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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
Citations:
Zbl 0940.13013
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References:
[1] Brodmann, M., Rotthaus, Ch., Sharp, R., On annihilators and associated primes of local cohomology modules, to appear in Journal of Pure and Applied Algebra · Zbl 0968.13010
[2] Brodmann, M, Sharp, R., Local cohomology - an algebraic introduction with geometric applications, Cambridge studies in advanced mathematics No. 60, Cambridge University Press, 1998 CMP 98:10 · Zbl 0903.13006
[3] Gerd Faltings, Der Endlichkeitssatz in der lokalen Kohomologie, Math. Ann. 255 (1981), no. 1, 45 – 56 (German). · Zbl 0451.13008
[5] Leif Melkersson and Peter Schenzel, Asymptotic prime ideals related to derived functors, Proc. Amer. Math. Soc. 117 (1993), no. 4, 935 – 938. · Zbl 0780.13003
[6] K. N. Raghavan, Local-global principle for annihilation of local cohomology, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 329 – 331. · Zbl 0818.13009
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