Homotopy invariant presheaves with framed transfers

TL;DR

This paper proves that certain homotopy-invariant framed presheaves become sheaves with similar invariance properties under the Nisnevich topology, extending Voevodsky's foundational work in algebraic geometry and motivic homotopy theory.

Contribution

It establishes invariance and stability properties of Nisnevich sheaves derived from framed presheaves over infinite fields, including characteristic 2 cases with additional module assumptions.

Findings

Nisnevich sheaf associated to A^1-invariant framed presheaves is A^1-invariant.

Such sheaves are strictly A^1-invariant and quasi-stable over infinite perfect fields.

Results extend to characteristic 2 with Z[1/2]-module assumptions.

Abstract

The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12]. Based on the theory, framed motives are introduced and studied in [7]. The main aim of this paper is to prove that for any $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$, the associated Nisnevich sheaf $\mathcal F_{nis}$ is $\mathbb A^1$-invariant whenever the base field $k$ is infinite of characteristic different from 2. Moreover, if the base field $k$ is infinite perfect of characteristic different from 2, then every $\mathbb A^1$-invariant quasi-stable Nisnevich framed sheaf of Abelian groups is strictly $\mathbb A^1$-invariant and quasi-stable. Furthermore, the same statements are true in characteristic 2 if we also assume that the $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$ is a presheaf of $\mathbb Z[1/2]$-modules. This result and the paper are inspired by Voevodsky's paper [13].