Strictly homotopy invariance of Nisnevich sheaves with GW-transfers

Abstract

The strictly homotopy invariance of the associated Nisnevish sheave ${\tilde{\mathcal{F}}}_{Nis}$ of a homotopy invariant presheave $\mathcal{F}$ with GW-transfers (or Witt-transfers) on the category of smooth varieties over a prefect field $k$, $chark\ne 2$, is proved, i.e. the isomorphism $$H_{Nis}^i({\mathbb{A}}^1\times X,{\tilde{\mathcal{F}}}_{Nis})\simeq H_{Nis}^i(X,{\tilde{\mathcal{F}}}_{Nis})$$ for any $X\in S{m}_k$ is obtained. This theorem is necessary for the construction of the triangulated category of GW-motives ${\mathbf{DM}}^{GW}(k)$ and Witt-motives ${\mathbf{DM}}^W(k)$ by the Voevodsky-Suslin method originally used for the construction of the category of motives $\mathbf{DM}(k)$. In particular, the result of the article gives the direct prove of the strictly homotopy invariance of the Nisnevich sheaves associated to hermitian K-theory and Witt-groups (without using of the representability of these cohomology theories in the motivic homotopy category ${\mathbf{H}}_{{\mathbb{A}}^1}(k)$ proved by Hornbostle [Horn_ReprKOWitt]); and on other side the strictly homotopy invariance theorem proved here and the representability criteria proved in [Horn_ReprKOWitt] implies that cohomologies $H_{nis}^i(-,{\tilde{\mathcal{F}}}_{nis})$ of the associated sheaf of a homotopy invariant presheave with GW-(Witt-)transfers $\mathcal{F}$ are representable in ${\mathbf{H}}_{{\mathbb{A}}^1}(k)$.