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

In this article, we prove some results on Witt, Grothendieck–Witt (GW) and K-theory of noetherian quasi-projective schemes X  , over affine schemes Spec(A)Spec(A). For integers k≥0k≥0, let CMk(X)CMk(X) denote the category of coherent OXOX-modules FF, with locally free dimension dimV(X)⁡(F)=k=grade(F)dimV(X)⁡(F)=k=grade(F). We prove that there is an equivalence Db(CMk(X))→Dk(V(X))Db(CMk(X))→Dk(V(X)) of the derived categories. It follows that there is a sequence of zig-zag maps K(CMk+1(X))⟶K(CMk(X))⟶∐x∈X(k)K(CMk(Xx))K(CMk+1(X))⟶K(CMk(X))⟶∐x∈X(k)K(CMk(Xx)) of the KK-theory spectra that is a homotopy fibration. In fact, this is analogous to the homotopy fiber sequence of the G-theory spaces of Quillen (see proof of [16, Theorem 5.4]). We also establish similar homotopy fibrations of GW-spectra and GWGW-bispectra, by application of the same equivalence theorem.