Localization problems of Quillen

Abstract

Let $X$ be a quasi projective scheme over a noetherian affine scheme $Spec(A)$, $U\subseteq X$ be an open subset, and $Z=X-U$.Assume that $Z$ is complete intersection, with $k=codimZ$. Consider the map $$q:\mathbb{K}\left(\mathcal{V}(X)\right)\to \mathbb{K}\left(\mathcal{V}(U)\right)$$ of the $\mathbb{K}$-theory spectra. We give a description of the homotopy fiber of $q$. Let $C{\mathbb{M}}^Z\left(X\right)$ denote the full subcategory of perfect modules $\mathcal{F}\in Coh(X)$ such that(1) ${\mathcal{F}}_{|U}=0$, (2) $grade(\mathcal{F})={\dim }_{\mathcal{V}(X)}\mathcal{F}=k$. It turns out that the homotopy fiber of $q$ is the $\mathbb{K}$-theory spectra $\mathbb{K}\left(C{\mathbb{M}}^Z\left(X\right)\right)$. Likewise, we compute the homotopy fiber of the pullback map $$g:\mathbb{G}W\left(\mathcal{V}(X)\right)\to \mathbb{G}W\left(\mathcal{V}(U)\right)$$ of Karoubi Grothendieck-Witt bispectra. Consequently, we obtain long exact sequences of $\mathbb{K}$-groups and of $\mathbb{G}W$-groups. These results settle some of the long standing open problems. We also inserted a conjecture.

Corrected, edited, inserted a conjecture