A finiteness result for associated primes of local cohomology modules. (English) Zbl 0955.13007

Let \(R\) denote a Noetherian ring, let \(\mathfrak a \subset R\) be an ideal of \(R\) and let \(M\) be a finitely generated \(R\)-module. For an integer \(i \geq 0\) let \(H^i_{\mathfrak a}(M)\) denote the \(i\)-th local cohomology module of \(M\) with respect to \(\mathfrak a.\) The aim of this note is to show that the set of associated primes \(\text{ Ass}_R H^i_{\mathfrak a}(M)\) is finite, whenever all of the modules \(H^j_{\mathfrak a}(M)\) for \(j < i\) are finitely generated. This generalizes the corresponding result for the special case of \(i \leq 2\) shown by M. Brodmann, C. Rotthaus and R. Y. Sharp [“On annihilators and associated primes of local cohomology modules”, J. Pure Appl. Algebra 153, No. 3, 197-227 (2000)]. The full result was shown independently with a different method by K. Khashyarmanesh and Sh. Salarian [Commun. Algebra 27, No. 12, 6191-6198 (1999; Zbl 0940.13013)].
But in fact the authors of the present paper show a slightly more general result: Under the same assumption as before let \(N \subseteq H^i_{\mathfrak a}(M)\) denote a finitely generated submodule. Then the set \(\text{ Ass}_R (H^i_{\mathfrak a}(M)/N)\) is finite. This turns out by a clever induction argument.


13D45 Local cohomology and commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators


Zbl 0940.13013
Full Text: DOI


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