Localization problems of Quillen

Abstract

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