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

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 $\mathbf{O}(q)$ of the form $q$ is a group scheme over $Y$ whose cohomology ring $H^(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})\simeq A_Y[HW_1(q),…, HW_n(q)]$isapolynomialalgebraoverthe\acute{e}talecohomologyring$A_Y:=H^(Y_{et},{\bf Z}/2{\bf Z})$ofthescheme$Y$\mathrm{.}Herethe$HW_i(q)$'s are Jardine’s universal Hasse-Witt invariants and $B_{\mathbf{O}(q)}$ is the classifying topos of $\mathbf{O}(q)$ as defined by Grothendieck and Giraud. The cohomology ring $H^{\ast }(B_{\mathbf{O}(q)},\mathbf{Z}\mathrm{/}2\mathbf{Z})$ contains canonical classes $\mathrm{d}\mathrm{e}\mathrm{t}[q]$ 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 $w_1(q),w_2(q)\in A_Y$. 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 $B_{\mathbf{O}(q)}$. 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.