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

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-Rülling-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.

19 pages. The introduction and the proof of Proposition 6.1 are largely modified