Colocalization and cotilting for commutative noetherian rings

TL;DR

This paper explores the relationship between tilting and cotilting modules over commutative noetherian rings, establishing correspondences via colocalization and localization techniques.

Contribution

It introduces a novel correspondence between n-cotilting modules and compatible families of localizations and colocalizations, expanding understanding of module relations over such rings.

Findings

Established a bijective correspondence for cotilting modules.

Constructed a similar but incomplete correspondence for tilting modules.

Provided explicit formulas for cotilting module correspondences.

Abstract

For a commutative noetherian ring R, we investigate relations between tilting and cotilting modules in Mod-R and Mod-R_m where m runs over the maximal spectrum of R. For each finite n, we construct a 1-1 correspondence between (equivalence classes of) n-cotilting R-modules C and (equivalence classes of) compatible families F of n-cotilting R_m-modules (m \in mSpec R). It is induced by the assignment C |-> (C^m ; m \in mSpec R) where C^m is the colocalization of C at m, and its inverse F |-> \prod_{M \in F} M. We construct a similar correspondence for n-tilting modules using compatible families of localizations; however, there is no explicit formula for the inverse.