Strictly homotopy invariance of Nisnevich sheaves with GW-transfers

Abstract

The strictly homotopy invariance of the associated Nisnevish sheave $\widetilde{\mathcal F}_{Nis}$ of a homotopy invariant presheave $\mathcal F$ with GW-transfers (or Witt-transfers) on the category of smooth varieties over a prefect field $k$, $char\,k \neq 2$, is proved, i.e. the isomorphism $$H^i_{Nis}(\mathbb A^1\times X,\widetilde{\mathcal F}_{Nis})\simeq H^i_{Nis}(X,\widetilde{\mathcal F}_{Nis})$$ for any $X\in Sm_k$ 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 $H^i_{nis}(-,\widetilde{\mathcal F}_{nis})$ of the associated sheaf of a homotopy invariant presheave with GW-(Witt-)transfers $\mathcal F$ are representable in $\mathbf H_{\mathbb A^1}(k)$.