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 $G(\operatorname {Spec} k) \to G(X)$ is an isomorphism for any $\mathbb {P}^1$-invariant Nisnevich sheaf with transfers G. This generalises a result of Binda, Rülling and 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.