Homotopy invariant presheaves with framed transfers

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category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12].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].