Triangulated category of effective $Witt$-motives $DW{M}_{eff}^{-}(k)$

Abstract

The category of effective $Witt$-motives $DW{M}^{-}(k)$ with functor $WM:S{m}_k\to DW{M}^{-}(k)$ defining motives of smooth affine varieties for perfect field $k$, $chark\ne 2$ is constructed. In the construction Voevodsky-Suslin method is applyed to a category of $Witt$-correspondence between affine smooth varieties $WCo{r}_k$ that morphisms are defined by class in $Witt$-group of quadratic space $(P,q_P)$ with $P$ being $k[X\times Y]$-module finitely generated projective over $k[X]$ and $q_P:P\to Ho{m}_{k[X]}(P,k[X])$ being $k[X\times Y]$-liner isomorphism. And the natural isomorphism $$Ho{m}_{DW{M}_{eff}^{-}(k)}(WM(X),\mathcal{F}[i])\simeq H_{Nis}^i(X,\mathcal{F})$$ for any smooth affine $X$ and homotopy invariant Nisnevich sheave $\mathcal{F}$ with $Witt$-transfers (that is presheave on the category $WCo{r}_k$ such that its restriction on the category $S{m}_k$ is a sheave) is proved.

Cited by in corpus (1)

• Effective Grothendieck-Witt motives of smooth varieties