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

TL;DR

This paper constructs the category of effective Witt-motives for smooth affine varieties over a perfect field, extending Voevodsky's approach using Witt-correspondences and establishing a key isomorphism with Nisnevich cohomology.

Contribution

It introduces a new category of Witt-motives using Witt-correspondences and proves a fundamental isomorphism relating morphisms in this category to Nisnevich cohomology.

Findings

Construction of the effective Witt-motive category $DWM^-_{eff}(k)$.

Establishment of an isomorphism between Hom in $DWM^-_{eff}(k)$ and Nisnevich cohomology.

Application of Voevodsky-Suslin method to Witt-correspondences.

Abstract

The category of effective $Witt$-motives $DWM^-(k)$ with functor $WM\colon Sm_k\to DWM^-(k)$ defining motives of smooth affine varieties for perfect field $k$, $char k\neq 2$ is constructed. In the construction Voevodsky-Suslin method is applyed to a category of $Witt$-correspondence between affine smooth varieties $WCor_k$ 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 $q_P\colon P\to Hom_{k[X]}(P,k[X])$ being $k[X\times Y]$-liner isomorphism. And the natural isomorphism $$Hom_{DWM^-_{eff}(k)}(WM(X),\mathcal F[i]) \simeq H^i_{Nis}(X,\mathcal F) $$ for any smooth affine $X$ and homotopy invariant Nisnevich sheave $\mathcal F$ with $Witt$-transfers (that is presheave on the category $WCor_k$ such that its restriction on the category $Sm_k$ is a sheave) is proved.