So vert c for all i n

Now assume vert c Then there exists a Weyl chamber d such that is a fundamental

dominantweight with resp ect to dWehave d c for some W W the Weyl group of

Then for some i nSince c and cweget b ecause c is a

i i i

fundamental domain for the action of W on V

Prop osition Let P be a facet of and let bethecorners of P Then

i is linearly independent

ii The ane support of P is span

iii P f V j There exist such that g

r i i

iv The corners of P are

Proof In case P is a chamb er the prop osition follows from lemma For the general case let

c be a chamb er suchthatP cLet b e the basis of corresp onding to c

n n

the corners of cWemay assume that

P f V j

h i for all i r and h i for all i r ng

i i

for some r nThenvert P f g and the prop osition is obvious

Corollary Let P be a facet of Then for every vert P vertP fg is again

the set of corners of a facet of Similarly vert P fg is the set of corners of a facet for any

vert P

Parab olic Sets of Ro ots

Recall that a parab olic set of ro ots is a subset R such that for every wehave R

or R

Lemma Let P beafacet of with corners Let

RP f jh i for all i rg

Then we have

f g f jh i for all RP g

The set RP is cal led the parabolic set of roots associated to the facet P

Proof The inclusion isobvious For the other inclusion cho ose a chamber c cham P giving

a basis of with dual basis Let satisfy h i for all RP

n n

c Since RP wehave h i and hence h i for all i nSo

n i i

ie for some j n Assume that jrThenh i for all i rThus

j j i

RP by the denition of RP andby the assumption on wegeth i

j j j j

whichisa contradiction

Lemma Let R beaparabolic set of roots in Dene

vert R f jh i for all Rg

Then there exists a unique facet P of such that vert P vert RMoreover R RP in the

notation of lemma

Proof There exists a Weyl chamber c suchthatR contains the set of p ositive ro ots with resp ect

to c Let b e the basis of dened by c and the corners of c Then there

n n

exists a subset say such that

R f jh i for all i rg

By corollary there exists a facet P suchthatvert P f gSoR RP and by

lemma wealsohave f g vert R

Corollary There is a bijective correspondence

ffacets of g fparab olic subsets of g

P RP

whose inverse is given by

R the unique P such that vert P vert R

This correspondenceisorder preserving

Another way of lo oking at this bijective corresp ondence is given by the following prop osition

Prop osition Let P be a facet of R the corresponding parabolic subset of Then

Proof Cho ose a Weyl chamber c cham P with basis and corners suchthat

n n

vert P f gLet b e the reection dened by for i nLet

r i i

Consider an i r permutes R as can easily b e seen by lo oking at the formula

i i

h i Hence and Therefore span So to prove

i i i r

that P it suces to prove that h i foralli rTo this end consider for

i r the set

R f R jh i g

i i

It is easily checked that R R and that R R Sowe can decomp ose R as

i i i i i

follows

R R R S f g

i i i i i

where the union is disjoint Wemaynow calculate

X X X

h i h i h i h i h i

i i i i i i

R S

i i

h i h i

i i i

which nishes the pro of

Reduction of a Ro ot System to a Facet

Let P b e a facet of P induces direct sum decomp ositions V spanP P and V

span P P WesetV P and V P Thenwe identify V as the dual of V Let

P P

p V V andp V V b e the pro jections given by the ab ove direct sum decomp ositions

Note thatp is the adjointofp V is a ro ot system in V Wecall the reduction

P P P P

of to the facet P The dual of in V is V Let C denote the set of Weyl

P P P

P P

chamb ers of the set of fundamental weights of

P P P

Lemma p induces a bijection

p fFacets Q of suchthatQ P g ffacets of g

Q pQ

For a facet Q of such that Q P we have dim pQ dimQ dim P Inparticular we get

a bijection

p cham P C

Proof Straightforward calculation

Lemma p induces a bijection

p vert P vert P

Proof Clear

Ro ot Systems with Complementary Convex Solids

Complementary Convex Solids

Nowwe are prepared to give the fundamental denition of this section

Denition Let d dc b e a family of vectors in V parametrized by the set of

Weyl chamb ers of This family d is called a complementary convex solid for or alternatively

the pair d is called a root system with complementary convex solid if the following two axioms

are satised

B If is a corner of b oth of the Weyl chamb ers c and d then

h dci h ddi

B If the Weyl chamb ers c and d have precisely n corners in common and if is the

unique ro ot that is p ositive with resp ect to c and negative with resp ect to d then

h dci h ddi

Let P b e a facet of Then we dene the dual solid F P of P by

F P conv dc

ccham P

We also dene F F fg F is the convex solid that gives rise to the name of the structure

dened ab ove

Note If then F IfP Q for facets P and Q of then F P F Q

For the next denition let b e endowed with a complementary convex solid dLetP b e a facet

Denition The complementary convex solid d for inducedbyd is dened by

P P

d cp dp c

for each c C Here p is the inverse of the map of lemma

It is easy to check that d satises axioms B and B

Note Let Q b e a facet of and F Q the dual solid of Q with resp ect to d Then

P P P

F Qp F p Q

In particular letting F F fg wehave F pF P

P P P

Characterization of F in Terms of Faces

Let d dc b e a complementary convex solid for For let H be the hyp erplane

in V dened as follows

H fx V jh xi h dcig

where c is anyWeyl chamber such that isacornerofc By B the denition of H do es not

dep end on the choice of cH is a nite set of hyp erplanes in V

We dene the p ositive half space asso ciated with

E fx V jh xi h dcig

of E E We will also have to consider the closure

Lemma Let P beafacet of Then

E F P H

vert P vert P

For example

F E

Proof Let P b e a facet of Consider the pro jectionp V V and letp b e its restriction

H to

vert P

p H V

vert P

First note thatp is a bijectionp F P F and for any vert P vert P wehave

p H E E

vert P

where p is considered as a fundamental weightof Hence

E E p H

vert P vert P

which reduces the pro of to the case P fg so the claim is now

conv dc E

Let us rst prove the inclusion To this end dene c for every c CWe

vert c

cho ose a fundamental weight and a Weyl chamber c suchthat vert c Let c b e an arbitrary

Weyl chamb er Weneedtoshow that h dcih dc iLet b e the simple ro ots

dened by c and the corners of c If for all i n wehave h i then

n i

c and we are done by B So wemay assume without loss of generality that h i

Let b e the reection dened by and let d c By B wegeth dcih ddi

Moreover dc dd R by B b ecause are corners of b oth c and d So in fact

dc dd x with x

Nowwehave

h dci h dc ddi h ddi

xh i h ddi

h ddi

So it suces to provethath ddih dc i But wehave d c and hence

h d i h c ih i

h c i

b ecause h i implies h i So since there are only nitely manyWeyl chambers we

are done

The other inclusion we will proveby induction on the rank n of So we assume to

hold for ro ot systems of rank n Then by the ab ove argumentinvolvingp the lemma holds

for all onedimensional facets of This implies for every

E conv dc H

to provecho ose an element x E Then cho ose any fundamental weight and

E is compact by considering for example consider the line L x t Obviously L

E and E It is also clearly convex Let t t b e the interval such that L E

and for i and x t H E fx t j t t t gThenwehave x t

x t H for suitable By x t conv dcfori Hence

i cC

x conv dc also b ecause x convx t x t

By lemma wecaninterpret dc as the family of vertices of the solid F and H

as the family of ane supp orts of the faces of F In particular d can b e recovered from the

nite set of hyp erplanes H

Stability of Ro ot Systems with Complementary Convex Solids

Again let b e a ro ot system in the ndimensional Rvector space V

Stability

For a facet P of welet

RP f j for all vert P h i g

b e the corresp onding parab olic set of ro ots see Corollary We also dene

U P f j there exists vert P h i g

Wehave U P RP and in fact RP is the disjointunionof U P and

Denition Assume that is endowed with a complementary convex solid dLetP be

a facet of R the corresp onding parab olic set of ro ots Then we dene the degreeofP resp

the degreeofR with respect to d as

X X

deg P deg R h dci h dci

U P

where c is anyWeyl chamber such that P c or equivalently c is anyWeyl chamber such that

R contains the ro ots p ositive with resp ect to c

By prop osition and prop erty B this denition do es not dep end on the choice of c

Prop osition If we have a root system with complementary convex solid d then the

fol lowing areequivalent

i For every facet P of we have deg P

ii For every onedimensional facet P of we have deg P

iii For every pair c C such that vert c we have h dci

iv F

Proof iii is obvious In view of prop osition iii is just a reformulation of ii The

equivalence of iii and iv follows from the characterization of F iii implies i by

prop osition

Denition We call a ro ot system with complementary convex solid d semistable if

the conditions of prop osition are satised

Remark Call d stable if any of the following equivalent conditions is satised

i For every p ositivedimensional facet P of wehavedegP

ii For every onedimensional facet P of wehave deg P

iii For every pair c C such that vert c wehave h dci

iv is an interior p ointof F

The b ehavior of the degree with resp ect to reduction is describ ed by the following lemma Cho ose

a complementary convex solid d for

Lemma Let P be a facet of and let Q P be another facet Let p V V bethe

reduction map Then we have

deg Q degpQ deg P

Proof Cho ose a Weyl chamber c cham Q Let b e the corners of P and

r s

the corners of QThen

deg Q deg P h dci

U QU P

but

U Q U P

f ji r s h i and i r h i g

i i

f j vert pQ h i g

U pQ

Hence

deg Q deg P h dci

U pQ

h pdci

U pQ

deg pQ

so that we are done

The Numerical Invariants

For what follows we will need numerical invariants for facets that are ner than the degree

Denition Let P beafacetofLet be a corner of P Then dene

P f jh i and vertP fg h i g

WecallP the elementary set of roots associatedtoP and

Lemma Let P beafacet of Thenforanycorner vert P we have

span

vert P

Proof Let c cham P Let b e the basis asso ciated to c the corners of

n n

c and the corners of P Let for i n b e the generators of the Weyl

r i

group asso ciated to Let P Consider for ir

h i

i i i

For j r wehave

h i h i

i j j

so that P also So for ir permutes P Hence is invariant

i i

under and therefore

h i

for all i r n This proves that span

Now assume that is endowed with a complementary convex solid d

Denition For a facet P and a corner of P dene

nP h dci

Here c is anyWeyl chamber in cham P WecallnP the numerical invariant of P with respect

to and d

This denition is justied by lemma and B

Note For any c C wehave

dc nc

vert c

So d can b e reconstructed from the numerical invariants of the chambers

The b ehavior of the numerical invariants with resp ect to reduction is describ ed by the following

lemma

Lemma Let P beafacet of and let Q P be another facet Let p V V bethe

reduction map Let vert Q vert P Then

nQ npQp

Proof This follows easily from Q pQp whichisalsoeasytosee

Sp ecial Facets

Denition A facet P of is called special with respect to the complementary convex

solid d if the following two conditions are satised

S For all vert P wehave nP

S d is semistable

P P

Our goal is to prove that there is exactly one sp ecial facet for a given ro ot system with comple

mentary convex solid

Let P b e a facet of Then P determines uniquely a p oint y P V by

fy P g H span P

vertP

Lemma The facet P of is special if and only if the fol lowing two conditions are

satised

S y P P

S y P F P

Proof Let b e the corners of the facet P Let be the average over P for

r i i

i rThenh i for all i j rSoby lemma is a basis

i j ij r

for span

By the denition of y P wehave h yP i h dci for i r Hence h yP i

i i i

h dci for i rNowwehave

y P h yP i

i i

h dci

i i

This makes the equivalence of S and S obvious

Let us prove the equivalence of S and S

d is semistable F

P P P

pF P

span P F P

y P F P

H since F P

Note Another way of formulating this result is

P is sp ecial P F P

In case P is sp ecial y P is the unique elementofP F P

Prop osition Endow V with a metric compatible with W the Weyl group of IfP is a

special facet of d then y P is the unique point of F closest to the origin of V Conversely

if y is the unique point of F closest to the origin and P is the facet such that y P then P is

special and y y P

Proof Let denote a scalar pro duct on V suchthatW acts through isometries on V Use

to identify V and V Letkk b e the asso ciated norm F b eing a closed convex and nonempty

set in V it contains a unique p ointofminimal distance from V Call this p oint y

Let P b e a sp ecial facet with corners and cho ose a chamber c cham P Toprove

that y P y rst note that y P F byS and note then let x F b e arbitrary and

prove that kxk ky P k

Cho ose i frg x being in F implies that h xi h dci By the denition of

i i

y P wehave h yP i h dcisowe get h x y P i

i i i

Since y P satises S wehave y P with for i r b ecause the

i i i

are simply p ositive scalar multiples of the Nowwehave

i i

kxk ky P k kx y P k y P x y P

ky P k y P x y P

ky P k x y P

i i

ky P k

by the ab ove remarks

Now let P b e the unique facet suchthaty P Consider the C function

f R

t ky tdc y k

For all t wehave y tdc y F since F is convex By assumption f assumes its

minimum at t Hence f Dierentiating f yields the result y dc y Since

P P

r r

dc y with fori rSowehave y P wehave y

i i i i i

i i

On the other hand y F implies dc y for all i r This certainly forces

y dc for i rSoy H In particular y F P P so that P

i i

vert P

is sp ecial and y y P

Corollary d has a unique special facet

Proof Clear

Prop osition If P is a facet that is maximal among the facets of maximal degreeof

d then P is special

Proof Let vert P Let Q b e the facet with corners vert P fg Let q V V b e the

reduction map Then nP nq P q by lemma Now b oth

q P q

and are p ositive scalar multiples of q Hence there exists an such that

Rq P

nq P q deg q P Nowby lemma wehave

deg P deg q P degQ

nP deg Q

Since Q P wehavedegPdeg Q by the assumptions on P Hence nP

Now let p V V b e the reduction map to P Let Q b e a facet of such that pQisa

facetof By lemma the maximalityofdegP implies deg pQ Thus d is

P P P

semistable

Corollary The special facet of d is the largest element in the set of facets of maximal

degreeof d

Proof Let P b e the sp ecial facet of d and let Q be any maximal facet of maximal degree

