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 ${\bf O}(q)$ of the form $q$ is a group scheme over $Y$ whose cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})\simeq A_Y[HW_1(q),..., HW_n(q)]$ is a polynomial algebra over the \'etale cohomology ring $A_Y:=H^*(Y_{et},{\bf Z}/2{\bf Z})$ of the scheme $Y$. Here the $HW_i(q)$'s are Jardine's universal Hasse-Witt invariants and $B_{{\bf O}(q)}$ is the classifying topos of ${\bf O}(q)$ as defined by Grothendieck and Giraud. The cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})$ contains canonical classes $\mathrm{det}[q]$ 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 $HW_{1}(q)$ and $HW_{2}(q)$ with coefficients in $A_Y$ written in terms of $w_1(q),w_2(q)\in A_Y$. This result is the source of numerous standard comparison formulas for classical Hasses-Witt invariants of quadratic forms. Our proof is based on computations with (abelian and non-abelian) Cech cocycles in the topos $B_{{\bf O}(q)}$. 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.