Homotopy invariant presheaves with framed transfers

Abstract

The category of framed correspondences $F{r}_{\ast }(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12].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].