文摘
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). For integers k≥0, let 6e18b4866c69bdffde86af8f1db8" title="Click to view the MathML source">CMk(X) denote the category of coherent OX-modules F, with locally free dimension dimV(X)(F)=k=grade(F). We prove that there is an equivalence e55e32d219b4cbed0fb27459c5aa" title="Click to view the MathML source">Db(CMk(X))→Dk(V(X)) of the derived categories. It follows that there is a sequence of zig-zag maps 6e46eb79e93a3c682663a646" title="Click to view the MathML source">K(CMk+1(X))⟶K(CMk(X))⟶∐x∈X(k)K(CMk(Xx)) of the K-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 46db0a61ff2" title="Click to view the MathML source">GW-bispectra, by application of the same equivalence theorem.