Kay Rülling (Universität Wuppertal): Tame cohomology of the structure sheaf in mixed characteristic

Abstract:
This is joint work in progress with Alberto Merici and Shuji Saito.
Let R be a complete discrete valuation ring of mixed characteristic with fraction field K.
We show that the tame cohomology of Hübner-Schmidt on smooth K-schemes relative to R of (a twist of) the structure sheaf is \A^1-invariant and is a finite R-module up to bounded torsion.
This induces a canonical R-lattice in the cohomology of the structure sheaf of smooth proper K-schemes.
If X has a regular model over R and resolutions of singularities in mixed characteristic hold, then this lattice would be the cohomology of this regular model. The interesting point is that we get the existence of such a lattice also in case X has no regular model and without using resolutions.
To this end we use classical results by Bartenwerfer and van der Put in rigid geometry.