On the cohomology of reciprocity sheaves

TL;DR

This paper establishes an action of Chow correspondences on the cohomology of reciprocity sheaves, generalizing classical results and providing new invariants and obstructions in algebraic geometry.

Contribution

It introduces a framework for the cohomology of reciprocity sheaves, including key structural formulas and applications to birational invariants and zero-cycle obstructions.

Findings

Existence of Chow correspondence action on reciprocity sheaf cohomology

Generalization of classical cohomological formulas

Construction of new birational invariants and obstructions

Abstract

In this paper we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence, and the existence of proper pushforward. In this way we recover and generalize analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves. We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero-cycles of degree one from the cohomology of reciprocity sheaves.