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

David Ssevviiri

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 $R/I$.

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

Paper Structure

This paper contains 9 sections, 18 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Some preliminary results
  3. Representability of functors
  4. Vanishing of local cohomology of $D_I(M)$
  5. Computation of local (co)homology
  6. Coherence of functors
  7. Further computation of local (co)homology
  8. Local (co)homology modulo exact zero-divisors
  9. Local (co)homology modulo regular elements

Key Result

Theorem 1.1

If every $R$-module is $I$-reduced, then:

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Corollary 2.8
  • ...and 22 more