zbMATH — the first resource for mathematics

Associated primes of local cohomology modules. (English) Zbl 1103.13010
For an ideal $$a$$ of a noetherian ring $$R$$ and $$R$$-modules $$M,N$$ generalized local cohomology is defined as $$H^i_a(M,N)=\varinjlim _n \text{Ext}^i_R(M/a^nM,N)$$; for $$M=R$$ one gets (usual) local local cohomology. Results about the associated primes of generalized local cohomology are proven, partially extensions of results known for the case of usual local cohomology. It is shown that for any natural number $$t$$ one has $\text{Ass}_R(H^t_a(M,N))\subseteq \bigcup_{i=0}^t \text{Ass}_R (\text{Ext}^i_R(M,H^{t-i}_a(N)));$ in addition, if both $$d=\text{pd}(M)$$ and $$n=\dim(N)$$ are finite, the set $$\text{Ass}_R(H^{n+d}_a(M,N))$$ is finite.

MSC:
 13D45 Local cohomology and commutative rings 13E99 Chain conditions, finiteness conditions in commutative ring theory
Full Text:
References:
 [1] Richard G. Belshoff, Edgar E. Enochs, and Juan Ramon García Rozas, Generalized Matlis duality, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1307 – 1312. · Zbl 0958.13005 [2] 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 [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] Carl Faith and Dolors Herbera, Endomorphism rings and tensor products of linearly compact modules, Comm. Algebra 25 (1997), no. 4, 1215 – 1255. · Zbl 0877.16001 [6] Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145 – 164. · Zbl 0196.24301 [7] 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 [8] Ken-Ichiroh Kawasaki, Cofiniteness of local cohomology modules for principal ideals, Bull. London Math. Soc. 30 (1998), no. 3, 241 – 246. · Zbl 0930.13013 [9] K. Khashyarmanesh and Sh. Salarian, On the associated primes of local cohomology modules, Comm. Algebra 27 (1999), no. 12, 6191 – 6198. · Zbl 0940.13013 [10] 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 [11] 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 [12] Anurag K. Singh, \?-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), no. 2-3, 165 – 176. · Zbl 0965.13013 [13] Weimin Xue, Rings with Morita duality, Lecture Notes in Mathematics, vol. 1523, Springer-Verlag, Berlin, 1992. · Zbl 0790.16009 [14] Ken-Ichi Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179 – 191. · Zbl 0899.13018
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.