Effective Grothendieck-Witt motives of smooth varieties

Abstract

The category of effective Grothendieck-Witt-motives ${\mathbf{DM}}_{\mathrm{eff},-}^{GW}(k)$ (and Witt-motives ${\mathbf{DM}}_{\mathrm{eff},-}^W(k)$) by Voevodsky-Suslin method starting with some category of GW-correspondences (and Witt-correspondences) over a perfect field $k$, $chark\ne 2$, is defined. The functor $M_{eff}^{GW}:S{m}_k\to {\mathbf{DM}}_{\mathrm{eff},-}^{GW}(k)$ of Grothendieck-Witt-motives of smooth varieties is computed and it is proved that for any smooth scheme $X$ and homotopy invariant sheave with GW-transfers $F$ $$Ho{m}_{{\mathbf{DM}}_{\mathrm{eff},-}^{GW}(k)}(M_{eff}^{GW}(X),F[i])\simeq H_{nis}^i(X,F)$$ naturally in $X$ and $F$.

arXiv admin note: text overlap with arXiv:1601.05383, one typos changed (in Grothendieck's name). Several sections are rewritten

References in corpus (5)

• Finite Chow-Witt correspondences
• On the effectivity of spectra representing motivic cohomology theories
• Strictly homotopy invariance of Nisnevich sheaves with GW-transfers
• Triangulated category of effective $Witt$-motives $DW{M}_{eff}^{-}(k)$
• Cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences

Cited by in corpus (3)

• On the effectivity of spectra representing motivic cohomology theories
• Cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences
• Zariski-local framed ${\mathbb{A}}^1$-homotopy theory