The Cartier operator on differentials of discretely ringed adic spaces and Purity in the tame cohomology

TL;DR

This paper extends the absolute purity theorem to tame cohomology for regular schemes over integers localized away from a prime, using Milne's approach with modifications, assuming resolution of singularities in positive characteristic.

Contribution

It proves a new isomorphism in tame cohomology analogous to Grothendieck's absolute purity, generalizing Milne's results to this setting.

Findings

Established the isomorphism Ri^! ν_m(n) ≅ ν_m(n−r)[−r] in tame cohomology.

Extended the absolute purity conjecture to cases involving tame cohomology.

Assumed resolution of singularities in positive characteristic for the proof.

Abstract

Let $X$ be a regular scheme over $\textrm{Spec}(\mathbb{Z}[1/p])$ where $p$ is prime. Let $i:Y\to X$ be a closed subscheme of pure codimension $r$. Let $n$ be a natural number prime to $p$. Let $\Lambda$ be a finite $\mathbb{Z}/n$-module over $X$. In this case, the absolute purity conjectured by Grothendieck and proved by Gabber states that \[ Ri^! \Lambda\cong \Lambda_Y(-r)[-2r]\in D_{\textrm{\'et}}(Y,\Lambda) \] For the $n=p^m$-case, a dualizing sheaf was proposed by Milne \cite{MilneValuesOfZeta}, namely the logarithmic de Rham-Witt sheaves $\nu_m(r)$. But this doesn't work for all degrees for the \'etale cohomology. It is however conjectured that this works for the tame cohomology. In this paper we make this work following the proof of Milne in loc. cit. by replacing \'etale by tame cohomology and assuming resolution of singularities in positive characteristic. We obtain the following isomorphism \[ Ri^! \nu_m(n)\cong \nu_m(n-r)[-r]. \]