Finiteness properties of extension functors of cominimax modules

Abstract

Let $(R,\mathfrak{m})$ be a complete commutative Noetherian local ring, I an ideal of R, M an R-module (not necessarily I-torsion) and N a finitely generated R-module with ${\text{Supp}}_R(N)\subseteq \mathrm{V}(I)$. It is shown that if M is I-ETH-cominimax (i.e. ${\text{Ext}}_R^i(R/I,M)$ is minimax (or Matlis reflexive), for all $i\ge 0$) and $\mathrm{dim\hspace{0.17em}}M\le 1$ or more generally $M\in {\text{FD}}_{\le 1}$, then the R-module ${\text{Ext}}_R^n(M,N)$ is finitely generated, for all $n\ge 0$. As an application to local cohomology, let $\Phi$ be a system of ideals of R and $I\in \Phi$, if $\mathrm{dim\hspace{0.17em}}M/\mathfrak{a}M\le 1$ (e.g., $\mathrm{dim\hspace{0.17em}}R/\mathfrak{a}\le 1$) for all $\mathfrak{a}\in \Phi$, then the R-modules ${\text{Ext}}_R^j({\mathrm{H}}_{\Phi }^i(M),N)$ are finitely generated, for all $i\ge 0$ and $j\ge 0$. Similar results are true for local cohomology defined by a pair of ideals and ordinary local cohomology modules.

Keywords:

• cominimax modules
• local cohomology
• Matlis reflexive modules
• minimax modules

2020 Mathematics Subject Classification:

• 13D45
• 13E05
• 14B15

Acknowledgments

The authors are deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions.