Journal of the Ramanujan Mathematical Society

Volume 30, Issue 1, March 2015 pp. 1–27.

Quadric invariants and degeneration in smooth-étale cohomology

Authors: Saurav Bhaumik and Nitin Nitsure
Author institution:Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India.

Summary: For a regular pair (X,Y) of schemes over Z[1/2] of pure codimension 1, we consider quadric bundles on X which are nondegenerate on X-Y, but are minimally degenerate on Y. We give a formula for the behaviour of the cohomological invariants (characteristic classes) of the nondegenerate quadric bundle on X-Y under the Gysin boundary map to the étale cohomology of Y with mod 2 coefficients. The results here are the algebro-geometric analogs of topological results for complex bundles proved earlier by Holla and Nitsure, continuing further the algebraization program which was commenced with a recent paper by Bhaumik. We use algebraic stacks and their smooth-étale cohomologies, A1-homotopies and Gabber's absolute purity theorem as algebraic replacements for the topological methods used earlier, such as CW complexes, real homotopies, Riemannian metrics and tubular neighbourhoods. Our results also hold for quadric bundles over algebraic stacks over Z[1/2].

Contents Full-Text PDF