Review ArticleNo Access

Coserreness with respect to specialization closed subsets and some Serre subcategories

Majid Rahro Zargar

Majid Rahro Zargar

Department of Engineering Sciences, Faculty of Advanced Technologies, University of Mohaghegh Ardabili, Namin, Iran

E-mail Address: zargar9077@gmail.com

E-mail Address: m.zargar@uma.ac.ir

Search for more papers by this author

https://doi.org/10.1142/S0219498823300015Cited by:0 (Source: Crossref)

Abstract

Let $𝒵$ be a specialization closed subset of $Spec R$ and $𝒮$ be a Serre subcategory of $Mod R$ . As a generalization of the notion of cofiniteness, we introduce the concept of $𝒵$ -coserreness with respect to $𝒮$ (see Definition 4.1). First, as a main result, for some special Serre subcategories $𝒮$ , we show that an $R$ -module $M$ with ${dim}_{R} M \leq 1$ is $𝒵$ -coserre with respect to $𝒮$ if and only if ${Ext}_{R}^{0, 1} (R / 𝔞, M) \in 𝒮$ for all ideals $𝔞 \in F (𝒵)$ . Indeed, this result provides a partial answer to a question that was recently raised in [K. Bahmanpour, R. Naghipour and M. Sedghi, Modules cofinite and weakly cofinite with respect to an ideal, J. Algebra Appl. 16 (2018) 1–17]. As an application of this result, we show that the category of $𝒵$ -coserre $R$ -modules $M$ with ${dim}_{R} M \leq 1$ is a full Abelian subcategory of $Mod R$ . Also, for every homologically bounded $R$ -complex $X$ whose homology modules belong to $𝒮$ we show that the local cohomology modules $H_{𝒵}^{i} (X)$ for all $i$ , are $𝒵$ -coserre in all the cases where ${dim}_{R} 𝒵 \leq 1$ , ${dim}_{R} X \leq 2 - sup X$ and $cd (𝒵, X) \leq 1 - sup X$ .

Communicated by D. Herbera

Keywords:

AMSC: 13D45, 13D07, 13D09

References

1. K. Bahmanpour, On the category of weakly Laskarian cofinite modules, Math. Scand. 115(1) (2014) 62–68. Crossref, Google Scholar
2. K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (2009) 1997–2011. Crossref, Web of Science, Google Scholar
3. K. Bahmanpour, R. Naghipour and M. Sedghi, On the category of cofinite modules which is Abelian, Proc. Amer. Math. Soc. 142 (2014) 1101–1107. Crossref, Web of Science, Google Scholar
4. K. Bahmanpour, R. Naghipour and M. Sedghi, Modules cofinite and weakly cofinite with respect to an ideal, J. Algebra Appl. 16 (2018) 1–17. Google Scholar
5. M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge University Press, Cambridge, 1998). Crossref, Google Scholar
6. L. W. Christensen, Gorenstein Dimensions, Lecture Notes in Mathematics, Vol. 1747 (Springer-Verlag, Berlin, 2000). Crossref, Google Scholar
7. L. W. Christensen, H.-B. Foxby and H. Holm, Derived category methods in commutative algebra, http://www.math.ttu.edu/lchriste/download/dcmca.pdf. Google Scholar
8. D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997) 45–52. Crossref, Web of Science, Google Scholar
9. K. Divaani-Aazar, H. Faridian and M. Tousi, A new outlook on cofiniteness, preprint (2017), arXiv:1701.07716 [math.AC]. Google Scholar
10. K. Divaani-Aazar, H. Faridian and M. Tousi, Stable under specialization sets and cofiniteness, J. Algebra Appl. 18(1) (2019) 1950015. Link, Web of Science, Google Scholar
11. K. Divaani-Aazar, H. Faridian and M. Tousi, Local homology, koszul homology and Serre classes, Rocky Mountain J. Math. 48(6) (2018) 1841–1869. Crossref, Web of Science, Google Scholar
12. K. Divaani-Aazar and M. Rahro Zargar, The derived category analogues of Faltings’ Local-global Principle and Annihilator Theorems, J. Algebra Appl. 18(7) (2019) 1950140. Link, Web of Science, Google Scholar
13. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra (Walter De Gruyter, Berlin, New York, 2000). Crossref, Google Scholar
14. A. Grothendieck, Cohomologie Locale des Faisceaux Coh érents et théorémes de Lefschetz Locaux et Globaux, (SGA 2) (North-Holland, Amsterdam, 1968). Google Scholar
15. R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970) 145–164. Crossref, Web of Science, Google Scholar
16. J. Herzog, Komplex auflösungen und dualität in der lokalen algebra, Habilitationsschrift, Universität Regensburg (1974). Google Scholar
17. Y. Irani, Cominimaxness with respect to ideals of dimension one, Bull. Korean Math. Soc. 54(1) (2017) 289–298. Crossref, Web of Science, Google Scholar
18. R. Kanda, Classifying Serre subgategories via atom spectrum, Adv. Math. 231 (2012) 1572–1588. Crossref, Web of Science, Google Scholar
19. K. I. Kawasaki, On the category of cofinite modules which is Abelian, Math. Z. 269 (2011) 587–608. Crossref, Web of Science, Google Scholar
20. J. Lipman, Lectures on local cohomology and duality, Local Cohomology and its Applications (Guanajuato, 1999), Lecture Notes in Pure and Applied Mathematics, Vol. 226 (Dekker, New York, 2002), pp. 39–89. Google Scholar
21. L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2) (2005) 649–668. Crossref, Web of Science, Google Scholar
22. M. Rahro Zargar and H. Zakeri, On injective and Gorenstein injective dimensions of local cohomology modules, Algebra Colloq. 22(1) (2015) 935–946. Link, Google Scholar
23. R. Takahashi, Classifying subcategories of modules over a commutative Noetherian ring, J. Lond. Math. Soc. 78(2) (2008) 767–782. Crossref, Google Scholar
24. R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra 213(4) (2009) 582–600. Crossref, Web of Science, Google Scholar
25. S. Yassemi, Generalized section functors, J. Pure Appl. Algebra 95(1) (1994) 103–119. Crossref, Web of Science, Google Scholar
26. K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997) 179–191. Crossref, Web of Science, Google Scholar
27. Y. Yoshino and T. Yoshizawa, Abstract local cohomology functors, Math. J. Okayama Univ. 53 (2011) 129–154. Google Scholar
28. T. Yoshizawa, Subcategories of extension modules by subcategories, Proc. Amer. Math. Soc. 140 (2012) 2293–2305. Crossref, Web of Science, Google Scholar