zbMATH — the first resource for mathematics

On the cofiniteness of local cohomology modules. (English) Zbl 1141.13014
Let $$I$$ be an ideal of a noetherian ring $$R$$ and let $$M$$ be a finitely generated $$R$$-module. The local cohomology modules $$H^l_I(M)$$ are not finitely generated in general. Even worse, in general, they have infinitely many associated prime ideals. However, there are certain positive finiteness results on local cohomology: For example, a result of M. P. Brodmann and A. Lashgari Faghani [Proc. Am. Math. Soc. 128, No. 10, 2851–2853 (2000; Zbl 0955.13007)] says that if $$H^0_I(M),\ldots H^{t-1}_I(M)$$ are finitely generated and $$N$$ is a finitely generated submodule of $$H^t_I(M)$$, then $$\text{Ass}_R(H^t_I(M)/N)$$ is finite. This paper contains a generalization of the Faghani-Brodmann theorem: The assumptions are weaker in the sense that $$H^0_I(M),\ldots ,H^{t-1}_I(M)$$ and $$N$$ are only required to be minimax (a module is called minimax if it has a finitely generated submodule such that the quotient by this submodule is Artinian; over a complete local ring the minimax modules are precisely the Matlis reflexive ones). The statement in the general version is that $$\operatorname{Hom}_R(R/I,H^t_I(M)/N)$$ is finitely generated (as an immediate consequence, $$H^t_I(M)/N$$ has only finitely many associated prime ideals).

MSC:
 13D45 Local cohomology and commutative rings 14B15 Local cohomology and algebraic geometry 13E05 Commutative Noetherian rings and modules
Keywords:
Local cohomology; cofiniteness
Full Text:
References:
 [1] M. P. Brodmann and A. Lashgari Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2851 – 2853. · Zbl 0955.13007 [2] M. Brodmann, Ch. Rotthaus, and R. Y. Sharp, On annihilators and associated primes of local cohomology modules, J. Pure Appl. Algebra 153 (2000), no. 3, 197 – 227. · Zbl 0968.13010 [3] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cambridge, 1998. · Zbl 0903.13006 [4] Edgar Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), no. 2, 179 – 184. · Zbl 0522.13008 [5] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. · Zbl 0237.14008 [6] Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (\?\?\? 2), North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. · Zbl 0159.50402 [7] Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145 – 164. · Zbl 0196.24301 [8] M. Hellus, On the set of associated primes of a local cohomology module, J. Algebra 237 (2001), no. 1, 406 – 419. · Zbl 1027.13009 [9] Craig Huneke, Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990) Res. Notes Math., vol. 2, Jones and Bartlett, Boston, MA, 1992, pp. 93 – 108. · Zbl 0782.13015 [10] Craig L. Huneke and Rodney Y. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765 – 779. · Zbl 0785.13005 [11] Mordechai Katzman, An example of an infinite set of associated primes of a local cohomology module, J. Algebra 252 (2002), no. 1, 161 – 166. · Zbl 1083.13505 [12] Gennady Lyubeznik, Finiteness properties of local cohomology modules (an application of \?-modules to commutative algebra), Invent. Math. 113 (1993), no. 1, 41 – 55. · Zbl 0795.13004 [13] Gennady Lyubeznik, A partial survey of local cohomology, Local cohomology and its applications (Guanajuato, 1999) Lecture Notes in Pure and Appl. Math., vol. 226, Dekker, New York, 2002, pp. 121 – 154. · Zbl 1061.14005 [14] Thomas Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscripta Math. 104 (2001), no. 4, 519 – 525. · Zbl 0987.13009 [15] Leif Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 417 – 423. · Zbl 0921.13009 [16] Leif Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649 – 668. · Zbl 1093.13012 [17] Le Thanh Nhan, On generalized regular sequences and the finiteness for associated primes of local cohomology modules, Comm. Algebra 33 (2005), no. 3, 793 – 806. · Zbl 1083.13007 [18] Anurag K. Singh, \?-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), no. 2-3, 165 – 176. · Zbl 0965.13013 [19] Thomas Zink, Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring, Math. Nachr. 64 (1974), 239 – 252 (German). · Zbl 0297.13015 [20] Helmut Zöschinger, Minimax-moduln, J. Algebra 102 (1986), no. 1, 1 – 32 (German). · Zbl 0593.13012 [21] Helmut Zöschinger, Über die Maximalbedingung für radikalvolle Untermoduln, Hokkaido Math. J. 17 (1988), no. 1, 101 – 116 (German). · Zbl 0653.13011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.