Then by prop osition and corollary P Q

The Canonical Parab olic Subgroup

Intro duction

Let k be a eld A curve will b e a smo oth pro jective geometrically connected k scheme of

dimension one We will apply the theory of ro ot systems with complementary convex solids to

the study of reductive group schemes over curves If G is a reductive group scheme over the

curve X with function eld K then for any generic maximal torus T G which is split we

get a ro ot system G TinV XT R The ro ot system is naturally endowed

with a complementary convex solid prop osition This complementary convex solid d is

dened in terms of the numerical invariants of the Borel subgroups of G containing T These

numerical invariants were already intro duced by Harder in we review the construction of

these invariants in a much more general context in section For the pro of that d is actually

a complementary convex solid for weneedtwo alternative constructions of these invariants

see page Once it is established that a generic split maximal torus gives rise to a ro ot

system with complementary convex solid it is easy to deduce our main result theorem

See also denitions and Everything dep ends on corollary which summarizes

the connection b etween canonical parab olic subgroups of G and sp ecial facets of the various

asso ciated ro ot systems with complementary convex solids

Note that for a Weyl chamber c of dc is simply the unique vector in V suchthat

h dci deg W B

for every Here B is the Borel subgroup of G corresp onding to c and deg W B is the

numerical invariantof B with resp ect to the ro ot

Note also that in the case G GLV where V is a vector bundle on X our result reduces

to the wellknown existence of a canonical ag in V the HarderNarasimhan ltration See

Prop osition Note that the degree of instability Denition which plays a central

role in our discussion is in this case simply twice the area of the HarderNarasimhan p olygon

Our results could also b e deduced from Prop osition in There Ramanathan states a

theorem on reduction of unstable Gbundles to a parab olic subgroup Unfortunately it seems

that his pro of has never b een published

Wewould liketodraw attention to Conjecture This conjecture to the eect that the

canonical parab olic is isolated would havemany nice consequences

Stability of Reductive Groups

Let k b e a eld and let X be a curveover k ieX is a onedimensional pro jective smo oth

geometrically connected scheme over k

Denition Let G b e a smo oth ane algebraic group scheme over X with connected

b ers Then we dene the degreeof G to b e

deg G deg g

where g is the scheme of Lie algebras of G considered simply as a vector bundle on X Note

that G is of nite typ e over X so that g is in fact a vector bundle on X

We will b e concerned with reductive group schemes over X ie smo oth ane group schemes

over X all of whose geometric b ers are connected reductive algebraic groups see Exp

XIX For the denition of parabolic subgroups see Exp XXVI For a parab olic

subgroup P of the reductive group scheme GX we denote its unip otent radical by R P It is

again a smo oth ane algebraic group scheme with connected b ers see Exp XXVI

Note A reductive group scheme has degree zero In particular if GX is a reductive

group scheme and P G is a parab olic subgroup then deg P degR P

Proof By passing to a nite etale cover of X wemay assume that G is an inner form This can

b e seen as follows

Let G b e the constant group on X of the same typ e as GThenG b eing an inner form means

E This is equivalent that there exists a G torsor E suchthatG E G Aut

G Ad

to Isomext G G having a section over X By Exp XXIV th IsomextG G isa

constant tordu X scheme Now from Exp XXI I cor G is quasiisotrivial Hence the

same is true of Isomext G G Now from Exp Xcor we get that the connected

comp onents of IsomextG G are nite and of course etale over X Replacing X byanyone

of them G b ecomes an inner form

So wemay assume that G E G for a G torsor E Theng E g Now

G Ad G Ad

the adjoint representation Ad G GLg factors through SLg Hence g has a reduction

of structure group to SLg So det g O

Lemma Let GX beareductive group scheme There exists an M such that deg P

M for al l parabolic subgroups P of G

Proof Wehave

deg P deg p

dim H X p rkpg by RiemannRo ch

X g rkg g since p is a subsheaf of g dim H

Hence M dim H X g rkg g where g is the genus of X will do

Denition Let GX b e a reductive group scheme

i We call G semistableifforevery parab olic subgroup P of G wehave deg P

ii We call G stable if for every prop er parab olic subgroup P of G wehavedegP

iii The largest integer d such that there exists a parab olic subgroup P of G with deg P d

is called the degree of instability of G denoted deg G

By Note wemay replace deg P by deg R P in this denition By Lemma the degree

of instability is nite

The Numerical Invariants of a Parab olic Subgroup

Let GX b e a reductive group scheme

Remark Let DynG b e the Dynkin diagram of G Exp XXIV For a parab olic

subgroup P G the typeof P tP is dened in Exp XXVI to b e a certain section of

P DynG the p ower scheme of DynG Of course there is a choice involved here since one

could also take the complementof tP as the typ e of P Note that P DynG indeed has an

automorphism given by taking complements In fact I b elieve that the authors of SGA made

the wrong choice Here we will adopt the alternative denition This is justied for example by

lemma iii Our tP contains more information ab out P than the one of SAG do es For

example if P B is a Borel subgroup the tP ofSGAisempty whereas our tP contains

information ab out how the corners of B are twisted

Another reason for adopting the complementary denition is to avoid having to talk ab out

the complementofthetyp e of P in what follows

Lemma Let G T bea reductive group scheme G over a connected scheme S witha

maximal torus T which is split Let G T be the corresponding root system

i There is a unique bijective correspondence

fparab olic subgroups of G containing T g fparab olic subsets of g

P RP

such that for any parabolic P such that T P the Lie algebra p is given by

p t g

ii There is a bijective correspondence

fparab olic subgroups of G containing T g ffacets of g

P P

iii Let P beaparabolic subgroup of G containing T There is a natural bijective correspon

dence

tP vert P

v v

Proof i Since S is connected T is the group of multiplicativetyp e asso ciated to its character

group XT via Exp X cor and there is a unique system of ro ots G T ofG in

XT Now the claim is lo cal with resp ect to the Zariski top ology on S sowemay assume that

g is trivial for every G T So in this situation T denes uniquely a deployment

Exp XXI I def of GNowifP is a parab olic subgroup of G containing T then by

Exp XXI I lem there exists a subset R suchthatp t g Clearly R is

uniquely determined by P By Exp XXVI prop R is a parab olic subset of The

same prop osition then also gives the bijectivityofthismap

ii Follows immediately from corollary

iii Without loss of generality assume that G is deployable ie that all g are trivial

Cho ose an epinglage E of G compatible with P Let B b e the Borel subgroup T B P dened

by E Letc be the Weyl chamber of suchthatc B Wehave c P Let b e the

simple ro ots dened by c the corners of c Let b e the corners of P Now by

n r

denition DynG is a constant S scheme Its set of comp onents is identied with f g

since S is connected tP DynGisthesubscheme dened by the subset f gSo

we dene our map simply by for i r Thisisobviously bijective It remains

i i

to show that it is indep endent of the choice of our epinglage E So let E b e another epinglage

of G compatible with P The inner automorphism of P mapping E to E induces an element

W theWeyl group of But P P whichentails for i r

i i

Construction Let P G b e a parab olic subgroup and let tP bethetyp e of P tP

is a nite etale X scheme Let tP b e the set of connected comp onents of tP and let T

b e the free ab elian group on tP For simplicity let us write tP fv v g Let

v n v b e a p ositive elementofT We will asso ciate with v avector bundle on X as

i i

follows

Let Y X b e an etale X scheme whichwe assume to b e connected for simplicity and let

T b e a maximal torus of G such that T P and such that T is split Let G T

Y Y Y

b e the corresp onding ro ot system and assume that g is trivial over Y for sothatG

is uniquely deployable with resp ect to T Let P b e the facet of corresp onding to P via

lemma The schemes v v form a partition of tP So via lemma weget

Y sY Y

a partition v v of vert P For i s let Now dene

vf jh i n for all i sg

i i

v is a closed set of ro ots satisfying v vLet

U vYT U

b e the unip otent subgroup scheme of G dened byv U vYT is smo oth with geometri

cally connected b ers and indep endentofany order chosen on v See Exp XXI I

Prop osition There exists a unique subgroup scheme U v of G such that for every con

nectedetale Y X and every split maximal torus T of G such that T P and such that G

Y Y Y

is deployable with respect to T wehave U v U vYT U v is a closed normal unipotent

subgroup of P with geometrical ly connectedbers Moreover W P vU v U v is a

v v

vector bund le over X in a natural way

Proof We just need to adopt the pro of of Exp XXVI prop to our situation Uniqueness

is clear in view of the fact that G can b e deployed lo cally with resp ect to the etale top ology

Exp XXVI lem For the existence consider a connected etale YX together with an

epinglage E of G adapted to P see Exp XXVI def Let T b e the maximal torus

Y Y

given by E

Claim U vYT is normal in P

Let R G T b e the parab olic set of ro ots corresp onding to P ThenP is

Y Y Y

generated by T and the various U for R T clearly normalizes U vYT To see that U

normalizes U vYT consider the commutation relation of Exp XXI I cor

n m

p xp y p x p y p c x y

nm nm

U x exp xA is the isomorphism given by E A denotes the element Here p G

of g given E Sowehavetoshow that for vand R wehave n m vif

it is in But this is only a trivial computation

If wecho ose another epinglage E there exists a unique inner automorphism of P mapping

E to E This inner automorphism will leave U vYTinvariantby the claim Hence U vYT

do es not dep end on the choice of T Thisproves that the U vYTindeedgluetogive a subgroup

scheme U vofG By the claim U v is normal in P Now consider for YXT

as ab ove

vf ji s h i n and i s h i n g

i i i i

We can construct a corresp onding unip otent normal subgroup scheme of P called U v just as

we did for v Clearly U v U v For a given epinglage E of G over YX etale

v v

we consider the following homomorphism of group schemes over Y

g U v U v

Y Y

v v

Y X

exp x x

By the commutator of two elements in U v is in U v sothatthisisindeedawell

Y Y

dened homomorphism Infactitiseven an isomorphism Via this isomorphism weintro duce

avector bundle structure on U v U v Changing the epinglage amounts to applying an

