Once again on an analogue of the certain Voevodsky theorem

TL;DR

This paper proves that for certain $A^1$-invariant presheaves, their Zariski and Nisnevich sheafifications coincide, and their cohomology groups agree on smooth schemes, extending Voevodsky's results.

Contribution

It establishes the equivalence of Zariski and Nisnevich sheafifications and cohomology for a class of $A^1$-invariant presheaves, generalizing Voevodsky's theorem.

Findings

Zariski and Nisnevich sheafifications coincide for the presheaves considered.

Cohomology groups in Zariski and Nisnevich topologies are equal for smooth schemes.

Extends Voevodsky's theorem to a broader class of presheaves.

Abstract

Suppose that $F$ is an $\mathbb{A}^{1}$-invariant quasi-stable $\mathbb{Z}F_{\ast}$-presheaf. Then its Zariski sheafification $F_{Zar}$ coincides with its Nisnevich sheafification $F_{Nis}$. Moreover, if $X\in Sm/k$ is $k$-smooth, then for any $n$ there is equality $H^{n}_{Zar}(X, F_{Zar})=H^{n}_{Nis}(X,F_{Nis})$.