Duality for relative logarithmic de Rham-Witt sheaves and wildly ramified class field theory over finite fields

TL;DR

This paper develops a duality theory for modified logarithmic de Rham-Witt sheaves on varieties over finite fields, providing new tools to analyze $p$-adic étale cohomology and wild ramification class field theory.

Contribution

It introduces new $p$-primary torsion sheaves depending on divisors and establishes a perfect duality with cohomology groups, advancing the understanding of ramification in positive characteristic.

Findings

Established a perfect duality between cohomology groups of modified sheaves and logarithmic de Rham-Witt sheaves.

Applied duality to study wild ramification class field theory over finite fields.

Provided a new framework for analyzing $p$-adic étale cohomology of open subvarieties.

Abstract

In order to study $p$-adic \'etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves. It is a modification of the logarithmic de Rham-Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wild ramification class field theory for the open subvariety $U$.