Y Y

inner automorphism coming from P Let R and v Then wehave

p xp y p xU v p y p c x y U v

Y n n Y

But y c x y is linear This shows that P acts linearly on U vU v via

inner automorphisms So changing the epinglage do esnt change the structure of vector bundle

on U vU v

Prop osition Let U U be the ltration of R P dened in Exp XXVI

prop Then for every i we have

U U W P v

i i

P P

s s

Here T fv T jIf v n v then n ig

i j j j

j j

Proof Let i Cho ose a v T Then by denition wehave U v U and U v U

i i i

for every v v This gives a natural map W P v U U By lo oking at this map lo cally

i i

it can easily b e seen that it identies W P vwithavector subbundle of U U The rest of

i i

the prop osition is also easily proved lo callyby reducing to questions ab out the corresp onding

sets of ro ots

Denition Let P b e a parab olic subgroup of G Then for anycomponent v of the typ e

tP ofP we dene nP v deg W P vtobethe numerical invariant of P with respect to the

component of the type vWe call the collection nP v the numerical invariants of P

v tP

WecallW P v the elementary vector bund les asso ciated with P

Denition The parab olic P G is called canonical if it satises the following conditions

C The numerical invariants of P are p ositive ie for every v tP wehave nP v

C PR P is semistable

Our goal is to prove that every reductive group scheme over X has a unique canonical parab olic

subgroup theorem

Rationally Trivial Reductive Group Schemes

Let K denote the function eld of X LetGX b e a rationally trivial reductive group scheme

over X This means that there exists a maximaltorus T G that is split In this case every

generic maximal torus is split

Note G is rationally trivial if and only if G is an inner form

Proof Let G b e the constant group over X of the same typ e as GIfG is rationally trivial

G Ghasaconstant generic b er This implies that IsomextG Gistrivial then Isomext

see the pro of of note and hence that G is an inner form ConverselyifG is an inner

form any maximal torus of G is split by Exp XXIV cor G has maximal tori by

K K

Exp XIV theoreme

Let DynGX b e the Dynkin diagram of G DynG X is a trivial covering b ecause it is so

generically In particular for any parab olic P G the typ e tP is a trivial covering of X

Prop osition Let T be a generic maximal torus of GLet G T bethecorre

sponding root system

i There exists a natural bijection

fparab olic subgroups of G containing T g ffacets of g

P P

with inverse

ffacets of g fparab olic subgroups of G containing T g

Q Q

This bijection is order preserving

ii Let P bea parabolic subgroup of G containing T There is a natural bijective correspon

dence

tP vert P

v v

with inverse

vert P tP

Proof i In view of lemma it suces to prove that every parab olic subgroup of G extends

uniquely to a parab olic subgroup of G But this follows immediately from the pro jectivityof

GX see Exp XXVI cor Par

ii This is clear from lemma b ecause tP and tP are in canonical bijection

By this prop osition the Borel subgroups containing a generic maximal torus T G corresp ond

to the Weyl chamb ers of the corresp onding ro ot system G T The maximal parab olics con

taining T corresp ond to the onedimensional facets of G T ie to the fundamental weights

If P is a parab olic subgroup of G containing T and vert P we let W P W P

and nP nP degW P

Alternative Descriptions of the Numerical Invariants

We will givetwo alternative constructions of the vector bundles W P v For simplicitywe

restrict to the case that P B is a Borel subgroup of G In this case all W B v will b e

line bundles We do not insist however on the fact that v T b e p ositive So cho ose a Borel

subgroup B G and x it throughout this discussion

First Construction Let T b e a generic maximal torus of G contained in B and

G T the corresp onding ro ot system Let c B be the Weyl chamberofdenedby B

let b e the corners of c c denes an ordering on by is p ositive with

resp ect to c

h i is an elementofT v is either p ositiveof Let b e a ro ot Then v

i i

negative according to whether is p ositive or negative We distinguish these two cases

i is p ositive Consider

f j g and f j g

Then dene

M M

V g and V g

K K

V and V are vector subspaces of

g g t

K K

The spaces V and V extend uniquely to vector subbundles of g Call these vector

bundles V andV Clearly V V is a line bundle on X

Lemma There is a natural isomorphism W B vV V

Proof Consider the group schemes U vandU v U v of Prop osition The

v v

corresp onding Lie algebras uvandu v are subbundles of gNowby construction wehave

that uv V and u v V Hence uvV and u vV and

K K

W B vU vU vuvu vV V

which nishes the pro of

ii is negative Consider

f j g and f j g

Then dene

M M

V t g and V t g

K K

Again we extend these vector spaces to subbundles of gWegetvector bundles V and

V Again V V is a line bundle on X In this case W B v is not dened yet

So we dene W B vV V

Second construction See Satz Let T b e a generic maximal torus of

G contained in B and G T the corresp onding ro ot system Let G b e a constant

reductive group scheme over X of the same typ e as GLetT B G b e a constant maximal

torus and a constant Borel subgroup of G Let G T b e the corresp onding ro ot

system Weidentify withG T making sure that the Weyl chamb ers corresp onding

to B and B b ecome equal Now the pair G B is asso ciated to an Aut G B torsor

IsomG B G B G B G B

Aut G B

Wehave that B Aut G B is a subgroup scheme A reduction of structure group of

ad ad

IsomG B G B toB is the same thing as a section of the sheaf IsomG B G B B

But there is a canonical isomorphism

IsomG B G B B IsomextG G

by Exp XXIV cor G b eing an inner form IsomextG G is trivial so that indeed

ad ad

we can reduce the structure group of IsomG B G B to B So there exists a B torsor

E such that E G B G B Every induces a character of B whichwe will

again denote So for every ro ot we get a line bundle

L E ad O

on X

Lemma For every we have L V V Inparticular the isomorphism

class of L does not depend on the choice of the B torsor E such that E ad G B

G B

Proof Cho ose an We distinguish two cases

i is p ositive Again consider the subsets and of dened in Then

dene

M M

V g and V g

These are vector subbundles of g the Lie algebra of G Clearly E ad V and E

B V V

Claim B acts on V and V via Ad The induced action on V V

is given by

Wehave an isomorphism

U B T

see Exp XXI I prop So it suces to consider the actions of T and U for

on V and V T clearly leaves b oth bundles invariant So consider for

the morphism

exp g U G

which maps onto U Exp XXI I So wehave to examine the action of Adexp xA

on V and V Here A X g is a xed nowhere vanishing section for

every By Exp XXI I lem wehave the following formula

Adexp xA A A c x A

i i

for any Now and implies i ifitisarootat

all The same is true for So B indeed acts on V and V Hence B

also acts on the quotient V V By U acts trivially on this quotient So

B acts via since T do es This nishes the pro of of the claim

So nowwehave

V V E ad V E ad V

B B

E V V

E ad O

ii is negative The pro of is analogous to the pro of in the rst case We use and

and dene

M M

V t g and V t g

The only complication is that and do es not imply that sowehave

to treat this case separatelyBut

Adexp xA A A mo d t g

by Exp XX So this complication causes no problem

Note If and are ro ots then L L L For anyroot wehave

L L O

Proof Obvious

The Ro ot System with Complementary Convex Solid Given by a Generic Maximal

Torus of G

Lemma Let P G bea parabolic subgroup and B P G aBorel subgroup Let v T

beapositive element in the freeabelian group on the components of the typeofP Let U v be

the corresponding unipotent group denedinproposition

Let T B be a maximal torus and the corresponding root system Let v be the

corresponding set of roots Then we have

deg U v deg W B v

h i where v

vert B

Proof Clear

Prop osition Let T G be a generic maximal torus and let G T bethe

K K

corresponding root system For c C let

dc nc

vert c

Then dc isacomplementary convex solid for

Proof Tocheck whether C is satised let beafundamental weight and c C aWeyl

chamb er of Let b e the simple ro ots dened by c the dual basis Let

n n

B c b e the Borel subgroup of G corresp onding to cletP R b e the maximal parab olic

in G corresp onding to and let v b e the comp onentofthetyp e of B corresp onding to

Note that the unip otent subgroup U v asso ciated to B and v in prop osition is nothing

but R P By lemma wehave

deg U v deg W B v

h i By lemmas and wehave W B v L By where v

i i

h i deg L FinallywehavedegL nB note wehavedegL

i i i i

Assembling these various facts weget

deg P deg R P

deg U v

deg W B v

deg L

X X

h i deg L

i i

X X

h inB

i i

X X

nB i h

i i

Now by prop osition there exists an suchthat Soweget h dci

deg P deg P do es not dep end on the choice of c but only on the choice of So C follows

For the pro of of C let c and d be Weyl chamb ers having precisely n corners in com

mon Let b e the simple ro ots dened by c the corresp onding corners of c

n n

Without loss of generality assume that d c so that the corners of d are

Then is the unique ro ot p ositive with resp ect to c and negative with resp ect to dWehave

h ddi nd h i nd h i

i i

Nowby lemmas and applied to dwehave that nd deg L By

note wehave deg L deg L By lemma again deg L deg V V

are dened by and Now is negative with resp ect to dsothatV and V

d d

resp ectively Here the index d means that the construction is taking place with resp ect

to d

Claim and

c d

d c

Toprove the claim it suces to prove the rst part So assume Then

with resp ect to c ie for all i n wehave h i Assume Then

i i d

with resp ect to d In particular h ih i for all i n since

i i n

are corners of dSoh i for all i n which forces to b e a scalar multiple of

But b ecause and since So we reachacontradiction

d c

and prove the claim

By the claim wehave V V and V V b ecause this is the case

c d

genericallySowehave a natural homomorphism of line bundles

V V V V

c d

which is nonzero b ecause generically it is the identity map on g Hence wehave

deg V V deg V V

c d

By lemma wehave

deg V V nc

But nc h dci This proves C

Denition dG T dc dened as in prop osition is called the complementary

convex solid of G with respect to T

Prop osition Let T G be a generic maximal torus and d the corresponding root

system with complementary convex solid Let P G bea parabolic subgroup scheme containing

T We have

deg P degP

where deg P is of course taken with respect to d

Moreover for every corner of P we have

nP nP

Proof These are easy calculations using lemma

The following prop osition explains the signicance of reduction

Prop osition Let P beaparabolic subgroup of G T a generic maximal torus of P

Then T is also a generic maximal torus of G and PR P Let d betheroot system with

complementary convex solid associatedwithG and T Let P be the facet of associatedto

P via proposition Then d is the root system with complementary convex solid

P P

associatedtoPR P with the generic maximal torus T

Proof Since a complementary convex solid is completely determined bythenumerical invariants

of the Weyl chamb ers note it will suce to check that the numerical invariants of the

Borel subgroups b ehave according to lemma In other words wehavetoshow that

for any Borel subgroup B P and any corner vert B vert P wehave deg W B

deg W BR P p where p is the corner of BR P asso ciated with But this

u u

follows immediately from the fact that R P U The notation U istaken from the

pro of of prop osition Indeed wehave

W B U U

U R P U R P

u u

W BR P p

which implies the result

The Main Theorem for Rationally Trivial Reductive Groups

Corollary Let P bea parabolic subgroup of G Then P is canonical if and only if for

every generic maximal torus of P the corresponding facet P of the corresponding root system

with complementary convex solid is special In particular G is semistable if and only if for every

generic maximal torus of Gthecorresponding root system with complementary convex solid is

semistable

Proof First assume that P is canonical Let T P b e a generic maximal torus Let P be

the corresp onding facet of G T Then prop ositions and immediately show

that P is sp ecial Conversely assume that for every T P the corresp onding P is sp ecial

weneedtocheckthatP satises C and C C is clear To prove C assume that

PR P is not semistable Then there exists a parab olic subgroup Q of G such that Q P

and degQR P Cho osing a maximaltorus T Q this contradicts the fact that the

u K

corresp onding P is sp ecial

Prop osition G contains a unique canonical parabolic It is maximal among the parabol

ics of maximal degree

Proof Toprove uniqueness let P and Q be two canonical parab olics By Exp XXVI

there exists a maximal torus T G such that T P Q Letd b e the corresp onding

K K K

ro ot system with complementary convex solid Then the corresp onding facets P and Q of

are b oth sp ecial by corollary and hence equal by corollary This implies P Q by

prop osition

Toprove existence let P b e maximal among the parab olics of maximal degree We claim

that P satises C and C Let T be any generic maximal torus of G contained in P Using

the notation of prop osition P is then maximal among the facets of maximal degree in

G T dG T by prop ositions and By prop osition P is sp ecial Hence P

is canonical by corollary

The Main Theorem

Lemma Let Y X be a nite morphism of curves over k Assume that YX is

Galois ie that the corresponding extension of function elds is Galois Let GX beareductive

group scheme Then if P G isaparabolic subgroup such that P P for every

Gal YX then there exists a unique parabolic Q G such that Q P

G X Exp XXVI cor Proof This follows immediately from the prop erness of Par

By this prop erness parab olic subgroups are completely characterized by their generic b ers

