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

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