Cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences

Abstract

The cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences between smooth varieties over an infinite prefect field $k$, $chark\ne 2$, is proved, the isomorphism $$Ho{m}_{{\mathbf{DM}}_{\mathrm{eff}}^{\mathrm{GW}}}(A^{\bullet },B^{\bullet })\simeq Ho{m}_{{\mathbf{DM}}_{\mathrm{eff}}^{\mathrm{GW}}}(A^{\bullet }(1),B^{\bullet }(1)),$$ for $A^{\bullet },B^{\bullet }\in {\mathbf{DM}}_{\mathrm{eff}}^{\mathrm{GW}}(k)$ in the category of effective Grothendieck-Witt-motives constructed in \cite{AD_DMGWeff} is obtained (and similarly for Witt-motives). This implies that the canonical functor ${\Sigma }_{{\mathbb{G}}_m^{\wedge 1}}^{\infty }:{\mathbf{DM}}_{\mathrm{eff}}^{\mathrm{GW}}(k)\to {\mathbf{DM}}^{\mathrm{GW}}(k)$ is fully faithful, where ${\mathbf{DM}}^{\mathrm{GW}}(k)$ is the category of non-effective GW-motives (defined by stabilization of ${\mathbf{DM}}_{\mathrm{eff}}^{\mathrm{GW}}(k)$ along ${\mathbb{G}}_m^{\wedge 1}$) and yields the main property of motives of smooth varieties in the category ${\mathbf{DM}}^{\mathrm{GW}}(k)$: $$Ho{m}_{{\mathbf{DM}}^{\mathrm{GW}}(k)}(M^{GW}(X),{\Sigma }_{{\mathbb{G}}_m^{\wedge 1}}^{\infty }\mathcal{F}[i])\simeq H_{Nis}^i(X,\mathcal{F}),$$ for any smooth variety $X$ and homotopy invariant sheave with GW-transfers $\mathcal{F}$ (and similarly for ${\mathbf{DM}}^{\mathrm{W}}(k)$).

1-th update. Added arXiv-hyperlink for the preprint on "Effective GW-motives of smooth varieties" in reference list. Some editorial changes are made and few typos are corrected. 2-th update. The most technical section 6 is rewritten and split into two sections. Some proofs in other sections are rewritten too. arXiv admin note: text overlap with arXiv:1709.05805

References in corpus (5)

• Finite Chow-Witt correspondences
• The Milnor-Witt motivic ring spectrum and its associated theories
• A cancellation theorem for Milnor-Witt correspondences
• Strictly homotopy invariance of Nisnevich sheaves with GW-transfers
• Effective Grothendieck-Witt motives of smooth varieties

Cited by in corpus (2)

• On the effectivity of spectra representing motivic cohomology theories
• Effective Grothendieck-Witt motives of smooth varieties