Lemma Let Y X be a nite separable morphism of curves over k and G areductive

group scheme over X Let P G bea parabolic subgroup The numerical invariants of P are

positive if and only if the numerical invariants of P arepositive

Proof Let v tP and let v v b e the elements of tP lyingover v In other

words v v Clearly wehave

W P v W P v

So if n P v degW P v ispositive for every i s then so is nP v deg W P v

i i

This proves one direction It also makes it p ossible to reduce the other direction to the case that

YX is Galois

v and hence on fv v gNow since v is connected the Gal YX acts naturally on

i s

action of Gal YX on fv v g is transitive For any Gal YX and i s wehave

W P v W P v and hence

i i

deg W P v deg W P v

i i

deg W P v

This proves that

deg deg W P v s deg W P v

for any i s which proves the lemma

Lemma Let Y X be a nite separable morphism of curves and G areductive group

scheme over X such that G is rational ly trivial Then G is semistable if and only if G is

semistable

Proof Clearly G semistable implies G semistable This allows us to reduce the other

direction to the case that YX is Galois So assume that G is semistable By prop osition

G has a canonical parab olic P G For every Gal YX P is also a canonical

parab olic in G Hence P P and P descends to a parab olic Q G such that Q P

NowdegQ so deg P which implies that P G and G is semistable

Theorem Let G bea reductive group scheme over the curve X ThenG has a unique

canonical parabolic subgroup P P is the largest element in the set of parabolics of maximal

degreein G

Proof Cho ose a nite Galois morphism of curves Y X such that G is rationally trivial

This can b e done bycho osing a nite Galois extension L of K over which G has a split maximal

torus Exp XIX prop and then taking Y to b e the integral closure of X in LNowby

prop osition G has a canonical parab olic P As in the pro of of lemma P descends

to X so there exists a unique parab olic Q G such that Q P Evidently Q is canonical

This proves the existence

NowifQ is any canonical parab olic in Gthen Q is canonical by lemmas and

This proves Q P and Q Q This proves uniqueness

For the last part let Q G b e maximal among the parab olics of maximal degree and let

P G b e the canonical parab olic Then deg P deg Q as can b e seen after pulling backvia

and using prop osition Hence deg P degQ and P Q by assumption on Q This

implies P Q again by pulling back via and using prop osition

Further Remarks

Denition Let G b e a reductive group scheme over the curve X LetV b e the free ab elian

group on the connected comp onents of DynG Let P b e the canonical parab olic subgroup of

G Assume that the typ e tP v q q v Then we call

nG nP v v V

i i

the type of instability of G

Corollary The canonical parabolic commutes with pul lbacks to separable curves over X

In particular semistability commutes with such pul lbacks

Proof Clear

Conjecture Let G bea reductive group scheme over the curve X Let P be the canon

ical parabolic subgroup of GLet g and p be the Lie algebras of G and P respectively Then

H X gp

The truth of this conjecture would imply that the canonical parab olic subgroup is a geometric

invariant ie that even after passing to inseparable extensions of the base eld k the canonical

parab olic remains canonical So it would make notions like the geometric degree of instability

sup eruous

Conjecture is easily seen to b e true if G GLV for a vector bundle V over X orif

P is a Borel subgroup It is also true if the genus of X is zero or one and G is rationally trivial

Conjecture is equivalent to claiming that the canonical parab olic is isolated in the

sense that the corresp onding k valued p ointof Par G here X Sp ec k is the structure

morphism is an isolated p ointof Par G with lo cal ring equal to k

Families of Reductive Group Schemes over Curves

Intro duction

In this section we study the b ehavior of the canonical parab olic subgroup in families of reductive

group schemes For this purp ose we consider the following setup Let S b e a lo cally no etherian

scheme and X S a curveover S This means that is a smo oth pro jective morphism

all of whose geometric b ers are onedimensional and connected We will study reductive group

schemes G over X

Our goal is to provethetwo fundamental Theorem and The rst theorem is a semi

continuity theorem It states that the geometric degree of instability is upp er semicontinuous

in families Under the assumption that the geometric degree of instability is constant wecan

almost prove that the canonical parab olics P in the various memb ers G of our family glue

s s

together to a parab olic subgroup P of G The signicance of Theorem is that all wehave

to do is pass to a cover S of S that is universal ly homeomorphic to S Note that the truth of

Conjecture would improve this result These theorems are nicely summarized in Corollary

Before considering the general case we consider the case that S is the sp ectrum of a discrete

valuation ring This is the contents of Section

In Section we apply our results to the following situation Wehave a reductive group

scheme G over a curve X over a eld k We study the algebraic k stackofGtorsors H Xk G

dened on page We asso ciate three invariants with a Gtorsor E

First there is the degreedegE ofE which is the homomorphism deg E G Z where

G is the character group of G dened by

deg E deg E O

G X

This degree generalizes the degree of a vector bundle

E E G The second and the third invariantof E are dened in terms of G Aut

GAd

the twist of G asso ciated to E The degree of instability of E is the degree of instabilityof G

dened in Denition The typ e of instability is dened in terms of the Dynkin diagram

Dyn G of GIfV is the free ab elian group on the connected comp onents of the scheme Dyn G

then the typ e of instability is a certain nonnegative elementofV See Section

For every homomorphism d G Zwelet H Xk G b e the substackofH Xk Gof

Gtorsors of degree d It is a closed and op en substack The stack H Xk G of Gbundles

of degree d and degree of instability less that or equal to m is op en in H Xk G The family

H Xk G denes an exhaustion of H Xk G If v V we let H Xk G

m m mv

d d d

b e the stackofGtorsors of degree d degree of instability m and typ e of instability v The

family H Xk G denes a stratication of H Xk G This is the analogue of the

mv mv

d d

stratication of Shatz see

The Case of a Discrete Valuation Ring

Lemma Let G bea reductive group scheme over the curve X over the eld k Let P G

bea parabolic subgroup such that

i the numerical invariants of P arepositive

ii deg P deg G

Then P is the canonical subgroup

Proof Let K b e the function eld of X By lemma and corollary wemaypasstoa

nite separable cover of X and thus wemay assume that G is rationally trivial Let T P

be any generic maximal torus Denote by P also the facet of G T corresp onding to

P By corollary it suces to provethatP is sp ecial with resp ect to the complementary

convex solid d on induced by GFor this it suces to prove that d is semistable So

P P

let Q P b e another facet By lemmas and it suces to show that deg Q deg P

which is clear

Lemma Let X beanintegral noetherian normal scheme G over X areductive group

scheme Let be the generic point of X IfP G is a generic parabolic subgroup such that

there exists a vector subbund le p g such that p is the Lie algebraofP then there exists a

parabolic subgroup P G such that p is the Lie algebraof P

Proof Let Grg b e the pro jective X scheme of vector subbundles of g There is a natural

X morphism f Par G Grg asso ciating to a parab olic subgroup of G its Lie algebra We

rst note that this morphism f is radicial To prove this let L b e an arbitrary eld together

with a morphism x SpecL X WeneedtoshowthatParGL GrgL is injective

So let P P ParGLbetwo parab olic subgroups of x G Gj Sp ec L By Exp XXVI

there exists a maximal torus T of x G such that T P P Iff P f P then the

Lie algebras p and p are equal as subbundles of x gThus by Exp XXI I Corollaire

wehave P P Thus f is indeed radicial

Nowtoprove the lemma note that p corresp onds to a section of q Grg X whichwe

will also denote by p Similarely P is a generic section of p ParG X LetZ b e the closure

of P f ginParG with the reduced subscheme structure Let i Z Par G b e the inclusion

morphism If we can show that p i is an isomorphism we are done b ecause q is separated

Towards this goal let us rst prove that p i is injective on the underlying top ological spaces

If z z Z both lie over x X then their images f z andf z also b oth lie over xNow

z and z are sp ecializations of P ParG Hence f z andf z are sp ecializations of

p Grg But the only sp ecialization of p lying over x is px Hence f z f z

px Because f is radicial this implies z z Thus p i is indeed injective

Since p i is clearly of nite typ e the injectivity implies that p i is quasinite Since p i is

also clearly pro jective Zariskis main theorem in the formulation of Grothendieck implies that

p i is nite in particular ane Now since p i is birational and X is normal p i is indeed an

isomorphism

Prop osition Let X S be a curve over the scheme S SpecR where R is a discrete

valuation ring and let G bea reductive group scheme over X Let be the generic point of S

the special point Then we have

i deg G deg G

i i

ii If deg G deg G then there exists a parabolic subgroup P G such that P is the

i i

canonical subgroup of G and P is the canonical subgroup of G

Proof Let P b e the canonical parab olic subgroup of G Letp b e the Lie algebra of P Then

wehave an exact sequence of vector bundles on X dening q

p g q

By the prop erness of QuotgXS there exists a coherent O mo dule qatover S together

with an epimorphism u g q suchthatujX is the map of the ab ove short exact sequence

Let p b e the kernel

p g q

Then the restriction of p to X is indeed p

Let y b e the generic p ointofX ThenO is a discrete valuation ring and R O is a

Xy Xy

lo cal morphism of discrete valuation rings making O a at Rmo dule The O mo dule q

Xy Xy y

is at over R By some trivial algebra using the fact that atness over a discrete valuation ring

just means that a lo cal parameter acts faithfully this implies that q is at over O Since it

y Xy

is also nitely generated it is free Thus letting U X b e the maximal op en subset over which

q is lo cally free wehave y U Clearlywe also have X U By lemma there exists a

parab olic subgroup P G such that the corresp onding Lie algebra is isomorphic to p as a

U U

vector subbundle of g

Let V U X Then P P jV is a parab olic subgroup of GjV Since X is a curve

e e

wemay extend P to a parab olic subgroup P of G GjX Let p b e the Lie algebra of P

e e

Then generically p pjX and p coincide so p is the vector subbundle of g generated by

the coherent subsheaf p Hence wehave

deg P deg p

deg p

deg G

Clearlywehavedeg G deg P This implies i

On the other hand deg G deg G implies that deg p degp and that p is already a

i i

vector subbundle of g Byawellknown result this implies that q is lo cally free over all of X

and hence that p is a vector subbundle of g In other words U X SoP is dened over all of

X To prove ii it suces to provethatP is the canonical parab olic subgroup of G

Let S b e the henselization of S ToprovethatP is canonical in G it suces to prove

that P is canonical in G Soby Prop osition wemay replace S by S and assume that S

is henselian By Zariskis connectedness theorem Arcata IV Prop wehave a bijection

DynG DynG This clearly implies that the numerical invariants of P are p ositive

noting that P b eing canonical its numerical invariants are p ositive This implies that P is

canonical by lemma

The Fundamental Theorems

The Scheme of Parab olic Subgroups

Lemma Let X S beaatprojective morphism of noetherian schemes Let Y bea

smooth projective X scheme Then the functor Y denedby

Y T Y X T

for any S scheme T isrepresentable We denote the S scheme representing this functor also by

Y Itislocal ly of nite type over S

Proof See c

Let GXS b e a reductive group scheme over a curve and let X S b e the structure

morphism Let ParGbethescheme of parab olic subgroups of G It is smo oth and pro jective

over X Consider the S scheme ParG It satises the universal mapping prop erty

Par GT fparab olic subgroups of GjX T g

for any S scheme T Also there is a universal parab olic subgroup P of GjX ParG

The Lie algebra p of P is a vector bundle on X ParG This vector bundle p induces a

decomp osition

ParG Par G

where Par G is the op en and closed subscheme of Par G characterized by the fact that

rkpjX r and deg pjX d for all p oints s of Par G Note that it do es not matter

s s

whether we think of p oints as geometric p oints p oints of the underlying top ological space or

eldvalued p oints in general Similarelywe dene Par G to b e the scheme of parab olic

subgroups of G that have degree d everywhere over S

Prop osition For any integer d the scheme Par G is of nite type over S Inpar

ticular the same holds for Par G for any r

Proof The arguments on pages of proving Satz of can b e adapted to

prove this prop osition

The Theorems

Lemma Let G bea reductive group scheme over the curve X over the noetherian scheme

