Cohomological invariants of finite Coxeter groups J´ erˆ ome Ducoat Universit´ e Grenoble 1 Ramification in Algebra and Geometry at Emory J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 1 / 8
Introduction Let k 0 be a field, Γ k 0 be the absolute Galois group of k 0 and C be a finite Γ k 0 -module. Let G be a smooth linear algebraic group over k 0 . J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 2 / 8
Introduction Let k 0 be a field, Γ k 0 be the absolute Galois group of k 0 and C be a finite Γ k 0 -module. Let G be a smooth linear algebraic group over k 0 . Definition A cohomological invariant a of G with coefficients in C is the datum of a map a k : H 1 ( k , G ) → H ∗ ( k , C ) for each extension k / k 0 , which are functorial in k / k 0 . J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 2 / 8
Introduction Let k 0 be a field, Γ k 0 be the absolute Galois group of k 0 and C be a finite Γ k 0 -module. Let G be a smooth linear algebraic group over k 0 . Definition A cohomological invariant a of G with coefficients in C is the datum of a map a k : H 1 ( k , G ) → H ∗ ( k , C ) for each extension k / k 0 , which are functorial in k / k 0 . We denote Inv k 0 ( G , C ) the set of all cohomological invariants of G over k 0 , with coefficients in C . Assume that char( k 0 ) is prime to the order of C . J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 2 / 8
Introduction Let k 0 be a field, Γ k 0 be the absolute Galois group of k 0 and C be a finite Γ k 0 -module. Let G be a smooth linear algebraic group over k 0 . Definition A cohomological invariant a of G with coefficients in C is the datum of a map a k : H 1 ( k , G ) → H ∗ ( k , C ) for each extension k / k 0 , which are functorial in k / k 0 . We denote Inv k 0 ( G , C ) the set of all cohomological invariants of G over k 0 , with coefficients in C . Assume that char( k 0 ) is prime to the order of C . Aim : determine all cohomological invariants of G with coefficients in C. J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 2 / 8
Introduction Let k 0 be a field, Γ k 0 be the absolute Galois group of k 0 and C be a finite Γ k 0 -module. Let G be a smooth linear algebraic group over k 0 . Definition A cohomological invariant a of G with coefficients in C is the datum of a map a k : H 1 ( k , G ) → H ∗ ( k , C ) for each extension k / k 0 , which are functorial in k / k 0 . We denote Inv k 0 ( G , C ) the set of all cohomological invariants of G over k 0 , with coefficients in C . Assume that char( k 0 ) 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 ∗ ( k 0 , C ) -module structure on Inv k 0 ( G , C ) ? J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 2 / 8
Introduction Let k 0 be a field, Γ k 0 be the absolute Galois group of k 0 and C be a finite Γ k 0 -module. Let G be a smooth linear algebraic group over k 0 . Definition A cohomological invariant a of G with coefficients in C is the datum of a map a k : H 1 ( k , G ) → H ∗ ( k , C ) for each extension k / k 0 , which are functorial in k / k 0 . We denote Inv k 0 ( G , C ) the set of all cohomological invariants of G over k 0 , with coefficients in C . Assume that char( k 0 ) 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 ∗ ( k 0 , C ) -module structure on Inv k 0 ( G , C ) ? Definition An invariant a ∈ Inv k 0 ( G , C ) is normalized if a k 0 (1) = 0, where 1 is the base point of H 1 ( k 0 , G ). J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 2 / 8
Cohomological invariants of O n Assume Char( k 0 ) � = 2. ∀ k / k 0 , H 1 ( k , O n ) ≃ { isomorphism classes of non degenerate ; quadratic forms of rank n over k } Theorem ( Serre, 2003) The H ∗ ( k 0 , Z / 2 Z )-module Inv k 0 ( O n , Z / 2 Z ) is free, with basis { w 0 , w 1 , ..., w n } . where ∀ i ≥ 0, ∀ k / k 0 , ∀ q = < α 1 , ..., α n > , w i ( q ) = � ( α j 1 ) ∪ ... ∪ ( α j i ) 1 ≤ j 1 <...< j i ≤ n define Stiefel-Whitney invariants. J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 3 / 8
