Applications of reduced and coreduced modules III: homological properties and coherence of functors

TL;DR

This paper explores how reduced and coreduced modules over a commutative ring can simplify the computation of local cohomology and homology, and establish conditions for functor coherence, linking module properties to cohomological dimensions.

Contribution

It extends the theory of reduced and coreduced modules by applying them to local (co)homology computations and functor coherence, connecting module properties with cohomological dimensions.

Findings

Reduced modules facilitate local cohomology computation.

Coreduced modules assist in local homology calculation.

Cohomological dimension equals projective or flat dimension under certain conditions.

Abstract

This is the third in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show in different settings that $I$-reduced (resp. $I$-coreduced) $R$-modules facilitate the computation of local cohomology (resp. local homology) and provide conditions under which the $I$-torsion functor as well as the $I$-transform functor (resp. their duals) become coherent. We show that whenever every $R$-module is $I$-reduced (resp. $I$-coreduced), the cohomological dimension (resp. dual of the cohomological dimension) of an ideal $I$ of a ring $R$ coincides with the projective (resp. flat) dimension of the $R$-module $R/I$.