A construction of tame sheaves and tame de Rham--Witt cohomology

TL;DR

This paper develops a new algebraic framework for tame sheaves and tame de Rham--Witt cohomology, enabling comparisons with syntomic cohomology and extending the theory to reciprocity sheaves.

Contribution

It introduces a general machinery to construct tame sheaves from étale sheaves and local tame sections, applied to de Rham--Witt sheaves and reciprocity sheaves.

Findings

Constructed tame sheaves from étale data and tame sections.

Compared tame syntomic cohomology with Nygaard filtration.

Extended the theory to reciprocity sheaves over fields.

Abstract

In this article, we consider an algebraic version of the tame site of a pair $(X,\widetilde{X})$. With this definition, we provide a general machinery to construct a tame sheaf from the data of an \'etale sheaf on $X$ and a family of local tame sections. We apply this construction to the big de Rham--Witt sheaves with tame sections defined by log poles and, over a field, to reciprocity sheaves, and deduce some consequences. As an application, we compare tame syntomic cohomology with the Nygaard filtration on the tame de Rham--Witt complex.