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 $k\ge 0$, 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 $\mathbf{GW}$-spectra and $\mathbb{G}W$-bispectra.

arXiv commentAdded a subsection on Grothendieck-Witt spectrum and bispectrum. Also added an appendix section on background on these concepts. Title was changed