Cohomology of presheaves with oriented weak transfers

TL;DR

This paper proves homotopy invariance and cohomology agreement for certain presheaves with weak transfers over fields of characteristic zero, extending Voevodsky's motives framework with explicit correspondence representatives.

Contribution

It generalizes a key result in Voevodsky's theory by establishing homotopy invariance and cohomology agreement for presheaves with oriented weak transfers.

Findings

Homotopy invariance of Nisnevich cohomology established

Zariski and Nisnevich cohomology agree for these presheaves

Explicit smooth representatives of correspondences constructed

Abstract

Over a field of characteristic zero, we establish the homotopy invariance of the Nisnevich cohomology of homotopy invariant presheaves with oriented weak transfers, and the agreement of Zariski and Nisnevich cohomology for such presheaves. This generalizes a foundational result in Voevodsky's theory of motives. The main idea is to find explicit smooth representatives of the correspondences which provide the input for Voevodsky's cohomological architecture.