Birational and ${\mathbf{A}}^1$-invariant lattices in the cohomology of the structure sheaf over non-archimedean fields

Abstract

We show that the cohomology of the structure sheaf of smooth and proper schemes over a complete non-archimedean field $K$ of characteristic zero, can be refined to an ${\mathbf{A}}^1$-invariant cohomology theory of smooth (not necessarily proper) schemes over $K$ with values in ${\mathcal{O}}_K$-lattices, and the same holds for $K$ of positive characteristic in dimensions at most $3$. As one application, we obtain that the automorphism group of the function field of a proper smooth variety $X$ of dimension at most 3 over a field of positive characteristic acts quasi-unipotently on the cohomology of the structure sheaf of $X$. The construction of the lattices relies on a variant of the tame cohomology of Hübner--Schmidt with coefficients in a twisted version of the tame structure sheaf and uses results from rigid analytic geometry on the cohomology of twisted integral rigid structure sheaves due to Bartenwerfer and van der Put.

25 pages