On Frölich twisted bundles

Introduction A symmetric bundle (E, f) over a noetherian Z[ 2 ]-scheme Y is a vector bundle E over Y equipped with a symmetric isomorphism f between E and its Y -dual E. A symmetric bundle can also be viewed as a quadratic form on E and we write (E, q) if we take this point of view, or, if the form is clear from the context, we might even just write E. It is well known how to describe the set of all twists of (E, f), that is the set of symmetric bundles which become isomorphic to (E, f) after an etale base extension. If O(E) denotes the orthogonal group (scheme) attached to (E, f) this set is H(Y,O(E)) (see [Mi], chapter 3 for a precise definition of this set). For α in H(Y,O(E)), let Eα be the twist of E corresponding to α. Every symmetric bundle of rank n is a twist of the standard symmetric bundle 1n = (O n Y , x 2 1 + ... + x 2 n). Let O(n) denote the automorphism group of this symmetric bundle. We write αE for the class of E in H (Y,O(n)) (n = rank(E)). Following Delzant [Dz] and Jardine [J1], for any symmetric bundle E over Y one can define a cohomological invariant, which generalizes the classical invariants of quadratic forms and which is known as the total Hasse-Witt class. This is a class wt(E) in the (graded) etale cohomology group H (Y,Z/2Z):

[1]  M. Taylor,et al.  Hasse–Witt invariants of symmetric complexes: an example from geometry , 2002 .

[2]  B. Erez,et al.  Invariants of a quadratic form attached to a tame covering of schemes , 2000 .

[3]  Takeshi Saito.  The sign of the functional equation of the L-function of an orthogonal motive , 1995 .

[4]  B. Kahn.  Equivariant Stiefel-Whitney classes , 1994 .

[5]  J. Jardine.  Higher spinor classes , 1994 .

[6]  H. Esnault,et al.  Coverings with odd ramification and Stiefel-Whitney classes. , 1993 .

[7]  M. Ojanguren,et al.  The Clifford algebra of a metabolic space , 1991 .

[8]  Max-Albert Knus,et al.  Quadratic and Hermitian Forms over Rings , 1991 .

[9]  V. Snaith.  Stiefel-Whitney Classes of a Symmetric Bilinear Form — A Formula of Serre , 1985, Canadian Mathematical Bulletin.

[10]  A. Fröhlich.  Orthogonal representations of Galois groups, Stiefel-Whitney classes and Hasse-Witt invariants. , 1985 .

[11]  Jean-Pierre Serre.  L'invariant de Witt de la forme Tr(x2) , 1984 .

[12]  M. Ojanguren.  A splitting theorem for quadratic forms , 1982 .

[13]  Manfred Knebusch,et al.  Symmetric bilinear forms over algebraic varieties , 1977 .

[14]  P. Deligne.  Les constantes locales de l'équation fonctionnelle de la fonctionL d'Artin d'une représentation orthogonale , 1976 .

[15]  O. Laborde.  Classes de Stiefel-Whitney en cohomologie étale , 1976 .

[16]  A. Fröhlich,et al.  On the functional equation of the ArtinL-function for characters of real representations , 1973 .