Pure and Applied Mathematics Journal

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A Study of Some Generalizations of Local Homology

Received: Jul. 04, 2023    Accepted: Jul. 25, 2023    Published: Jul. 28, 2023
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Abstract

Tate local cohomology and Gorenstein local cohomology theory, which are important generalizations of the classical local cohomology, has been investigated. It has been found that they have such vanishing properties and long exact sequences. However, for local homology, what about the duality? In this paper we are concerned with Tate local homology and Gorenstein local homology. In the first part of the paper we generalize local homology as Tate local homology, and study such vanishing properties, artinianness and some exact sequence of Tate local homology modules. We find that for an artianian R-module M and a finitely generated R-module N with finite Gorenstein projective dimension, the Tate local homology module of M and N with respect to an ideal I is also an artinian module. In the second part of the paper we consider Gorenstein local homology modules as Gorenstein version. We discuss vanishing properties and some exact sequences of Gorenstein local homology modules and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. Finally, as an applicaton of the exact sequence connecting these local homology modules, we find that for finitely generated R-modules with finite projective dimension and admitting Gorenstein projective proper resolution respectively, Gorenstein local homology coincides with generalized local homology in certain cases.

DOI 10.11648/j.pamj.20231202.12
Published in Pure and Applied Mathematics Journal ( Volume 12, Issue 2, April 2023 )
Page(s) 34-39
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Tate Homology, Local Homology, Generalized Local Homology, Artinian Module

References
[1] Asadollahi, J. and Salarian, S. (2006). Cohomology theries based on Gorenstein injective modules. Trans Amer Math Soc 358 (5), 2183-2203.
[2] Avramov, L. L. and Martsinkovsky, A. (2002). Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc London Math Soc 85, 393- 440.
[3] Brodmann, M. P. and Sharp, R. Y. (1998). Local cohomology: an algebraic introduction with geometric applications. Cambridge: Cambridge University Press.
[4] Celikbas, O., Liang, L., Sadeghi, A. et al (2023). A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula. Acta Mathematica Sinica, English Series 39 (03), 439-458.
[5] Christensen, L. W. and Jorgensen, D. A. (2014). Tate (co)homology via pinched complexes. Trans Amer Math Soc 366, 667-689.
[6] Christensen, L. W. and Jorgensen, D. A. (2015). Vanishing of Tate homology and depth formula over local rings. J Pure Appl Algebra 219, 464-481.
[7] Enochs, E. E. and Jenda, O. M. G. (2000). Relative homology algebra, Berlin: Walter de Gruyter.
[8] Fatemeh, M. and Kamran, D. (2012). Local homology and Gorenstein flat modules. J Algbra Appl 11 (2), 1250022 (8pages).
[9] Frankild, A. (2003). Vanishing of local homology. Math Z 244, 615-630.
[10] Greenlees, J. P. C. and May, J. P. (1992). Derived functors of I-adic completion and local homology. J Algebra 149, 438-453.
[11] Grothendieck, A. (1966). Local cohomology.New York: Springer-Verlag.
[12] Hartshorne, R. (1977). Algebra Geometry, New York: Springer-Verlag.
[13] Herzog, J. and Zamani, N. (2003). Duality and vaninishing if generalized local cohomology. Arch Math 81, 512-519.
[14] Holm, H. (2004). Gorenstein homological dimensions. J Pure Appl Algebra 189, 167-193.
[15] Iacob, A. (2007). Absolute, Gorenstein, and Tate torsion modules. Comm. Algebra 35, 1589-1606.
[16] Mao, X. and Wang, H. (2023). Local cohomology for Gorenstein homologically smooth DG algebras. Science China (Mathematics) 66 (6), 1161-1176.
[17] Matlis, E. (1974). The Koszul complex and duality. Comm Algebra 1, 87-144.
[18] Matlis, E. (1978). The higher properties of R-sequences. J Algebra 50, 77-112
[19] Nam, T. T. (2012). Generalized local homology for artinian modules. Algbra Colloq 19, 1205-1212.
[20] Moslehi, K. and Ahmadi, M. R. (2014). On the vanishing of Generalized local homology modules and its duality. Rom J Math Comput Sci 4 (1), 44-49.
[21] Nam, T. T. (2010). Left-derived functors of Generalized I-adic completion and Generalized local homology. Comm. Algbra 38 (2), 440-453.
[22] Tarrio, L. A., Lopez, A. J. and Lipman, J. (1997). Local homology and cohomology on schemes. Ann Sci Ecole Norm Sup 30 (1), 1-39.
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    Yanping Liu. (2023). A Study of Some Generalizations of Local Homology. Pure and Applied Mathematics Journal, 12(2), 34-39. https://doi.org/10.11648/j.pamj.20231202.12

