Journal of Mathematics of Kyoto University

Some results on local cohomology modules defined by a pair of ideals

Abstract

Let $R$ be a commutative Noetherian ring, and let $I$ and $J$ be two ideals of $R$. Assume that $R$ is local with the maximal ideal ${\mathfrak{m}}$, we mainly prove that (i) there exists an equality ${\text{inf}}\{i\, \mid H_{I,J}^i(M)\, {\text{ is not Artinian}} \}={\text{inf}}\{ {\text{depth}}M_{\mathfrak{p}} \mid \, {\mathfrak{p}}\in W(I, J)\backslash \{{\mathfrak{m}}\} \}$ for any finitely generated $R-$module $M$, where $W(I, J)=\{{\mathfrak{p}} \in {\text{Spec}}(R) \mid \, I^n \subseteq {\mathfrak{p}}+J\,\, {\text{for some positive integer}} \,n \}$; (ii) for any finitely generated $R-$module $M$ with ${\text{dim}}M=d$, $H_{I,J}^d(M)$ is Artinian. Also, we give a characterization to the supremum of all integers $r$ for which $H_{I,J}^r(M) \neq 0$.

Article information

Source
J. Math. Kyoto Univ., Volume 49, Number 1 (2009), 193-200.

Dates
First available in Project Euclid: 30 July 2009

https://projecteuclid.org/euclid.kjm/1248983036

Digital Object Identifier
doi:10.1215/kjm/1248983036

Mathematical Reviews number (MathSciNet)
MR2531134

Zentralblatt MATH identifier
1174.13024

Subjects