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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.

##### MSC:
 13D45 Local cohomology and commutative rings 13E15 Commutative rings and modules of finite generation or presentation; number of generators
Zbl 0940.13013
Full Text:
##### References:
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