Unramified logarithmic Hodge-Witt cohomology and $\mathbb{P}^1$-invariance

TL;DR

This paper proves that certain cohomology theories and sheaves are invariant under the projective line, extending previous results and providing new proofs for unramified logarithmic Hodge-Witt cohomology.

Contribution

It generalizes the invariance result from reciprocity sheaves to all P^1-invariant Nisnevich sheaves with transfers and offers a direct proof for the invariance of unramified logarithmic Hodge-Witt cohomology.

Findings

G(X) is isomorphic to G(Spec k) for P^1-invariant sheaves.

Unramified logarithmic Hodge-Witt cohomology is P^1-invariant.

Extension of invariance results to broader classes of sheaves.

Abstract

Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for any $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers $G$. This generalize a result of Binda-R\"ulling-Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge-Witt cohomology is a $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers.