On the Cofiniteness of Certain Local Cohomology Modules for a Pair of Ideals
Main Article Content
Abstract
Let be non-negative integers and be an -module such that is finitely generated for all and for all , where is a class of modules. We hence prove that (1) if then is -cofinite for all ; (2) if and is finitely generated for all , then is -cofinite for all . These extend the results of Khazaei-Sazeedeh [10, Thm 2.10, Thm 2.11] for local cohomology modules for a pair of ideals. Finally, we prove that is -cofinite for all whenever is principal, is an -module satisfying is finitely generated for all , and is in dimension . This extends a theorem [13, Thm 1] of Kawasaki.
Keywords:
: cofinite module, local cohomology, local cohomology for a pair of ideals, in dimension < 2 module. MSC2020-Mathematics Subject Classification System: 13D45, 14B15, 13E05.
References
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Vol. 147, 1997, pp. 179-191, https://doi.org/10.1017/S0027763000006371.
D. Delfino, T. Marley, Cofinite Modules and Local Cohomology, J. Pure Appl. Alg. Vol. 121, No. 1, 1997,
pp. 45-52, https://doi.org/10.1016/S0022-4049(96)00044-8.
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N. V. Hoang, N. T. Ngoan, On the Cofiniteness of Local Cohomology Modules in Dimension < 2, Hokkaido Math. J, Vol. 52, No. 1, 2023, pp. 65-73, https://doi.org/10.14492/hokmj/2020-428.
M. Khazaei, R. Sazeedeh, A Criterion for Cofiniteness of Modules, Rend. Sem. Mat. Univ. Padova, Vol. 151, 2024, pp. 201-211, https://doi.org/10.4171/rsmup/128.
A. Tehranian, A. P. E. Talemi, Cofiniteness of Local Cohomology Based on A Nonclosed Support Defined by A Pair of Ideals, Bull. Iranian Math. Soc. Vol. 36, No. 02, 2010, pp. 145-155, http://bims.iranjournals.ir/article_16_5c327b3a154ef3140eee1b10988a0689.pdf (accessed on: June 21st, 2024).
D. Asadollahi, R. Naghipour, Faltings’ Local-global Principle for the Finiteness of Local Cohomology Modules, Comm. Alg. Vol. 43, No. 3, 2015, pp. 953-958, https://doi.org/10.1080/00927872.2013.849261.
K. I. Kawasaki, Cofiniteness of Local Cohomology Modules for Principal Ideals, Bull. London Math. Soc,
Vol. 30, No. 3, 1998, pp. 241-246, https://doi.org/10.1112/S0024609397004347.
N. V. Hoang, A Note on the Cofiniteness of Local Cohomology Modules for A Pair of Ideals, TNU Journal of Science and Technology, Vol. 229, No. 6, 2024, pp. 75-81, https://doi.org/10.34238/tnu-jst.9517.
M. Aghpournahr, Cofiniteness of Local Cohomology Modules for A Pair of Ideals for Small Dimensions, J. Alg and Its App, Vol. 17, No. 2, 2018, pp. 1850020, https://doi.org/10.1142/S0219498818500202.
N. T. Ngoan, N. V. Hoang, N. H. Hoang, On the Cofiniteness of In Dimension < 2 Local Cohomology Modules for A Pair of Ideals, East-West J. of Math, Vol. 24, No. 2, 2023, pp. 118-127, http://eastwestmath.org/index.php/ewm/article/view/490/392 (accessed on: June 21st, 2024).
M. P. Brodmann, R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometry Applications, Second Edition, Cambridge University Press, 2013.
L. Melkersson, Modules Cofinite with Respect to an Ideal, J. Alg, Vol. 285, No. 2, 2005, pp. 649-668, https://doi.org/10.1016/j.jalgebra.2004.08.037.