S Then there exists an M such that deg G M for al l s Speck S where k is a eld

Proof By the pro of of Lemma wehave

deg G dim H X g rkg g X

s s s s s

Now since rk g andg X are lo cally constantonS we mightaswell assume that they are

s s

constant say equal to r and g resp ectively By the semicontinuity theorem dim H X g

s s

is upp er semicontinuous on S Nowevery increasing sequence of op en subsets of S b ecomes

stationary This nishes the pro of

Theorem Let X S be a curve over the local ly noetherian scheme S Let G

bea reductive group scheme over X Then the geometric degree of instability of G is upper

semicontinuous on S

Proof Without loss of generalitylet S b e no etherian Let d be an integer Let

S fs S j for every extension k of swehavedeg G dg

d k

We need to show that S is op en in S LetM b e as in Lemma Then

Par G

is by Prop osition an S scheme of nite typ e Its image in S is precisely the complement

of S By Chevalleys theorem this implies that S is constructible in S Hence it suces to

d d

prove that S is stable under generization

Todothiswemay assume without loss of generalityby I I Exc that S is a discrete

valuation ring and that for the generic p oint S wehavedeg G deg G forany eld

i i

extension k of Then we are done by Prop osition i

Theorem Let X S be a curve over the local ly noetherian scheme S Let G bea

reductive group scheme over X ofconstant geometric degree of instability d Then there exists

a scheme S nite radicial and surjective over S andaparabolic subgroup P of G such that

for every k valuedpoint s of S where k is a eld the canonical parabolic subgroup of G is P

Proof Without loss of generality assume that S is no etherian First let us prove that the

dimension of the geometric canonical parab olic subgroup is a lo cally constant function on S

This means that for any r the set

S fs S j the dimension of the canonical parab olic subgroup

of G is equal to r g

s denotes the algebraic closure of s Consider the S scheme is op en and closed in S Here

S and contains S By induction wemay Par G Its image in S is contained on

r rd

thus assume that S is constructible So it remains to provethatS is stable under sp ecialization

r r

and generization For this wemay assume that S is a discrete valuation ring Wemayalso

assume that the geometric canonical parab olic subgroups are already dened over the two p oints

S Then the claim follows immediately from Prop osition ii So we will assume that

S S for some r

Consider the scheme Par G By Prop osition it is of nite typ e over S Hence

by Prop osition Par G is prop er over S By the characterization of the canoni

cal parab olic as the unique parab olic subgroup of degree d and dimension r we get that

Par G S is radicial and surjective in particular quasinite Now from Zariskis main

theorem it follows that a prop er quasinite morphism is nite So Par G is a nite radicial

surjective S scheme and taking S Par G will prove our theorem

Note Let S b e a lo cally no etherian scheme and S an S scheme Then the following are

equivalent

i The scheme S is nite radicial and surjectiveover S

ii The morphism S S is radicial surjective universally closed and of nite typ e

S is lo cally of nite typ e and a universal homeomorphism iii The morphism S

Remark The truth of Conjecture would imply that at least if S is integral no ethe

rian we could take S S in Theorem

The Typ e of Instability

Let X S b e a curveover the lo cally no etherian scheme S whichwe will assume to

b e connected for simplicityLetG b e a reductive group scheme over X and let DynGbe

the scheme of Dynkin diagrams of G This scheme Dyn G is nite and etale over X Let

v v b e the connected comp onents of Dyn G and let V b e the free ab elian group on the

generators v v Whenever T S is a connected S scheme we denote the corresp onding

free ab elian group on the connected comp onents of Dyn G by V The natural morphism

T T

Dyn G Dyn G induces a homomorphism V V

T T

Now let P b e a parab olic subgroup of GLettP b e the typ e of P See Remark It

is a closed and op en subscheme of Dyn Gsoby renumb ering if necessarywemay assume that

tP v q q v LetVP V b e the subgroup generated by v v Nowwe can do the

s s

construction of page in this context We get for every i s avector bundle W P v

over X Thenthenumerical invariantof P with resp ect to v denoted nP v is the degree of

i i

W P v for some p oint s S By the connectedness of S this denition is indep endent of the

i s

choice of sWe dene

nP nP v v

i i

which is an elementofVP V

More generallyif P is a parab olic subgroup of G where T is a connected S scheme weget

nP V SonP has a natural image in V

Now let s S b e a settheoretic p ointofS Cho ose an algebraically closed eld k over s

Let P b e the canonical parab olic subgroup of G Asabove wegetnP V The image of

k k k k

nP inV do es obviously not dep end on the choice of k over s Thus wehave constructed

a function n S V

Denition We will call this function n the geometric type of instability of the reductive

group scheme G over the curve X over the connected lo cally no etherian scheme S

Wewillnow summarize the results of our considerations in the following corollary

Corollary Let X S be a curve over the connectedlocal ly noetherian scheme S

Let G beareductive group scheme over X Let V bethefreeabelian group on the connected

components of the scheme of Dynkin diagrams of G Then the geometric degree of instability of

dg is a G is upper semicontinuous on S So for any d Zthe set S fs X j deg G

local ly closed subset of S Moreover for any d Zthe geometric type of instability n S V

is continuous when restrictedtoS

Proof The rst part of the claim is just Theorem For the second part wemay replace

S by some induced subscheme structure on some connected comp onentof S and thus assume

that the geometric degree of instability is constantonS Invoking Theorem wemayeven

assume that there exists a parab olic subgroup P of Gsuch that P is the canonical parab olic

subgroup of G for every geometric p oint s of S Then n is clearly constant and equal to nP

Applications to Gbundles

Let S b e a connected lo cally no etherian scheme X S acurveover S and G a reductive

group scheme over X We will study the S stack H XS G of families of Gtorsors For the

denition see page By Corollary H XS G is a smo oth algebraic S stack if S is the

sp ectrum of a eld Whenever we refer to H XS Gasanalgebraic S stack we will tacitly

assume that S is the sp ectrum of a eld

Let V b e the free ab elian group on the set of connected comp onents of the Dynkin diagram

DynGofG

Stabilityof Gbundles

Let E be a Gtorsor Then G Aut E E G is again a reductive group scheme on

G GAd

X This follows from descent theory b ecause G is a sheaf on X that is lo cally with resp ect to

the fppf top ology an ane scheme This argumentshows that G is a scheme over X Then

it is clearly a reductive group scheme b ecause this prop erty is lo cal with resp ect to the fppf

top ology

Note that there is a canonical isomorphism of X schemes DynG Dyn G This comes

E E

ab out as the image of the canonical section of IsomextG GinIsomDynG Dyn G See

Exp XXIV Hence we can canonically identify the free ab elian group on the connected

comp onents of Dyn G with V

Denition The Gtorsor E is called geometrical ly semistable if for every geometric

E E

p oint s of S the reductive group scheme G G on X is semistable See Denition

s s s

The geometric degree of instability of E denoted deg E is the function on S asso ciating

E E

of G where to every set theoretic p oint s S the degree of instabilitydeg G

s s

s is the algebraic closure of the residue eld sofs By Theorem deg E isupper

semicontinuous on S

The geometric type of instability of E denoted nE is simply the geometric typ e of insta

bilityof G see Denition By the ab ove remarks we can consider nE as a function

nE S V By Corollary nE iscontinuous over subsets of S over whichdegE is

constant

The Natural Stratication of the StackofGbundles

Denition Let m beaninteger Then by H XS G we will denote the S stack

of Gtorsors of geometric degree of instability less than or equal to m This stack H XS G

is clearly an op en substackofH XS G in particular also algebraic and smo oth over S We

denote the complementof H XS G in H XS G by H XS G and endowit

m m m

with the reduced induced substack structure see Lemme So H XS G is a closed

substackofH XS G and hence algebraic and lo cally of nite typ e

For an element v V such that v weletH XS G b e the substackofH XS G

mv m

of Gtorsors of geometric typ e of instability v More preciselyletT be any S scheme whichwe

may without loss of generality assume to b e connected Then a morphism T H XS G

denes a G torsor E via the induced morphism T H XS G Noting that the degree of

T T

instabilityof E is necessarely constant equal to mwe see that the geometric typ e of instability

nE T V is also constant WedeneT H XS G to factor through H XS G

T m mv

if nE is equal to v Clearly H XS G is an op en and closed substackofH XS G

T mv m

For example H XS G is the S stack of geometrically semistable Gtorsors The only el

ement v V suchthatH XS G is nonemptyis v Wehave H XS G

H XS G

The family H XS G denes a stratication of H XS G We call it the strati

m m

cation with resp ect to the degree of instability The family H XS G is an increasing

m m

family of op en substacks covering the stack H XS G ie an exhaustion of H XS G For

every m we get a decomp osition

H XS G H XS G

m mv

Remark Let E be a Gtorsor Since for any further Gtorsor F the group schemes

Aut F and Aut E Hom E F are canonically isomorphic the Change of Origin morphism

G G

H XS G H XS G see Prop osition resp ects all the ab ove substacks

The Degree of Gbundles

Let S b e a connected lo cally no etherian scheme and X S acurveover S Let G b e briey

more generally a smo oth ane group scheme of nite typ e over X Let G HomG G

b e the character group of Gwhichwe consider simply as an ab elian group If A is an ab elian

group we denote by A HomA Z its dual group

Denition Let E be a Gtorsor Then the degreeofE denoted deg E is the homo

morphism deg E G Zdened bydegE degE O Since E O is a

G X G X

line bundle on X and S is connected the degree of E O is indep endentofs S So

G X s

deg E iswelldened The function deg E is a homomorphism since multiplying gluing data

for line bundles corresp onds to adding degrees

Denition Let d G Then we dene H XS GtobetheS stackofGtorsors of

degree d Here we view the degree of a G torsor E over a connected S scheme T as an element

T T

of G via the natural homomorphism G G The stack H XS G is clearly an

X X X

op en and closed substackofH XS G Wehave a decomp osition

XS G H H XS G

d G

Denition Now assume G to b e reductive Then combining the degree the degree of

instability and the typ e of instabilitywe get lo cally closed substacks H XS G of H XS G

d d

and closed and op en substacks H XS G of H XS G

mv m

d d

Remark Returning to the case that G is an arbitrary smo oth and ane group scheme

over X note that if E is a Gtorsor then there is a canonical isomorphism G G

X X

So we can compare degrees of Gtorsors and Gtorsors Then the Change of Origin mor

E E

XS G for every XS G with H phism H XS G H XS Gidenties H

ddeg E d

d G Checking this is a straightforward calculation So for a given d G any

X X

Gtorsor E with deg E d induces an isomorphism

XS G XS G H H

Reduction to a Parab olic Subgroup

Let G b e a reductive group scheme over the curve X over the connected lo cally no etherian

scheme S LetP G b e a parab olic subgroup of GLettP bethetyp e of P see Remark

This is a closed and op en subscheme of DynG Hence the comp onents v v of

tP generate a subgroup of V This subgroup will b e denoted VP As on page we get for

every i s avector bundle W P v over X By construction P acts on W P v linearly

i i

via the action induced from inner automorphisms see the pro of of Prop osition Taking

the determinant of this representation of P gives a character of P This pro cess denes a

homomorphism VP P Identifying VP with its dual via the basis v v weget

a homomorphism

v P VP

i i

Denote the image of this homomorphism v by VP

Naturallywehave a homomorphism G P and hence a homomorphism

X X

d P G

X X

j G

Taking the determinant of the action of P on its unip otent radical R P wegetacharacter

of P This makes sense b ecause R P has a ltration that is invariant under the action of P

and whose factors are vector bundles

Lemma There exist positive rational numbers y y such that

i i

Proof Wemay assume that G admits a split maximal torus T Lo oking at the corresp onding

ro ot system our lemma follows from Prop osition

Evaluating at denes a homomorphism m P ZFrom Lemma it easily follows

that m factors through VP Sowemay when the need arises consider m as a homomorphism

m VP ZWehave

s s

X X

n y n v m

i i i i

i i

Note If E is a P torsor of degree then the asso ciated Gtorsor E G has degree d

If v n v then n nE P v where we consider E P as parab olic

i i i PAd i P Ad

subgroup of E G Also deg E P m

P Ad P Ad

Let R P b e the unip otent radical of P and H PR P Wehave a natural morphism

u u

H XS P H XS H

Let H XS P b e the preimage of H XS H under this morphism In other words we call

a P torsor semistable if its asso ciated H torsor is

Denition We call an element P positive if for all i sWe

denote the set of p ositive elements of P by P

X X

Prop osition The natural homomorphism

d v P G VP

X X

is an injection with nite cokernel

Proof Let RGandRP b e the radicals of G and P resp ectivelyWehave G RG

X X

and P RP So since RG P it makes sense to consider the sequence of ab elian

groups

VP P RG

X X

Note that we do not claim that it is exact

By construction the characters vanish on the center of G After all they are

induced by inner automorphisms of GSinceRGiscontained in the center of G we see that