Cohomological invariants of S n : the splitting principle ∀ k / k 0 , H 1 ( k , S n ) ≃ { isomorphism classes of ´ etale algebras ; of rank n over k } J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 4 / 8
Cohomological invariants of S n : the splitting principle ∀ k / k 0 , H 1 ( k , S n ) ≃ { isomorphism classes of ´ etale algebras ; of 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 ∈ Inv k 0 ( S n , C ). If, for every k / k 0 and every multiquadratic ´ etale k -algebra E , a k ( E ) = 0, then a = 0. J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 4 / 8
Cohomological invariants of S n : the splitting principle ∀ k / k 0 , H 1 ( k , S n ) ≃ { isomorphism classes of ´ etale algebras ; of 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 ∈ Inv k 0 ( S n , C ). If, for every k / k 0 and every multiquadratic ´ etale k -algebra E , a k ( E ) = 0, then a = 0. Corollary For any normalized a ∈ Inv k 0 ( S n , C ), 2 a = 0. J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 4 / 8
Cohomological invariants of S n Assume Char( k 0 ) � = 2 and set C = Z / 2 Z . Theorem ( Serre, 2003) The H ∗ ( k 0 , Z / 2 Z )-module Inv k 0 ( S n , Z / 2 Z ) is free with basis { w 0 , ..., w [ n / 2] } . where ∀ 0 ≤ i ≤ n ∀ k / k 0 , ∀ ( E ) ∈ H 1 ( k , S n ) � Tr E / k ( x 2 ) � w i ( E ) = w i define Stiefel-Whitney invariants. J´ erˆ ome 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 < r 1 , .., r s | ∀ i , j ∈ { 1 , .., s } , ( r i r j ) m i , j = 1 > , where ∀ i , j ∈ { 1 , ..., s } , m i , j ∈ N ∪ { + ∞} and m i , i = 1 for every i ∈ { 1 , ..., s } . J´ erˆ ome 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 < r 1 , .., r s | ∀ i , j ∈ { 1 , .., s } , ( r i r j ) m i , j = 1 > , where ∀ i , j ∈ { 1 , ..., s } , m i , j ∈ N ∪ { + ∞} and m i , i = 1 for every i ∈ { 1 , ..., s } . Let W be a finite Coxeter group and k 0 be a field of characteristic zero. Assume that W admits a linear representation over k 0 . J´ erˆ ome 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 < r 1 , .., r s | ∀ i , j ∈ { 1 , .., s } , ( r i r j ) m i , j = 1 > , where ∀ i , j ∈ { 1 , ..., s } , m i , j ∈ N ∪ { + ∞} and m i , i = 1 for every i ∈ { 1 , ..., s } . Let W be a finite Coxeter group and k 0 be a field of characteristic zero. Assume that W admits a linear representation over k 0 . Theorem ( D.,2010 ) Let a ∈ Inv k 0 ( W , C ). Assume that for every abelian subgroup H of W generated by reflections, Res H W ( a ) = 0. Then a = 0. Corollary For every normalized a ∈ Inv k 0 ( W , C ), 2 a = 0. J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 6 / 8
Splitting principle for cohomological invariants of Weyl groups of type B n Let n ≥ 2 and W be a Weyl group of type B n . ∀ k / k 0 , H 1 ( k , W ) ≃ { isomorphism classes of pointed ´ etale algebras ( L , α ) . with α ∈ L ∗ / L ∗ 2 } Corollary Assume that a ∈ Inv k 0 ( W , C ) vanishes on any pointed algebra of the form k ( √ t 1 ) × ... × k ( √ t q ) × k n − 2 q , ( u 1 , ..., u q , α ′ ) � � with u 1 , ..., u q ∈ k ∗ , for any k / k 0 and for every 0 ≤ q ≤ m . Then a is the zero invariant. J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 7 / 8
Cohomological invariants of Weyl groups of type B n Let us now set C = Z / 2 Z . Theorem ( D.,2011 ) If − 1 , 2 ∈ k ∗ 2 0 , then the H ∗ ( k 0 , Z / 2 Z )-module Inv k 0 ( W , Z / 2 Z ) is free with basis { w i ∪ ˜ w j } 0 ≤ i ≤ m , 0 ≤ j ≤ 2( m − i ) . where : ∀ 0 ≤ i ≤ n , ∀ k / k 0 , ( w i ) k : H 1 ( k , W ) → H ∗ ( k , Z / 2 Z ) ( L , α ) �→ w i (Tr L / k ( x 2 )) ∀ 0 ≤ j ≤ n , ∀ k / k 0 , w j ) k : H 1 ( k , W ) → H ∗ ( k , Z / 2 Z ) (˜ ( L , α ) �→ w j (Tr L / k ( α x 2 )) J´ erˆ ome Ducoat (Universit´ e Grenoble 1) May 18, 2011 8 / 8
Recommend
More recommend
