Abstract
References
- M. Aghapournahr and L. Melkersson, Local cohomology and Serre subcategories, J. Algebra, 320(3) (2008), 1275-1287.
- M. Asgharzadeh and M.Tousi, A unified approach to local cohomology modules using Serre classes, Canad. Math. Bull., 53(4) (2010), 577-586.
- N. T. Cuong and T. T. Nam, The I-adic completion and local homology for Artinian modules, Math. Proc. Cambridge Philos. Soc., 131(1) (2001), 61-72.
- N. T. Cuong and T. T. Nam, A local homology theory for linearly compact modules, J. Algebra, 319(11) (2008), 4712-4737.
- K. Divaani-Aazar and A. Hajikarimi, Generalized local cohomology modules and homological Gorenstein dimensions, Comm. Algebra, 39(6) (2011), 2051-2067.
- Y. N. Do, T. M. Nguyen and N. T. Tran, Some finiteness results for co-associated primes of generalized local homology modules and applications, J. Korean Math. Soc., 57(5) (2020), 1061-1078.
- S. O. Faramarzi and Z. Barghsouz, Serre subcategory, local homology and local cohomology, J. Algebr. Syst., 7(2) (2020), 301-314.
- J. Herzog, Komplexe, Auosungen und Dualitat in der Localen Algebra, Habilitationschrift Univ. Regensburg, 1970.
- C. U. Jensen, Les Foncteurs Derives de $\varprojlim$ et Leurs Applications en Theorie des Modules, Lecture Notes in Mathematics, 254, Springer-Verlag, Berlin-New York, 1972.
- I. G. Macdonald, Duality over complete local rings, Topology, 1 (1962), 213-235.
- L. Melkersson and P. Schenzel, The co-localization of an Artinian module, Proc. Edinburgh Math. Soc., 38(1) (1995), 121-131.
- T. T. Nam, A finiteness result for co-associated and associated primes of generalized local homology and cohomology modules, Comm. Algebra, 37(5) (2009), 1748-1757.
- T. T. Nam, Generalized local homology for Artinian modules, Algebra Colloq., 19(1) (2012), 1205-1212.
- R. N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings, Quart. J. Math. Oxford Ser. (2), 26(103) (1975), 269-273.
- S. Yassemi, Generalized section functors, J. Pure Appl. Algebra, 95(1) (1994), 103-119.
- D. N. Yen and T. T. Nam, Generalized local homology and duality, Internat. J. Algebra Comput., 29(3) (2019), 581-601.
- H. Zoschinger, Minimax-moduln, J. Algebra, 102(1) (1986), 1-32.
Abstract
This paper deals with generalized local homology and generalized local cohomology modules belong to a Serre category of the category of $R$-modules under some conditions. For an ideal $I$ of $R$, the concept of the condition $C^I$ on a Serre category which is dual to the condition $C_I$ of Melkersson is defined. As a main result, it is shown that for a finitely generated $R$-module $M$ with $pd(M) <\infty$ and a minimax $R$-module $N$ of any Serre category $\mathcal{S}$ satisfying the condition $C^I$, the generalized local homology $\text{H}^I_i(M,N)$ belongs to $\mathcal{S}$ for all $i>pd(M)$. Also, if $\mathcal{S}$ satisfies the condition $C_I$, then the generalized local cohomology module
$\text{H}^i_I(M,N)\in \mathcal{S}$ for all $i>pd(M)$.
Keywords
Generalized local homology module generalized local cohomology module Serre subcategory minimax module condition C^I
References
- M. Aghapournahr and L. Melkersson, Local cohomology and Serre subcategories, J. Algebra, 320(3) (2008), 1275-1287.
- M. Asgharzadeh and M.Tousi, A unified approach to local cohomology modules using Serre classes, Canad. Math. Bull., 53(4) (2010), 577-586.
- N. T. Cuong and T. T. Nam, The I-adic completion and local homology for Artinian modules, Math. Proc. Cambridge Philos. Soc., 131(1) (2001), 61-72.
- N. T. Cuong and T. T. Nam, A local homology theory for linearly compact modules, J. Algebra, 319(11) (2008), 4712-4737.
- K. Divaani-Aazar and A. Hajikarimi, Generalized local cohomology modules and homological Gorenstein dimensions, Comm. Algebra, 39(6) (2011), 2051-2067.
- Y. N. Do, T. M. Nguyen and N. T. Tran, Some finiteness results for co-associated primes of generalized local homology modules and applications, J. Korean Math. Soc., 57(5) (2020), 1061-1078.
- S. O. Faramarzi and Z. Barghsouz, Serre subcategory, local homology and local cohomology, J. Algebr. Syst., 7(2) (2020), 301-314.
- J. Herzog, Komplexe, Auosungen und Dualitat in der Localen Algebra, Habilitationschrift Univ. Regensburg, 1970.
- C. U. Jensen, Les Foncteurs Derives de $\varprojlim$ et Leurs Applications en Theorie des Modules, Lecture Notes in Mathematics, 254, Springer-Verlag, Berlin-New York, 1972.
- I. G. Macdonald, Duality over complete local rings, Topology, 1 (1962), 213-235.
- L. Melkersson and P. Schenzel, The co-localization of an Artinian module, Proc. Edinburgh Math. Soc., 38(1) (1995), 121-131.
- T. T. Nam, A finiteness result for co-associated and associated primes of generalized local homology and cohomology modules, Comm. Algebra, 37(5) (2009), 1748-1757.
- T. T. Nam, Generalized local homology for Artinian modules, Algebra Colloq., 19(1) (2012), 1205-1212.
- R. N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings, Quart. J. Math. Oxford Ser. (2), 26(103) (1975), 269-273.
- S. Yassemi, Generalized section functors, J. Pure Appl. Algebra, 95(1) (1994), 103-119.
- D. N. Yen and T. T. Nam, Generalized local homology and duality, Internat. J. Algebra Comput., 29(3) (2019), 581-601.
- H. Zoschinger, Minimax-moduln, J. Algebra, 102(1) (1986), 1-32.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | February 16, 2025 |
| Publication Date | |
| Submission Date | October 13, 2024 |
| Acceptance Date | January 20, 2025 |
| Published in Issue | Year 2025 Early Access |