Duality for Relative Logarithmic de Rham-Witt Sheaves on Semistable Schemes over ${\mathbb{F}}_q[[t]]$

Abstract

We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes $X$ over a local ring ${\mathbb{F}}_q[[t]]$, where ${\mathbb{F}}_q$ is a finite field. As an application, we obtain a new filtration on the maximal abelian quotient ${\pi }_1^{ab}(U)$ of the étale fundamental groups ${\pi }_1(U)$ of an open subscheme $U\subseteq X$, which gives a measure of ramification along a divisor $D$ with normal crossing and $Supp(D)\subseteq X-U$. This filtration coincides with the Brylinski-Kato-Matsuda filtration in the relative dimension zero case.