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

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 $R/I$.