Deforming motivic theories I: Pure weight perfect Modules on divisorial schemes

TL;DR

This paper introduces weight r pseudo-coherent modules on divisorial schemes and establishes a derived Morita equivalence linking perfect complexes supported on a closed subscheme to these modules, impacting K-theory and cyclic homology computations.

Contribution

It defines weight r pseudo-coherent modules and proves a canonical derived Morita equivalence with perfect complexes, connecting various K-theories and cyclic homology theories.

Findings

Established a canonical derived Morita equivalence for divisorial schemes.

Demonstrated isomorphisms between different K-theories and cyclic homology theories.

Applied results to identify generators of topological filtrations on K-theory.

Abstract

In this paper, we introduce a notion of weight r pseudo-coherent Modules associated to a regular closed immersion i:Y -> X of codimension r, and prove that there is a canonical derived Morita equivalence between the DG-category of perfect complexes on a divisorial scheme X whose cohomological support are in Y and the DG-category of bounded complexes of weight r pseudo-coherent O_X-Modules supported on Y. The theorem implies that there is the canonical isomorphism between the Bass-Thomason-Trobaugh non-connected K-theory [TT90], [Sch06] (resp. the Keller-Weibel cyclic homology [Kel98], [Wei96]) for the immersion and the Schlichting non-connected K-theory [Sch04] associated to (resp. that of) the exact category of weight r pseudo-coherent Modules. For the connected K-theory case, this result is just Exercise 5.7 in [TT90]. As its application, we will decide on a generator of the topological filtration on the non-connected K-theory (resp. cyclic homology theory) for affine Cohen-Macaulay schemes.