Let U X be an etale morphism with U connected and T a split maximal torus of G

U U

contained in P Let G T b e the corresp onding ro ot system Let P also denote

U U U

the facet of corresp onding to P via Lemma For a corner of P let P be

the corresp onding elementary set of ro ots see Denition Let for

vert P There exists a natural map f vert P fv v g Restricting the characters

U s

to T weget

s U

f v

for every i s

b e a relation among the characters Wehave Now let

i i s

X X

i i

f v

vert P

By abuse of notation wehave written n instead of n here Now since is linearly

v i vert P

indep endent wegetn for all vert P and hence This proves the injectivity

of the homomorphism in the sequence

Nowlet P and assume that Considering again U and T as ab ove this

implies that restricting to T weget V where V span T R is the vector

U U

space in which our ro ot system lives In fact by considering the action of the Weyl group on

it is easily seen that span We can write

vert P

since is a basis for span Replacing bysomeinteger multiple wemay

vert P

assume that all vert P areintegers

Claim For all vert P wehave f f

U U

To prove the claim cho ose an epinglage E of G compatible with T Then DynG

U U U U

is constant equal to the Dynkin diagram of E G The typ e tP of P is a subscheme of

U U

DynG whose connected comp onents are in canonical bijection with the corners of P So

wemay consider asacontinuous function on tP DynG Cho osing a

vert P U U U

dierent maximal torus and a dierentepinglage do es not change this function Hence the

for U connected and etale over X suchthatG is epinglable over U glue together and dene

acontinuous function on tP Dyn G This function is then necessarily constantonevery

connected comp onent v fori s This proves the claim

Now consider the element

v v

v f

of VP Here we mean by anumber for some f v By the claim it do es

f v

not matter which wecho ose This element v maps to under Thus wehaveproven that

vP has nite index in ker

It is wellknown that G has nite index in RG this implies that im has nite index

X X

So is a complex of nitely generated free Zmo dules At the rst place it is exact and at

the other two places it has nite cohomologyTaking the dual of gives a complex with the

same prop erties

RG P VP

X X

Taking the dual of the injective map with nite cokernel G RG we get another such

X X

ie RG is a subgroup of nite index in G

X X

Together these facts imply that P injects into the direct pro duct We also see that

P VP and P G have nite cokernel

X X X

Prop osition Let P If E is a geometrical ly semistable P torsor of degree

then the associated Gtorsor E G has degree d and geometric type of instability v

Moreover its geometric degree of instability is equal to m Hence we get a natural morphism

of S stacks

H XS P H XS G

m v

Under the assumption that the connectedcomponents of DynG aregeometrical ly connected

this morphism is nite and a universal homeomorphism

Proof Directly from the denitions and Note we get the existence of a morphism

H XS P H XS G

This morphism actually factors through H XS G because H XS P is reduced

b eing smo oth Thus wehave the existence of our morphism

Toprove the second claim wecho ose a connected S scheme T and a morphism

E T H XS G

m v

whichweinterpret as a G torsor By Prop osition wehave a cartesian diagram

EP T

T T

H XS P H XS G

Let EP b e the op en and closed subscheme of EP dened by the following

T T T T

cartesian diagram

EP EP

T T T T

H XS P H XS P

v v

Then wehave the diagram

EP EP

T T T T

H XS P H XS P

This follows from Prop osition and the fact that E has degree d We also use the fact

that a reduction of structure group of E to P of degree with v v is necessarily semi

t t

stable for any geometric p oint t of T Herewetake advantage of the fact that the comp onents

of DynG corresp ond bijectively to the comp onents of DynG

Nowby Exp XXVI Lemme wehave a canonical isomorphism

EP Par G

T T

Applying we get

EP Par G

T T

Letting Par G b e the image of EP under this canonical isomorphism

T T

tP v v

we get a cartesian diagram

Par G T

H XS P H XS G

m v

Weknow that the geometric degree of instabilityof E is equal to m m The geometric

dimension of the canonical parab olic asso ciated to E is also constant say equal to r Then from

the pro of of Theorem we knowthat Par G is nite radicial and surjectiveover

T T rm

T Now it is easy to see that

E E

Par G Par G

T T rm T

This shows that our morphism is representable and nite radicial and surjective In view of Note

this nishes the pro of

Remark If Conjecture is true the morphism of Prop osition can b e shown

to b e an isomorphism

The Trace Formula for the StackofGBundles

Intro duction

In this section we search for op en substacks of H Xk G that are of nite typ e Our main

result is that H Xk G ford G and m is of nite typ e see Theorem

In section weprove that H Xk G is the quotient of a DeligneMumford stackbyan

ane algebraic group Wedothisby studying levelD structures where D is an eective divisor

on X For a Gtorsor E alevelD structure s is just a section of E over D s E D The stack

of Gtorsors with levelD structure is denoted by H Xk G D and the natural morphism

H Xk G D H Xk G

given by forgetting the levelD structure is a principal b er bundle with structure group G

where D Sp ec k is the structure morphism See Corollary If wecho ose D large

Xk G D is a DeligneMumford stack So wemay apply the results of Section enough H

to H Xk G and get the trace formula Corollary

Finally in Section we prove the trace formula for H XF G where wehave to makean

assumption on the group GWe call this assumption x see Denition This assumption

amounts to the truth of Conjecture for G and all its twists Gwhere E is a Gtorsor

Since H XF G is in general not of nite typ e b oth sides of the trace formula are innite

sums

The Case of Vector Groups

Prop osition Let X S be a curve of genus g over the local ly noetherian scheme

S Let V beavector group over X ofconstant degree d and rank r Assume that R V is a

vector group and commutes with base change The same is then automatical ly true for V

Then the natural morphism

H XS V R V

makes H XS V a gerbeoverR V If T is an ane R V scheme then the T gerbe

H XS V is trivial isomorphic to B V Inparticular H XS V is an algebraic S

T T

stack of nite typeandsmoothofrelative dimension r g d over S

Clear

Proof First we will dene the morphism For any S scheme T wehave H XS V T

V T We also have natural maps X V H X V X V andR V T R

T T T T T T T T

X V R V T The map T istaken to b e the comp osition and H

T T T T

Claim The morphism makes H XS V a gerb e over R V

To prove the claim let rst T be an S scheme and let E F ob X V suchthatE

T T

F in R V T WehavetoprovethatE and F are lo cally isomorphic with resp ect to

T T

the at top ology on T Wemay assume that T is ane Then R V T H X V and

T T T T

E F implies that E and F are isomorphic

Secondly let T be an S scheme and R V T Wehavetoprovethat may lo cally

T T

b e lifted So again wemay assume hat T is ane and hence that H X V Taking a

T T

V torsor E representing the cohomology class we get an ob ject E ob X V mapping

T T T

to under This proves the claim

NowletT b e an ane scheme with a morphism t T R V Considering T as an S

scheme wehave R V T H X V so that t induces a cohomology class H X V

T T T T

Cho osing a V torsor E representing we get an element E X V whichgives rise to

T T T

a morphism t T H XS V Clearlywehave ttSoH XS V is trivial over

the R V scheme T Nowwehave Aut E V so that the T gerb e H XS V is

T T T

isomorphic to B V

The last part of the prop osition is lo cal in S Sowemay assume that S is ane Then R V

is an ane scheme Hence by what we just proved wehave H XS V B V R V

Another way of writing this is

H XS V R VR V

where R V acts on R V trivially The rest of the prop osition is then obvious using

RiemannRo ch to compute the dimension

Corollary Let X be a curve over the eld k and V avector group over X Then we have

H Xk V H X V H X V

where H X V for i is consideredasavector group over k and H X V acts trivial ly

on H X V In particular H Xk V is a smooth algebraic k stack of nite type and of

dimension rk V g deg V

Proof Since the base scheme is a eld the hyp otheses of Prop osition are satised Note

in particular formula

Corollary Let X be a curve over the local ly noetheraian scheme S and V avector group

on X ThenH XS V is a smooth algebraic S stack of nite type and of relative dimension

rk V g deg V

Proof Let E b e a lo cally free coherent O mo dule suchthatV VE Then we consider the

homomorphism of group schemes over X

V GLEO

v v

where is the identityonEO and v acts on EO via v e v In this way V

X X

b ecomes a closed subgroup of GLEO and by Prop osition wehavethatH XS V

is an algebraic S stack lo cally of nite presentation The smo othness of H XS V follows

from Prop osition The calculation of the dimension can now b e done b erwise ie wemay

assume that X Speck is a eld and apply Corollary

Applications to Parab olic Subgroups

Lemma Let X be a curve over the eld k and

V G H

a short exact sequenceofgroup schemes on X where V is a vector group Then the natural mor

phism H Xk G H Xk H is a smooth epimorphism of nite typeandrelative dimension

rk V g deg V

Proof Since any pullbackandanytwist of a vector group is again a vector group and R ap

plied to a vector group is always zero Prop osition implies that H Xk G H Xk H

is an epimorphism Then the other prop erties are lo cal on the base and the second half of

Prop osition implies the result making use of Corollary

Corollary Let X be a curve over the eld k and V agroup scheme over X having a

ltration V V V such that al l factors V V for i n arevector

n i i

groups Then H Xk V is a smooth algebraic k stack of nite type and dimension dim V g

deg V

Proof By induction on nLetin b e an index suchthatV Then V is a vector group

i i

We consider the short exact sequence of groups on X

V V VV

i i

By Lemma H Xk V H Xk V V is smo oth of nite typ e and relative dimension

rk V g deg V By induction hyp othesis wehave H Xk V V smo oth of nite typ e and

i i i

relative dimension dim VV g deg VV The corollary follows

X i i

Corollary Let X be a curve over the eld k and

V G H

a short exact sequenceofgroup schemes on X where V agroup scheme over X having a ltration

V V V such that al l factors V V for i n arevector groups

n i i

Then the natural morphism H Xk G H Xk H is a smooth epimorphism of nite type

and relative dimension dim V g deg V where E is the universal Gtorsor

Proof Use Prop ositions and Note also that the relative dimension of this morphism

is not necessarily constant

Prop osition Let G beareductive group scheme over the curve X over the eld k Let

P G beaparabolic subgroup and let H PR P Then the natural morphism of k stacks

H Xk P H Xk H is a smooth epimorphism of nite type and of relative dimension

dim R P g deg P where E is the universal Gtorsor

X u

Proof By Exp XXVI Prop R P has a ltration such that the quotients are vector

groups So wemay apply Corollary

Calculating the Dimension of the StackofGBundles

Lemma Let X beasmooth algebraic k stack of dimension n Then TX X is smooth of

relative dimension n

Proof

Corollary Let G beareductive algebraic group scheme over the curve X over the eld k

Then H Xk G is a smooth algebraic k stack of dimension dim Gg

Proof Note that by Corollary all wehavetocheck is that the formula for the dimension is

correct Consider the short exact sequence of group schemes on X

g TG G

Then Lemma implies that H Xk TG H Xk G is smo oth of relative dimension

rk gg Note that deg g by Note But directly from the denitions wehave

H Xk TG TH Xk G So the corollary follows from Lemma

Algebraic Stacks of Finite Typ e

Quasi Compact Stacks

Lemma Let f X Y beacontinuous map of topological spaces X Y IfX is quasi

compact and f is surjective then Y is quasicompact

Proof Clear

Lemma Let X be an algebraic S stack where S is an ane scheme Then X is quasi

compact if and only if jXj the set of points of X with the Zariski topology is a quasi compact

topological space

Proof If X is quasi compact there exists a presentation p X X suchthatX is quasi

compact over S The morphism p induces a continuous and surjectivemapp jX j jXj of the

corresp onding Zariski top ologies Since S is ane jX j is quasi compact and by lemma jXj

is quasi compact

Conversely assume that jXj is quasi compact Cho ose a presentation p X X of X Let

