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

TL;DR

This paper computes canonical cohomology classes of symmetric bundles over schemes with invertible 2, expressing them in terms of classical invariants, and develops a general cohomology theory for classifying topoi of group schemes.

Contribution

It provides explicit formulas for determinant and Clifford classes in the cohomology ring of the classifying topos of an orthogonal group scheme, linking them to classical Hasse-Witt invariants.

Findings

Explicit formulas for det[q] and [C_q] in terms of HW invariants.

Development of a cohomology framework for classifying topoi of group schemes.

Applications to classical comparison formulas for quadratic form invariants.

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.