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 $k=codim Z$. Consider the map $$ q:{\mathbb K}\left({\mathscr V}(X)\right) \rightarrow {\mathbb K}\left({\mathscr V}(U)\right) $$ 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 ${\mathscr F} \in Coh(X)$ such that(1) ${\mathscr F} _{|U}=0$, (2) $grade({\mathscr F} )=\dim_{{\mathscr V}(X)}{\mathscr 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 $$ g: {\mathbb G}W\left({\mathscr V}(X)\right) \rightarrow {\mathbb G}W\left({\mathscr V}(U)\right) $$ 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.