V b e a family parametrizing all ane op en subsets of X For every i I let U pV

i iI i i

which is an op en substackofX and let U b e the op en subset of jXj corresp onding to U Then

i i

U is a covering so that there exist n I such that U U cover jXjNow let

i iI n

X V and let p pjX Then p X X is a presentation of X and X is quasi

compact as a nite union of ane schemes

Corollary Let f X Y be a morphism of S stacks where S is quasi compact If X is

quasi compact and f is surjective then Y is quasi compact

Proof Since S is quasi compact the question of Y b eing quasi compact is lo cal in S sowe

may assume that S is ane Then we consider the induced map f jXj jYj of the asso ciated

sets of p oints with the resp ective Zariski top ologies The corollary then follows immediatly from

Lemmas and

Passing to Galois Covers

Let X b e a curveover the eld k Let Y X b e a nite Galois extension of X This means

that Y is another curveover k and is a nite morphism of curves making K Y a Galois

extension of K X Let GalYX

Let G b e a reductive group scheme over X with character group G Then Gisa

X X

mo dule and G G Wehave the trace map tr G G dened by

X X X X

Taking the dual wegettr G G With these notations tr

X X

wehave the following prop osition

Prop osition Let G beareductive group scheme over the curve X over the eld k Let

Y X beaGaloiscover of X Then we have for every d G anatural morphism of

algebraic k stacks

Yk G H Xk G H

tr d

induced by pul ling back via This morphism is ane of nite type

Proof Let n Let d G Ifd isainvariant extension of nd G to G

X X X

then for any Gwehave

nd d

and hence d tr d In particular d is uniquely determined by dIfE is a Gtorsor of degree

deg E wehave for G

deg E deg E O

G Y

deg E O

G X

n deg E O

G X

n deg E

so that deg E extends n deg E We also havefor and G

deg E deg E

deg E

so that deg E is invariant Hence deg E tr deg E bytheabove remarks This proves

that we get a natural morphism

H Xk G H Yk G

tr d

It is ane by Theorem

Borel Subgroups

Prop osition Let G bearational ly trivial reductive group scheme over the curve X over

the eld k Assume that X has a k valuedpoint Let B G be a Borel subgroup and let

B Then H Xk B is an algebraic k stack of nite type

Proof The quotientgroup T BR B is a torus over X The torus T is split since G is

rationally trivial By Prop osition weonlyneedtoprovethatH Xk T isofnitetyp e

where d is the image of under the natural map B T Write T as T G Then

X X

T Z Let d d d Then wehave

H Xk G H Xk G H Xk G

m m m

d d d

Thus we are reduced to the case T G By Remark a line bundle L of degree d induces

an isomorphism H H Xk G Xk G since any inner form of G is equal fo G Now

m m m m

cho osing a k valued p oint P of X wehave H Xk G P Pic X Jac X Hence wehave

H Xk G Jac XG where G acts trivially on Jac X

m m m

The Theorem

Theorem Let G beareductive group scheme over the curve X over the eld k Let m

beaninteger and d G Za homomorphism Then H Xk G is a smooth algebraic

k stack of nite type

Proof By Corollary all wehavetoprove is that H Xk G is of nite typ e If

Y X is a Galois covering of X thenby Prop osition and Corollary wehavea

nite typ e morphism

H Xk G H Yk G

m m

tr d

So for the purp ose of proving our theorem wemay pass to a Galois cover of X an assume that

G is rationally trivial and that X admits k valued p oints By Corollary we need only nd

an algebraic k stack of nite typ e Y together with a surjective morphism

Y H Xk G

Cho ose a Borel subgroup B GLet be the characters of B given by the action of

B on its asso ciated elementary line bundles Consider the natural homomorphisms d B

G and m B Zsee pages and following Consider nally the morphism

X X

Xk G Xk B H H

where the disjoint sum is taken over all B satisfying the following three conditions

i d d

ii m m

iii g for i s

By Note this morphism indeed exists Also if E is a B torsor of degree then

are the numerical invariants of the Borel subgroup E B of E G

B Ad B Ad

Claim The number of B satisfying the ab ove three conditions is nite

By Prop osition all we need to do is checkthat

fn n Z j n g for i s and n y mg

s i i i

is nite Here y y Q are given as in Lemma But this is clear

Hence by Prop osition the stack on the left hand side of is of nite typ e So it only

remains to prove the surjectivity of the morphism in Letting E SpecK H Xk G

representapointofH Xk G wemay replace k by K and thus assume that E is a G

torsor over X of degree d such that deg G m By Satz there exists a Borel

subgroup B G such that the numerical invariants n B n B satisfy n B g

s i

for i s This Borel subgroup B will dene a reduction of structure group E of E to

B ie E is a B torsor suchthatE G E and E B B Let degE b e the

B BAd

degree of E Then by Note wehave d degE d m degB deg G m

and n B g for i sThus E denes a p oint of the stack on the left hand

i i

side of mapping to E This proves the surjectivity of the morphism in and thus the

theorem

Level Structures

Let X be a scheme and D X ascheme over X LetGX b e a group scheme

Denition We dene the groupoid of Gtorsors with levelD structure denoted X G D

to b e the category whose ob jects are pairs E s where E is a right Gtorsor over X and

s is a section of E E The set of morphisms from E stoE s for E s E s

ob X G D is dened to b e the set of all isomorphisms of Gtorsors E E suchthat

Note Clearlywehave a morphism of group oids f X G D X G given by

forgetting the levelD structure We also have a natural action of D Gontherightof

X G D making the following diagram a cartesian and co cartesian diagram of group oids

X G D D G X G D

p f

X G D X G

Proof Straightforward checking

Note If E is a Gtorsor then wehave a cartesian diagram of group oids

D E fg

X G D X G

where is given by sE sforevery s D E

Proof Clear

Let X be a scheme over the base scheme S Let D X be an X scheme and D S the

structure morphism Let G be a group scheme over X

Denition We dene the S stack of Gtorsors with levelD structure denoted H XS G D

by setting

H XS G D U X G D

U U U

for every S scheme U

Note Wehave a natural morphism of S stacks

f H XS G D H XS G

and a natural action of the sheaf of groups G on H XS G D yielding a cartesian

diagram of S stacks

H XS G D G H XS G D

p f

H XS G D H XS G

The question of whether this diagram is co cartesian ie whether f is an epimorphism is a

little more subtle

Proof Follows immediatly form Note

Lemma Let Y X be a nite at morphism of schemes Let U Y be a faithful ly

smooth etale Y scheme of nite presentation Then U is a faithful ly smooth etale X

scheme local ly of nite presentation

Proof This follows easily using formal smo othness and the formal prop erties of the functor

Note Let U be an S scheme and E a G torsor Then wehave a cartesian diagram of

S stacks

E U

H XS G D H XS G

where is given by sE s for any s E D So if G is of nite presentation over X

U U

and D is prop er and at over S then f is representable and lo cally of nite presentation If in

addition G is smo oth over X and D is nite over S then f is in addition smo oth and surjective

So in this case f is an epimorphism and the diagram of Note is co cartesian

Proof The fact that the diagram is cartesian follows immediatly from Note If G is of

nite presentation over X then E is an algebraic space of nite presentation over X Hence

the same is true for E over D Now D U is prop er and at so by E is

U U U U

U U

an algebraic space lo cally of nite presentation over U This proves that f is representable

lo cally of nite presentation NowifG is smo oth over X then E is smo oth over D Hence

the claim follows form Lemma

Corollary If G is a smooth group scheme over X and D is nite and at over S then

G is a smooth group scheme over S and

f H XS G D H XS G

is a principal homogeneous Gbund le

Proof Immediate from Notes and

Applications to the StackofGBundles

Lemma Let k be eld and X a curve over k Let H beak stack of nite type Let

g bealocal ly freecoherent sheaf on X H Then there exists a nite set of closedpoints

D fx x g X such that gD where X H H is the structure morphism

Proof We will use no etherian induction If H there is nothing to prove Otherwise cho ose a

p oint of H and let SpecK H b e a represetativeof Letg b e the pullbackof g to X

K K

Then wecho ose nitely many closed p oints D fx x g of X suchthatthevector bundle

gD on the curve X over the eld K has no glob el sections By cohomology and base

K K

change Chap I I I Theorem there exists an op en substack U Hcontaining such

that gD j U Let H H U with the reduced substack structure Then H is a prop er

closed substackofHSoby the induction hyp othesis there exists D fy y gX such

that g D Let D D D Then again by cohomology and base change wehave

gD as required

We will now calculate the Zariski tangent space to the scheme of automorphisms of a Gtorsor

with levelD structure E s We will need the following lemma

Lemma Let S bealocal ly noetherian scheme and X S aprojective at S scheme

Let G beananesmooth group scheme over X Let D X be a closed subscheme of X

that is nite and at over S and Let E beaGtorsor and s D E asection of E

over D Then we have a short exact sequenceofgroup schemes of nite type over S

E s G G Aut

E E

Proof First note that G is an ane X scheme so that G is a group scheme of nite over

S by Prop osition For the same reason G is a nite typ e group scheme over S We

will now construct the homomorphism

G G

Let g GS Wemay consider g as an automorphism g E E of the Gtorsor E Then

let g GS GD b e the unique elementsuchthats g g s Since s gh

s s s

ghsg s h g s h s g hwe see that wehave dened a homomorphism of

s s s s

group schemes Directly from the denition wehave

g g ss

g AutE s

E s is a closed subgroup scheme This proves the exactness of the ab ove sequence Hence Aut

of G in particular of nite typ e over S

Under the same hyp otheses as in the Lemma let us now assume that S Speck where k

is a eld Let AutE s b e an automorphism of E s The element itself induces an

isomorphism of the Zariski tangent space of Aut E sat with the Zariski tangent space of

AutE s at Denote these Zariski tangent spaces by T AutE sandT AutE s

resp ectivelyNow directly from wehave an exact sequence of k vector spaces

E s H X g H D g T Aut

So wehave

T AutE sH X gD

Lemma Let G bea reductive group scheme over the curve X over the eld k Let

p X H Xk G H Xk G bethesecond projection Let E be the universal Gtorsor

on X H Xk G Let m beaninteger and d G Z a homomorphism Let

D fx x gX be closedpoints such that p gD j H Xk G Such points

n m

exist by Lemma Then H Xk G D is a DeligneMumford stack over k

Proof Let

E sSpecK H Xk G D

represent a p ointofH Xk G D SoE is a G torsor on X of degree d and geometric

m K K

degree of istability less than or equal to m and s is a section of E over D We need to

E s where X Sp ec k is the structure morphism is Aut show that the K scheme

unramied over Sp ec K Without loss of generalitywemay assume that K k Bywe

have T AutE sH X gD which is equal to zero by assumption

Prop osition Let G bea reductive group scheme over the curve X over the eld k Let

m and d G be given Then there exists a DeligneMumford stack X over k and a

nonsingular ane algebraic group over k acting on X such that

H Xk G X

Proof Cho ose D as in Lemma and X H Xk G D Take G where

D X is the inclusion and D Sp ec k the structure morphism Then is an ane

algebraic k group by Prop osition and nonsingular by Lemma By Corollary we

have

X H Xk G

Done

Corollary Let G beareductive group scheme over the curve X over the eld k Let

Xk G is a smooth algebraic k stack of nite type m and d G be given Then H

and dimension dim Gg and if k F is nite the Lefschetz trace formula holds for this

X q

F stack

dim Gg

XF G C q tr jH H

q m q

AutE

E XG

where the sum is taken over al l isomorphism classes of Gtorsors E such that deg E d and

deg E m

Proof Combine Theorem with the dimension calculation Corollary Then apply

Theorem which is p ossible b ecause of Prop osition Note also that F is p erfect and

