Witt, $GW$, $K$-theory of quasi-projective schemes

Abstract

In this article we continue our investigation of the Derived Equivalences over noetherian quasi-projective schemes $X$, over affine schemes $\spec{A}$. For integers $k\geq 0$, let $C{\mathbb M}^k(X)$ denote the category of coherent ${\CO}_X$-modules ${\mathcal F}$, with locally free dimension $proj\dim(\CF)=k=grade({\mathcal F})$. We prove that there is a zig-zag equivalence ${\mathcal D}}^b\left(C{\mathbb M}^k(X)\right) \to {\mathcal D}^k\left({\mathcal V}(X)\right)$ of the derived categories. It follows that there is a sequence of zig-zag maps ${\mathbb K}\left(C{\mathbb M}^{k+1}(X)\right) \to {\mathbb K}\left(C{\mathbb M}^{k}(X)\right) \to \coprod_{x\in X^{(k)}} {\mathbb K}\left(C{\mathbb M}^{k}(X_x)\right) \\ $of the $\K$-theory spectra that is a homotopy fibration. In fact, this is analogous to the fibrations of the $G$-theory spaces of Quillen (see proof of \cite[Theorem 5.4]{Q}). We also establish similar homotopy fibrations of ${\bf GW}$-spectra and ${\mathbb G}W$-bispectra.