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

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 \[ Ri^! \nu_m(n)\cong \nu_m(n-r)[-r]. \]