Finiteness of extension functors of generalized local cohomology modules

Abstract

Let R be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of R, M and X finite R–modules, and t a non-negative integer such that ${\mathrm{H}}_{\mathfrak{a}}^i(M,X)$ is $\mathfrak{a}$–cofinite for all i < t. In this article, we show that ${\text{Hom}}_R(R/\mathfrak{a},{\mathrm{H}}_{\mathfrak{a}}^t(M,X))$ and ${\text{Ext}}_R^1(R/\mathfrak{a},{\mathrm{H}}_{\mathfrak{a}}^t(M,X))$ are finite, and ${\text{Ext}}_R^2(R/\mathfrak{a},{\mathrm{H}}_{\mathfrak{a}}^t(M,X))$ is finite if and only if ${\text{Hom}}_R(R/\mathfrak{a},{\mathrm{H}}_{\mathfrak{a}}^{t+1}(M,X))$ is finite. As a consequence, we observe that ${\text{Ass}}_R({\mathrm{H}}_{\mathfrak{a}}^t(M,X))$ is a finite set.

Keywords:

• Associated prime ideals
• cofinite modules
• extension functors
• generalized local cohomology modules

2010 Mathematics Subject Classification:

• 13D07
• 13D45

Acknowledgements

The authors would like to thank the referee for the invaluable comments on the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research of Alireza Vahidi was in part supported by a grant from Payame Noor University.