Cohomological invariants of finite Coxeter groups J er ome Ducoat - - PowerPoint PPT Presentation

Cohomological invariants of finite Coxeter groups

J´ erˆ

• me Ducoat

Universit´ e Grenoble 1

Ramification in Algebra and Geometry at Emory

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 1 / 8

Introduction

Let k0 be a field, Γk0 be the absolute Galois group of k0 and C be a finite Γk0-module. Let G be a smooth linear algebraic group over k0.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 2 / 8

Introduction

Let k0 be a field, Γk0 be the absolute Galois group of k0 and C be a finite Γk0-module. Let G be a smooth linear algebraic group over k0.

Definition

A cohomological invariant a of G with coefficients in C is the datum of a map ak : H1(k, G) → H∗(k, C) for each extension k/k0, which are functorial in k/k0.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 2 / 8

Introduction

Let k0 be a field, Γk0 be the absolute Galois group of k0 and C be a finite Γk0-module. Let G be a smooth linear algebraic group over k0.

Definition

A cohomological invariant a of G with coefficients in C is the datum of a map ak : H1(k, G) → H∗(k, C) for each extension k/k0, which are functorial in k/k0. We denote Invk0(G, C) the set of all cohomological invariants of G over k0, with coefficients in C. Assume that char(k0) is prime to the order of C.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 2 / 8

Introduction

Let k0 be a field, Γk0 be the absolute Galois group of k0 and C be a finite Γk0-module. Let G be a smooth linear algebraic group over k0.

Definition

A cohomological invariant a of G with coefficients in C is the datum of a map ak : H1(k, G) → H∗(k, C) for each extension k/k0, which are functorial in k/k0. We denote Invk0(G, C) the set of all cohomological invariants of G over k0, with coefficients in C. Assume that char(k0) is prime to the order of C. Aim : determine all cohomological invariants of G with coefficients in C.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 2 / 8

Introduction

Let k0 be a field, Γk0 be the absolute Galois group of k0 and C be a finite Γk0-module. Let G be a smooth linear algebraic group over k0.

Definition

A cohomological invariant a of G with coefficients in C is the datum of a map ak : H1(k, G) → H∗(k, C) for each extension k/k0, which are functorial in k/k0. We denote Invk0(G, C) the set of all cohomological invariants of G over k0, with coefficients in C. Assume that char(k0) is prime to the order of C. Aim : determine all cohomological invariants of G with coefficients in C. Questions : Are there other invariants than the constant ones ? What is the H∗(k0, C)-module structure on Invk0(G, C) ?

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 2 / 8

Introduction

Let k0 be a field, Γk0 be the absolute Galois group of k0 and C be a finite Γk0-module. Let G be a smooth linear algebraic group over k0.

Definition

A cohomological invariant a of G with coefficients in C is the datum of a map ak : H1(k, G) → H∗(k, C) for each extension k/k0, which are functorial in k/k0. We denote Invk0(G, C) the set of all cohomological invariants of G over k0, with coefficients in C. Assume that char(k0) is prime to the order of C. Aim : determine all cohomological invariants of G with coefficients in C. Questions : Are there other invariants than the constant ones ? What is the H∗(k0, C)-module structure on Invk0(G, C) ?

Definition

An invariant a ∈ Invk0(G, C) is normalized if ak0(1) = 0, where 1 is the base point

• f H1(k0, G).

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 2 / 8

Cohomological invariants of On

Assume Char(k0) = 2. ∀k/k0, H1(k, On) ≃ {isomorphism classes of non degenerate quadratic forms of rank n over k} ;

Theorem (Serre, 2003)

The H∗(k0, Z/2Z)-module Invk0(On, Z/2Z) is free, with basis {w0, w1, ..., wn}. where ∀i ≥ 0, ∀k/k0, ∀q =< α1, ..., αn >, wi(q) =

• 1≤j1<...<ji≤n

(αj1) ∪ ... ∪ (αji) define Stiefel-Whitney invariants.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 3 / 8

Cohomological invariants of Sn : the splitting principle

∀k/k0, H1(k, Sn) ≃ {isomorphism classes of ´ etale algebras

• f rank n over k}

;

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 4 / 8

Cohomological invariants of Sn : the splitting principle

∀k/k0, H1(k, Sn) ≃ {isomorphism classes of ´ etale algebras

• f rank n over k}

;

Definition

We call multiquadratic ´ etale algebra an ´ etale k-algebra which is isomorphic to a product of ´ etale k-algebras of rank 1 or 2.

Theorem (Serre, 2003)

Let a ∈ Invk0(Sn, C). If, for every k/k0 and every multiquadratic ´ etale k-algebra E, ak(E) = 0, then a = 0.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 4 / 8

Cohomological invariants of Sn : the splitting principle

∀k/k0, H1(k, Sn) ≃ {isomorphism classes of ´ etale algebras

• f rank n over k}

;

Definition

We call multiquadratic ´ etale algebra an ´ etale k-algebra which is isomorphic to a product of ´ etale k-algebras of rank 1 or 2.

Theorem (Serre, 2003)

Let a ∈ Invk0(Sn, C). If, for every k/k0 and every multiquadratic ´ etale k-algebra E, ak(E) = 0, then a = 0.

Corollary

For any normalized a ∈ Invk0(Sn, C), 2a = 0.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 4 / 8

Cohomological invariants of Sn

Assume Char(k0) = 2 and set C = Z/2Z.

Theorem (Serre, 2003)

The H∗(k0, Z/2Z)-module Invk0(Sn, Z/2Z) is free with basis {w0, ..., w[n/2]}. where ∀0 ≤ i ≤ n ∀k/k0, ∀(E) ∈ H1(k, Sn) wi(E) = wi

• TrE/k(x2)
• define Stiefel-Whitney invariants.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 5 / 8

Cohomological invariants of finite Coxeter groups : a vanishing principle

Definition

A Coxeter group W is a group with a given presentation of type < r1, .., rs | ∀i, j ∈ {1, .., s}, (rirj)mi,j = 1 >, where ∀i, j ∈ {1, ..., s}, mi,j ∈ N ∪ {+∞} and mi,i = 1 for every i ∈ {1, ..., s}.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 6 / 8

Cohomological invariants of finite Coxeter groups : a vanishing principle

Definition

A Coxeter group W is a group with a given presentation of type < r1, .., rs | ∀i, j ∈ {1, .., s}, (rirj)mi,j = 1 >, where ∀i, j ∈ {1, ..., s}, mi,j ∈ N ∪ {+∞} and mi,i = 1 for every i ∈ {1, ..., s}. Let W be a finite Coxeter group and k0 be a field of characteristic zero. Assume that W admits a linear representation over k0.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 6 / 8

Cohomological invariants of finite Coxeter groups : a vanishing principle

Definition

A Coxeter group W is a group with a given presentation of type < r1, .., rs | ∀i, j ∈ {1, .., s}, (rirj)mi,j = 1 >, where ∀i, j ∈ {1, ..., s}, mi,j ∈ N ∪ {+∞} and mi,i = 1 for every i ∈ {1, ..., s}. Let W be a finite Coxeter group and k0 be a field of characteristic zero. Assume that W admits a linear representation over k0.

Theorem (D.,2010)

Let a ∈ Invk0(W , C). Assume that for every abelian subgroup H of W generated by reflections, ResH

W (a) = 0. Then a = 0.

Corollary

For every normalized a ∈ Invk0(W , C), 2a = 0.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 6 / 8

Splitting principle for cohomological invariants of Weyl groups of type Bn

Let n ≥ 2 and W be a Weyl group of type Bn. ∀k/k0, H1(k, W ) ≃ { isomorphism classes of pointed ´ etale algebras (L, α) with α ∈ L∗/L∗2} .

Corollary

Assume that a ∈ Invk0(W , C) vanishes on any pointed algebra of the form

• k(√t1) × ... × k(√tq) × kn−2q, (u1, ..., uq, α′)
• with u1, ..., uq ∈ k∗, for any k/k0 and for every 0 ≤ q ≤ m.

Then a is the zero invariant.

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 7 / 8

Cohomological invariants of Weyl groups of type Bn

Let us now set C = Z/2Z.

Theorem (D.,2011 )

If −1, 2 ∈ k∗2

0 ,

then the H∗(k0, Z/2Z)-module Invk0(W , Z/2Z) is free with basis {wi ∪ ˜ wj}0≤i≤m,0≤j≤2(m−i). where : ∀0 ≤ i ≤ n, ∀k/k0, (wi)k : H1(k, W ) → H∗(k, Z/2Z) (L, α) → wi(TrL/k(x2)) ∀0 ≤ j ≤ n, ∀k/k0, (˜ wj)k : H1(k, W ) → H∗(k, Z/2Z) (L, α) → wj(TrL/k(αx2))

J´ erˆ

• me Ducoat (Universit´

e Grenoble 1) May 18, 2011 8 / 8