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    Yanping Liu. A Study of Some Generalizations of Local Homology. Pure Appl. Math. J. 2023, 12(2), 34-39. doi: 10.11648/j.pamj.20231202.12

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    AMA Style

    Yanping Liu. A Study of Some Generalizations of Local Homology. Pure Appl Math J. 2023;12(2):34-39. doi: 10.11648/j.pamj.20231202.12

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  • @article{10.11648/j.pamj.20231202.12,
      author = {Yanping Liu},
      title = {A Study of Some Generalizations of Local Homology},
      journal = {Pure and Applied Mathematics Journal},
      volume = {12},
      number = {2},
      pages = {34-39},
      doi = {10.11648/j.pamj.20231202.12},
      url = {https://doi.org/10.11648/j.pamj.20231202.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20231202.12},
      abstract = {Tate local cohomology and Gorenstein local cohomology theory, which are important generalizations of the classical local cohomology, has been investigated. It has been found that they have such vanishing properties and long exact sequences. However, for local homology, what about the duality? In this paper we are concerned with Tate local homology and Gorenstein local homology. In the first part of the paper we generalize local homology as Tate local homology, and study such vanishing properties, artinianness and some exact sequence of Tate local homology modules. We find that for an artianian R-module M and a finitely generated R-module N with finite Gorenstein projective dimension, the Tate local homology module of M and N with respect to an ideal I is also an artinian module. In the second part of the paper we consider Gorenstein local homology modules as Gorenstein version. We discuss vanishing properties and some exact sequences of Gorenstein local homology modules and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. Finally, as an applicaton of the exact sequence connecting these local homology modules, we find that for finitely generated R-modules with finite projective dimension and admitting Gorenstein projective proper resolution respectively, Gorenstein local homology coincides with generalized local homology in certain cases.},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - A Study of Some Generalizations of Local Homology
    AU  - Yanping Liu
    Y1  - 2023/07/28
    PY  - 2023
    N1  - https://doi.org/10.11648/j.pamj.20231202.12
    DO  - 10.11648/j.pamj.20231202.12
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 34
    EP  - 39
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20231202.12
    AB  - Tate local cohomology and Gorenstein local cohomology theory, which are important generalizations of the classical local cohomology, has been investigated. It has been found that they have such vanishing properties and long exact sequences. However, for local homology, what about the duality? In this paper we are concerned with Tate local homology and Gorenstein local homology. In the first part of the paper we generalize local homology as Tate local homology, and study such vanishing properties, artinianness and some exact sequence of Tate local homology modules. We find that for an artianian R-module M and a finitely generated R-module N with finite Gorenstein projective dimension, the Tate local homology module of M and N with respect to an ideal I is also an artinian module. In the second part of the paper we consider Gorenstein local homology modules as Gorenstein version. We discuss vanishing properties and some exact sequences of Gorenstein local homology modules and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. Finally, as an applicaton of the exact sequence connecting these local homology modules, we find that for finitely generated R-modules with finite projective dimension and admitting Gorenstein projective proper resolution respectively, Gorenstein local homology coincides with generalized local homology in certain cases.
    VL  - 12
    IS  - 2
    ER  - 

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Author Information
  • College of Economics, Northwest Normal University, Lanzhou, PR China

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