Applications of reduced and coreduced modules I

TL;DR

This paper explores the applications of reduced and coreduced modules over a commutative ring, demonstrating their role in establishing key dualities and equivalences in module theory.

Contribution

It introduces the use of $I$-reduced and $I$-coreduced modules to realize important dualities and equivalences traditionally known in derived categories.

Findings

Establishes that $I$-reduced and $I$-coreduced modules support MGM equivalence and GM duality.

Realizes $I$-torsion and $I$-adic completion functors as representable.

Computes natural transformations between these functors under certain conditions.

Abstract

This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.