Abstract
We present some finiteness results for co-associated primes of generalized local homology modules. Let M be a finitely generated R-module and N a linearly compact R-module. If N and H I i (N) satisfy the finiteness condition for co-associated primes for all i < k, then Coass R (H I k (M,N)) is a finite set. On the other hand, if H I i (N) = 0 for all i < t and Tor R j (M,H i t (N)) = 0 for all j < h, then Tor R h (M,H I t (N)) ≅ H I h+t (M,N). Moreover, Coass(H I h+t (M,N)) is also a finite set providedN satisfies the finiteness condition for co-associated primes. Finally, N is a semi-discrete linearly compact R-module such that 0: N I ≠ 0. Let t=Width I (N) and h = tor_(M,H I t (N)); it follows that Width I+Ann(M)(N) = t + h and Coass(H I h+t (M,N)) is a finite set.
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References
M. P. Brodmann and A. L. Faghani, “A finiteness result for associated primes of local cohomology modules,” Proc. AMS. 128(10), 2851–2853 (2000).
N. T. Cuong and T. T. Nam, “A Local homology theory for linearly compact modules,” J. Algebra 319, 4712–4737 (2008).
G. Faltings, “Der Endlichkeitssatz in der lokalen Kohomologie,” Math. Ann. 255(1), 45–56 (1981).
J. Herzog, “Komplexe, Auflösungen und dualität in der localen Algebra,” Habilitationsschrift Univ. Regensburg (1970).
C. Huneke, “Problems on local cohomology,” in: Free Resolutions in Commutative Algebra and Algebraic Geometry (Sundance, Utah, 1990), Research Notes in Mathematics (Boston, MA., 1992), Vol. 2, pp. 93–108.
I. G. Macdonald, “Duality over complete local rings,” Topology 1, 213–235 (1962).
A. Mafi, “On the associated primes of generalized local cohomology modules,” Comm. Algebra 34(7), 2489–2494 (2006).
L. Melkersson, “Some applications of a criterion for Artinianness of amodule,” J. Pure Appl. Algebra 101(3), 291–303 (1995).
T. T. Nam, “Left derived functors of the generalized I-adic completion and generalized local homology,” Comm. Algebra 38, 440–453 (2010).
T. T. Nam, “A finiteness result for co-associated and associated primes of generalized local homology and cohomology modules,” Communications in Algebra 37, 1748–1757 (2009).
T. T. Nam, “Generalized local homology,” Proceedings of the 4th Japan-Vietnam Joint Seminar, Meiji University, Japan, 217–235 (2009).
T. T. Nam, “Generalized local homology and cohomology for Artinian modules,” Algebra Colloquium 19(Spec 1), 1205–1212 (2012).
T. T. Nam, “On the finiteness of co-associated primes of local homology modules,” J. Math. Kyoto Univ. (JMKYAZ) 48(3), 521–527 (2008).
A. Ooishi, “Matlis duality and the width of a module,” Hiroshima Math. J. 6, 573–587 (1976).
J. J. Rotman, An introduction to homological algebra, Academic Press, (1979).
A. M. Simon, “Adic-completion and some dual homological results,” PublicationsMatemtiques 36, 965–979 (1992).
A. M. Simon. “Some homological properties of complete modules,” Math. Proc. Cambr. Phil. Soc. 108, 231–246 (1990).
A. K. Singh, I. Swanson, “Associated primes of local cohomology modules and of Frobenius powers,” Int. Math. Res. Not. 33, 1703–1733 (2004).
S. Yassemi, “Coassociated primes,” Comm. Algebra 23, 1473–1498 (1995).
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Nam, T.T., Yen, D.N. The finiteness of coassociated primes of generalized local homology modules. Math Notes 97, 738–744 (2015). https://doi.org/10.1134/S0001434615050089
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DOI: https://doi.org/10.1134/S0001434615050089