Corollary

Passing to the Limit

Exhausting Top oi

Denition Let E b e a top os and X an ob ject of E Acovering U U of X is called

i iN

an exhaustion of X ifU is a sub ob ject of X for all i N and U U for all i N

i i i

Lemma Let E be a topos X an object of E and U U an exhaustion of X Let M

i iN

bea presheaf of abelian groups on E such that M U is nite for al l i N Then H UM

for al l p

Proof N is an ordered set so wemay consider N as a categoryWe endow N with the chaotic

top ology Then N the category of presheaves of sets on N is the top os of sheaves on the site

b b

N The category ab N of ab elian sheaves on the top os N is the category of pro jective systems of

ab elian groups indexed by NFor an ab elian sheaf F on N wehave the CartanLeray sp ectral

sequence corresp onding to the covering N n of the nal ob ject of N

p q pq

H N H F H NF

For any n N wehave H n F forq b ecause the functor F F n is exact N having

the chaotic top ologySosinceN is closed under direct pro ducts the Cech complex C N F

is zero if q

q p q

H minn n F C N H F

n n N

p p

So from the sp ectral sequence wegetH NF H NF Now the global section

b b

functor N is nothing but the inverse limit functor So wehave H NF lim F and

hence lim F H NF Since M M U is a pro jective system of ab elian groups we

i iN

can apply this result and get

f f

lim M H N M

p p

Obviously H NM H UM by lo oking at the Cech complexes So wehave

H UM lim M U

Since our indexing category is N lim always vanishes for p Since M U is nite for all

f f

M i N M satises the MittagLeer condition and hence lim

Prop osition Let E bea topos that can be exhaustedbynoetherian open subtopoi that

satisfy the niteness theorem with respect to N Then for any constructible abelian sheaf F on

E with NF we have

p p

H E F limH U F

where U is an exhaustion of E such that for al l i N E is a noetherian topos satisfying

i iN

the niteness theorem with respect to N

Proof Let U U b e an exhaustion of E such that for all i N E is a no etherian

i iN

top os satisfying the niteness theorem with resp ect to N Let F b e a constructible ab elian sheaf

on E with NF Then for any i N the sheaf F jU is no etherian and so H U F is nite

i i

q p q

So we can apply Lemma to the presheaf H F on E obtaining H U H F for

p So the CartanLeray sp ectral sequence of the covering U gives us

q q

H E F H U H F

But this is what wewanted to prove

Corollary Let k beaseparably closed eld and char k IfX is an algebraic k stack

that can be exhaustedbyopen substacks X of nite type then

i iN

p p

H X F lim H X F

sm i

for any constructible abelian sheaf F on X with F for some n

Proof By Theorem the top os X satises the niteness theorem with resp ect to for

every n So the Corollary follows form Prop osition

The Condition x

If X is a curveover an algebracically closed eld k then the following lemma shows that the

index m in the notation H X G is redundant so wemay write H X G instead

mv v

d d

Lemma Let G bea reductive group scheme over the curve X over the algebraical ly closed

eld k Let V be the freeabelian group on the connectedcomponents of Dyn G Then there

exists a function m V Z such that if E is a Gtorsor of type of instability v V then

mv deg GIfP is a parabolic subgroup of G then we have a commutative diagram

P V

m m

where the horizontal map v and the diagonal map m are dened asonpage

Proof If there exists no Gtorsor of typ e of instability vthenwemay assign an arbitrary

value to mv So let E and F be Gtorsors of typ e of instability vWeneedtoshow that

E F

deg G deg G Without loss of generalitywemay assume that G contains a parab olic

i i

subgroup P of typ e where

n v Let E and F b e the canonical reductions of E and F to P Then and v

i i

vdeg E v vdeg F so our lemma follows from the fact that m factors through VP

Lemma Let G bea reductive group scheme over the curve X over the algebraical ly closed

eld k Then there exists a function r V Zsuch that if E is a Gtorsor of type of instability

v V then r vdim R P where P is the canonical parabolic subgroup of G

X u

Proof This follows from the fact that parab olic subgroups of the same typ e are forms of each

other

Denition Let G b e a reductive group scheme over the curve X over the eld k We will

say that G satises condition x if the following is satised Whenever K is a eld extension of

k and E is a G torsor wehave

H X gp

where p denotes the Lie algebra of the canonical parab olic subgroup of G

Note If Conjecture is true then every reductive group scheme over a curve satises

condition x

Prop osition Let G beareductive group scheme over the curve X over the algebraical ly

closed eld k Assume that G satisties Condition x Let P bea parabolic subgroup of G and

let P Let H PR P Then we have a natural morphism of algebraic k stacks

XH H XG H

The morphism is a smooth epimorphism of ntie type and relative dimension dim R P g

P X u

m Itinduces isomorphisms

i i

XH Q XG Q H H H H

for every i

Proof By Prop osition wehave a natural morphism

H X P H X G

Using condition x this can b e shown to b e an isomorphism Now comp osing the inverse of

with the morphism of Prop osition weget The condition x implies that we can

apply Prop osition to the factors of the ltration of R P This implies that induces

u P

isomorphismson adic cohomology

Lemma Let G bea reductive group scheme over the curve X of genus g over the alge

braical ly closed eld k Assume that G satises Condition x Let d G be xed Then

for every v V the natural closed immersion

H X G H X G

d d

is a smooth pair of algebraic k stacks of codimension cvr vg mvFor the denition

of r v and mv seeLemmas and

Proof If H X G there is nothing to prove Otherwise there exists mayb e only after

achange of origin a parab olic subgroup P of G and an element P suchthatd d

and v v Then as we already noted in the pro of of Prop osition wehave H X G

H X P This proves that H X G is smo oth of dimension dim P g m So the

v X

co dimention is

cv dim Gg dim P g m

X X

dim R P g m

which nishes our pro of

Denition Let G b e a reductive group scheme over the curve X of genus g over the

algebraically closed eld k Let b e the number of roots of GThenwe dene for every

integer i thenumber i to b e the smallest integer satisfying

if g

Prop osition Let G beareductive group scheme over the curve X of genus g over the

algebraical ly closed eld k Assume that G satises Condition x Let i be an integer Then

if m i the canonical homomorphism

i i

X G Q X G Q H H H H

d d

is an isomorphism

Proof By Lemma we get for every m i an isomorphism

i i

H H X G Q H H X G Q

m m

d d

since wemay estimate the co dimension c of H X G in H X G for v such that mvm

v m

d d

as follows

m if g

c r vg m

if g m

so that for m iwehave i c By Corollary this implies our prop osition

Eigenvalues of Frob enius

Lemma Let G be a nonsingular linear algebraic group over F acting on a smooth

equidimensional DeligneMumford stack X of nite type over F Let X XG be the cor

responding smooth algebraic F stack of nite type Then for any embedding Q C and any

i i

eigenvalue of on H X C we have jjq

q sm

Proof As in the pro of of Theorem we reduce to the case that G GL Then using the

sp ectral sequence on page we reduce to proving the lemma for the DeligneMumford

stack X Then using the Gysin sequence Corollary and Prop osition we reduce to

the case that X is nice Then by Corollary we reduce to the case that X is a smo oth

varietyover F Again using the Gysin sequence wemayeven reduce to the case that X is an

ane smo oth F varietyNow using Poincare duality as in the pro of of Prop osition our

lemma translates to the following statement If is an eigenvalue of the geometric Frob enius F

i i

on H X C then wehave jj q But this follows from Theoreme

Corollary Let G beareductive group scheme over the curve X over the eld F Let

d G be given Then if is an eigenvalue of on H H X G C for some m

q m

i i

or on H H X G C thenwehavejj q

X G follows from Lemma using Prop osition Now Proof The statement for H

the statementforH X G follows from Prop osition

Torsors under the Radical of G

Lemma Let X be a curve over the eld k and F an etale abelian group scheme over X

Then H X F is torsion

Proof Let E be an F torsor Clearly there exists a Galois extension of curves Y X such that

E has a section s over Y Let GalYX and let n Since F is ab elian wemay form

the nfold twist of E with itself

E E E

F F

Letting b e the elements of it is obvious that s s is a section of E

n n

over X Thus E is trivial

Lemma Let T be a torus over the curve X over the algebraical ly closed eld k Let

M f T j H X T g

Then M is a subgroup of nite index in T

Proof Let T Weneedtoshow that there exists a T torsor E suchthatdegE n for

T bethecharacter sheaf of T and consider the character group T ofT some n Let

X X

as a constant sheaf on X Thenwehave a short exact sequence of etale ab elian group schemes

over X

T F T

X X

where F is dened so as to make this sequence exact It es easily seen that T is lo cally a

F T H X Hom T so that F is lo cally free Hence Ext direct summand of F T

X X X

which is torsion by Lemma So the sequence is quasi split The dual of is a

sequence of tori over X

S T G

for some integer r By the ab ove arguments T G has a quasisection Wemayidentify

r r

T with G and it is clear that there exists a G torsor of degree Taking the

X X

m m

asso ciated T torsor via a quasisection of T G we get a T torsor E suchthatdegE n

for some n

Lemma Let G beareductive group scheme over the curve X over the algebraical ly closed

eld k Let R be the radical of G Then there exist nitely many elements d d G

such that for every d G there exists an i fng and an Rtorsor E of degree d d

Proof Apply Lemma to R so wehave M R of nite index Now as noted

in the pro of of Prop osition R has nite index in G soM has nite index in

X X

G Letd d b e a set of representatives for G M Soifd G there exists an

X X X

i fng such that d d M

Passing to the Limit

From now on we let G b e a reductive group scheme over the curve X over the algebraically closed

eld k Assume that G satises Condition x This is for example the case if G is rationally

trivial and g f gif G GL or if Dyn G is connected Let V b e the free ab elian group on

the comp onents of the Dynkin diagram of G

For a closed and op en subscheme Dyn G andaninteger welet

C fv V j mvg

Here m is dened as in Lemma and V is the set of linear combinations of comp onents

contained in all of whose co ecients are p ositive

Lemma We have C O where s is the number of connectedcomponents of

DynG

Proof Use Lemma

Let A b e the set of closed and op en subschemes Dyn G such that there exists a Gtorsor E

such that the canonical parab olic subgroup of G has typ e For every A cho ose sucha

Gtorsor and call it E Let P G b e the canonical parab olic subgroup whichisoftyp e

and let H P R P For each A cho ose n H according to

Lemma We get a nite family of smo oth algebraic k stacks

H X H

parametrized by A fng By the niteness theorem Theorem wehave that

H H X H Q is nite dimensional over Q for every pair j A fng and

for every i ZSoifweset

b j dim H H X H Q

i Q

then b j is a family of nonnegativeintegers Let B i for i Zbechosen such that

B i b j for every j A fng and suchthatB ifor i

Lemma There exists an integer N such that B iO i

Proof This dep ends essentially on the sp ectral sequence on page

Wenow dene for i

X X

D i C B i r g

where r is dened as in Lemma noting that r v for v V dep ends only on the

comp onents of Dyn G with resp ect to which v has nonzero co ecients

Lemma There exists an integer M such that D iO i

Proof Follows form Lemmas and

Lemma Let i be given Then we have

dim H H X G Q D i

Proof By Prop osition wehave

i i

dim H H X G Q dim H H X G Q

Q Q

d d

By Lemma and the Gysin sequence wehave

dim H H X G Q

ir vg mv

dim H H X G Q

Q v

mv i

X X X

ir vg

dim H H X G Q

Q v

Nowifv V for Awehave

X G X G H H

v v

ddeg E d

by Remark Let P b e the unique element suchthatd d deg E and

v vThenby Prop osition wehave

E i i

X G Q H H X H Q H H

ddeg E

Note that if no such exists then H X G Nowby construction and Lemma

there exists a j fng such that

X H H X H H

So assembling these remarks together we see that

i i

X G Q H H X H Q H H

and hence

dim H H X G Q b j B i

Q v i

So wehave

X X X

dim B i r g H H X G Q

X X

C B i r g

D i

what was to b e proven

Theorem Let G beareductive group scheme over the curve X over the eld F Assume

X G C converges absolutely and we have that G satises Condition x Then tr jH H

tr jH H X G C

Aut E

E H XG

X G C converges absolutely follows from Lemma Proof The fact that tr jH H

Corollary and Prop osition b ecause

D iq

converges absolutely This follows for example from the quotient criterion and Lemma

Wewillnow prove the convergence of

lim

AutE

E H XG

and calculate its limit

So let b e given and cho ose i is suchaway that

D iq

Then let m i Cho ose m m and i i such that m i Then wehave

D iq

For every i i wehave i m Sowemay calculate as in the pro of of Lemma

X X

X G Q C B i r g dim H H

X X

C B i r g

D i

Hence wehave

X X

i i i

D iq dim H H X G Q q

ii ii

So wemaynow estimate

tr jH H X G C

AutE

E H XG

X X

i i

X G C tr jH H

AutE

E H XG

by Lemma

X X

i i

X G C tr jH H

m q

AutE

E H XG

by Prop osition

X G C tr jH H

m q

AutE

E H XG

by Corollary

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