Triangulated Matlis equivalence

TL;DR

This paper extends classical module category equivalences to a triangulated setting, connecting derived categories of torsion modules and contramodules over a Matlis domain, generalizing to broader ring contexts.

Contribution

It introduces a triangulated equivalence between derived categories of torsion modules and contramodules over a Matlis domain, generalizing previous results to any commutative ring with specific properties.

Findings

Establishes a triangulated equivalence for a broad class of rings.

Connects complexes of modules with torsion and contramodule cohomology.

Simplifies to an equivalence when S consists of nonzero-divisors or torsion is bounded.

Abstract

This paper is a sequel to arXiv:1503.05523 and arXiv:1605.03934. We extend the classical Harrison-Matlis module category equivalences to a triangulated equivalence between the derived categories of the abelian categories of torsion modules and contramodules over a Matlis domain. This generalizes to the case of any commutative ring $R$ with a fixed multiplicative system $S$ such that the $R$-module $S^{-1}R$ has projective dimension $1$. The latter equivalence connects complexes of $R$-modules with $S$-torsion and $S$-contramodule cohomology modules. It takes a nicer form of an equivalence between the derived categories of abelian categories when $S$ consists of nonzero-divisors or the $S$-torsion in $R$ is bounded.