The classifying topos of a group scheme and invariants of symmetric bundles. Issue 5 (8th July 2014) - Cassou‐Noguès, Ph.; Chinburg, T.; Morin, B.; Taylor, M. J.
Abstract : LetY be a scheme in which2 is invertible and letV be a rankn vector bundle onY endowed with a non‐degenerate symmetric bilinear formq . The orthogonal groupO ( q ) of the formq is a group scheme overY whose cohomology ringH * ( B O ( q ), Z / 2 Z ) ≃ A Y [ H W 1 ( q ), …, H W n ( q ) ] is a polynomial algebra over the étale cohomology ringA Y : = H * ( Y et, Z / 2 Z ) of the schemeY . Here, theH W i ( q ) 's are Jardine's universal Hasse–Witt invariants andB O ( q ) is the classifying topos ofO ( q ) as defined by Grothendieck and Giraud. The cohomology ringH * ( B O ( q ), Z / 2 Z ) contains canonical classesdet [ q ] and[ C q ] of degree1 and2, respectively, which are obtained from the determinant map and the Clifford group ofq . The classical Hasse–Witt invariantsw i ( q ) live in the ringA Y . Our main theorem provides a computation ofdet [ q ] and[ C q ] as polynomials inH W 1 ( q ) andH W 2 ( q ) with coefficients inA Y written in terms ofw 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 toposB 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.