×

zbMATH — the first resource for mathematics

Finiteness properties of local cohomology modules for \(\mathfrak a\)-minimax modules. (English) Zbl 1157.13014
Let \(\mathfrak a\) be an ideal of a commutative noetherian ring \(R\) and \(N\) a finitely generated \(R\)-module. In 1992 Huneke has conjectured that for any nonnegative integer \(t\), the local cohomology module \(H^t_{\mathfrak{a}}(N):={\varinjlim}_n\text{Ext}^t_R(R/{\mathfrak{a}}^n,N)\) has only finitely many associated prime ideals. A. K. Singh [Math. Res. Lett. 7, No. 2–3, 165–176 (2000; Zbl 0965.13013)] gave a counterexample to this conjecture. However in many situations, it is shown that local cohomology modules of finitely generated modules have finitely many associated prime ideals. For instance, Brodmann and Lashgari and independently Salarian and Khashyarmanesh have shown that if for a nonnegative integer \(t\), \(H^i_{\mathfrak{a}}(N)\) is finitely generated for all \(i<t\), then the set of associated prime ideals of \(H^t_{\mathfrak{a}}(N)\) is finite [see M. P. Brodmann and A. Lashgari Faghani, Proc. Am. Math. Soc. 128, No. 10, 2851–2853 (2000; Zbl 0955.13007) and K. Khashyarmanesh and Sh. Salarian, Commun. Algebra 27, No. 12, 6191–6198 (1999; Zbl 0940.13013)].
The main achievement of the paper under review is a generalization of the above mentioned result. The authors introduce the notion of \(\mathfrak{a}\)-minimax modules and they examine some of the properties of these modules. According to their definition, an \(R\)-module \(M\) is said to be \(\mathfrak{a}\)-minimax if the Goldie dimension of \(H^0_{\mathfrak{a}}(M/L)\) is finite for all submodules \(L\) of \(M\). The authors prove that if \(M\) is \(\mathfrak{a}\)-minimax \(R\)-module such that for a nonnegative integer \(t\), \(H^i_{\mathfrak{a}}(M)\) is \(\mathfrak{a}\)-minimax for all \(i<t\), then \(\text{Hom}_R(R/\mathfrak{a},H^t_{\mathfrak{a}}(M))\) is \(\mathfrak{a}\)-minimax. This immediately implies that the Goldie dimension of \(H^t_{\mathfrak{a}}(M)\) is finite, and so the set of associated prime ideals of \(H^t_{\mathfrak{a}}(M)\) is finite.

MSC:
13D45 Local cohomology and commutative rings
14B15 Local cohomology and algebraic geometry
13E05 Commutative Noetherian rings and modules
PDF BibTeX XML Cite
Full Text: DOI
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] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. · Zbl 0788.13005
[5] Kamran Divaani-Aazar and Mohammad Ali Esmkhani, Artinianness of local cohomology modules of ZD-modules, Comm. Algebra 33 (2005), no. 8, 2857 – 2863. · Zbl 1090.13012
[6] Edgar Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), no. 2, 179 – 184. · Zbl 0522.13008
[7] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. · Zbl 0237.14008
[8] Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145 – 164. · Zbl 0196.24301
[9] M. Hellus, On the set of associated primes of a local cohomology module, J. Algebra 237 (2001), no. 1, 406 – 419. · Zbl 1027.13009
[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] 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
[12] 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
[13] 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
[14] Gennady Lyubeznik, Finiteness properties of local cohomology modules for regular local rings of mixed characteristic: the unramified case, Comm. Algebra 28 (2000), no. 12, 5867 – 5882. Special issue in honor of Robin Hartshorne. · Zbl 0981.13008
[15] 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
[16] 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
[17] Leif Melkersson, Some applications of a criterion for Artinianness of a module, J. Pure Appl. Algebra 101 (1995), no. 3, 291 – 303. · Zbl 0842.13014
[18] Leif Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649 – 668. · Zbl 1093.13012
[19] 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
[20] Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. · Zbl 0441.18018
[21] Anurag K. Singh, \?-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), no. 2-3, 165 – 176. · Zbl 0965.13013
[22] Wolmer V. Vasconcelos, Divisor theory in module categories, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1974. North-Holland Mathematics Studies, No. 14; Notas de Matemática No. 53. [Notes on Mathematics, No. 53]. · Zbl 0296.13005
[23] Thomas Zink, Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring, Math. Nachr. 64 (1974), 239 – 252 (German). · Zbl 0297.13015
[24] Helmut Zöschinger, Minimax-moduln, J. Algebra 102 (1986), no. 1, 1 – 32 (German). · Zbl 0593.13012
[25] 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.