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 $k$, $char k \neq 2$, is proved, the isomorphism $$Hom_{\mathbf{DM}^\mathrm{GW}_\mathrm{eff}}(A^\bullet,B^\bullet) \simeq Hom_{\mathbf{DM}^\mathrm{GW}_\mathrm{eff}}(A^\bullet(1),B^\bullet(1)),$$ for $A^\bullet,B^\bullet\in \mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)$ 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 $\Sigma_{\mathbb G_m^{\wedge 1}}^\infty\colon \mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)\to \mathbf{DM}^\mathrm{GW}(k)$ is fully faithful, where $\mathbf{DM}^\mathrm{GW}(k)$ is the category of non-effective GW-motives (defined by stabilization of $\mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)$ along $\mathbb G_m^{\wedge 1}$) and yields the main property of motives of smooth varieties in the category $\mathbf{DM}^\mathrm{GW}(k)$: $$ Hom_{\mathbf{DM}^\mathrm{GW}(k)}(M^{GW}(X), \Sigma_{\mathbb G_m^{\wedge 1}}^\infty\mathcal F[i]) \simeq H^i_{Nis}(X,\mathcal F) ,$$ for any smooth variety $X$ and homotopy invariant sheave with GW-transfers $\mathcal F$ (and similarly for $\mathbf{DM}^\mathrm{W}(k)$).