Finite generation conjectures for cohomology over finite fields

TL;DR

This paper introduces an intermediate cohomology bridging motivic and Weil-etale cohomology, linking key conjectures on their finite generation properties.

Contribution

It constructs a new cohomology theory that connects motivic and Weil-etale cohomology, providing a framework to relate their finite generation conjectures.

Findings

Established a new intermediate cohomology theory.

Linked Bass and Beilinson-Tate conjectures through this framework.

Provided insights into the finite generation of cohomology groups.

Abstract

We construct an intermediate cohmology between motivic cohomology and Weil-etale cohomology. Using this, the Bass conjecture on finite generation of motivic cohomology, and the Beilinson-Tate on the finite generation of Weil-etale cohomology are related.