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 ≥ 0, let CM k (X) denote the category of coherent O X -modules F , with locally free dimension dim V (X) (F ) = k = grade(F ). We prove that there is a zig-zag equivalence D b CM k (X) → D k (V (X)) of the derived categories. It follows that there is a sequence of zig-zag maps K CM k+1 (X)of the Ktheory spectra that is a homotopy fibration. In fact, this is analogous to the fibrations of the G-theory spaces of Quillen (see proof of [Q, Theorem 5.4]). We also establish similar homotopy fibrations of GW-spectra and GW -bispectra. * Partially supported by a General Research Grant (no 2301857) from U. of Kansas localization theorems, applicable to non-regular schemes. Further, while developments in Grothendieck-Witt theory (GW -theory) and Witt theory followed the foot prints of Ktheory ([S3, B1]), due to lack of any natural duality on Coh(X), the situation in these two areas appear even less complete. When, X is non-regular, the category M(X) of coherent sheaves with finite V (X)-dimension differs from Coh(X). There appears to be a gap in the literature of K-theory, GW -theory, and Witt theory, with respect to the place of the category M(X). One can speculate, whether this lack of completeness is attributable to this gap? The goal of this one and the related articles is to work on this gap and attempt to establish the said literature on non-regular schemes at the same pedestal as that of regular schemes. For quasi-projective schemes over noetherian affine schemes, this goal is accomplished to up some degree of satisfaction. The special place of the full subcategory CM k (X) ⊂ M(X) would also be clear subsequently, where for integers k ≥ 0, CM k (X) will denote the full subcategory of objects F in M(X), with dim V (X) (F ) = grade (F ) = k.With respect to a certain facets of these three areas, namely Algebraic K-theory, Grothendieck-Witt (GW ) theory and Witt theory, a common thread between them is their invariance properties with respect equivalences of the associated Derived categories. We review some of the results on such invariances. For example, recall the theorem of Thomanson-Trobaugh ([TT, Theorem 1.9.8]): suppose A → B is a functor of complicial exact categories with weak equivalences. Assume that the associated functor of the triangulated categories T A → T B is an equivalence. Then, the induced map K(A) → K(B) of the K-theory spaces is a homotopy equivalence (see [S1, 3.2.24]). The non-connective version of this theorem was given by Schlichting ([S2, Theorem 9], also see [S1, 3.2.29]): which states that under the relaxed hypothesis that, if T A → T B is an equivalence up to factors, then it induces a homotopy equivalence K(A) → K(B) of the K-theory spectra. While K-theory is defined for complicial exact categories with weak equivalences, Schlichting defined Grotherndieck Witt (GW ) spectra and bispectra ([S3], also see §A) of pointed dg categorie...

Convergence of two obstructions for projective modules

“…are referred to[M1, Remark 4.14(1)], and the GW analogues of the diagram corresponding to[M1, Remark 4.5(1)].…”