Abstract
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.
Highlights
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
In [3], Auel et al used Br(−) [2∞] in characteristic p = 2 to obtain a similar result for conic bundles over P2
K ∈ Fldgkm, f is an isomorphism in HINis [22, Corollary 11.2]. (V5) Given F ∈ PST, we denote by FZar the Zariski sheaf associated to the presheaf on Sm obtained by restricting F along the graph functor Sm → Cor
Summary
(V5) Given F ∈ PST, we denote by FZar the Zariski sheaf associated to the presheaf on Sm obtained by restricting F along the graph functor Sm → Cor. K ∈ Fldgkm, f is an isomorphism in HINis [22, Corollary 11.2]. (V5) Given F ∈ PST, we denote by FZar the Zariski sheaf associated to the presheaf on Sm obtained by restricting F along the graph functor Sm → Cor It does not admit a structure of presheaf with transfers, but if F ∈ HI, we have FZar = FNis by [22, Theorem 22.2], and FZar acquires transfers by (V1). This is an analogue of [9, Définition 1.1], where a proper morphism f : X → Y is said to be universally CH0-trivial if the induced map fK∗ : CH0 (XK ) → CH0 (YK ) is an isomorphism for each K ∈ Fldk. A proper birational morphism in Sm is universally H0S-trivial
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