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.
Categories: math.AC math.AG math.KT