The classifying topos of a group scheme and invariants of symmetric bundles

Abstract

Let Y be a scheme in which 2 is invertible and let V be a rank n vector bundle on Y endowed with a non-degenerate symmetric bilinear form q. The orthogonal group O ( q ) of the form q is a group scheme over Y whose cohomology ring H * ( B O ( q ) , Z / 2 Z ) ≃ A Y [ H W 1 ( q ) , … , H W n ( q ) ] is a polynomial algebra over the étale cohomology ring A Y : = H * ( Y et , Z / 2 Z ) of the scheme Y. Here, the H W i ( q ) 's are Jardine's universal Hasse–Witt invariants and B O ( q ) is the classifying topos of O ( q ) as defined by Grothendieck and Giraud. The cohomology ring H * ( B O ( q ) , Z / 2 Z ) contains canonical classes det [ q ] and [ C q ] of degree 1 and 2 , respectively, which are obtained from the determinant map and the Clifford group of q. The classical Hasse–Witt invariants w i ( q ) live in the ring A Y . Our main theorem provides a computation of det [ q ] and [ C q ] as polynomials in H W 1 ( q ) and H W 2 ( q ) with coefficients in A Y written in terms of w 1 ( q ) , w 2 ( q ) ∈ A Y . This result is the source of numerous standard comparison formulas for classical Hasse–Witt invariants of quadratic forms. Our proof is based on computations with (abelian and non-abelian) Cech cocycles in the topos B O ( q ) . This requires a general study of the cohomology of the classifying topos of a group scheme, which we carry out in the first part of this paper.