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

Let X be a smooth proper variety over a field k and suppose that the degree map CH0(X ⊗k K) → Z is isomorphic for any field extension K/k. We show that G(Spec k)→ G(X) is an isomorphism for any P-invariant Nisnevich sheaf with transfers G. This generalizes 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 P-invariant Nisnevich sheaf with transfers.

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