Finite homological dimension and a derived equivalence

TL;DR

This paper establishes a derived equivalence between categories of modules over Cohen-Macaulay rings, leading to new isomorphisms in cohomology theories and improvements in spectral sequence computations.

Contribution

It demonstrates a novel derived equivalence for Cohen-Macaulay rings, connecting categories of finite length and projective modules with finite homological dimensions.

Findings

Derived equivalence of certain subcategories over Cohen-Macaulay rings

Isomorphisms of generalized cohomology groups like K-theory

Enhanced spectral sequence and Gersten complex results

Abstract

For a Cohen-Macaulay ring $R$, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite projective dimension modules with the bounded derived category of projective modules with finite length homologies. This yields isomorphisms of various generalized cohomology groups (like K-theory) and improves on terms of a spectral sequence and Gersten complexes.