zbMATH — the first resource for mathematics

Cohomological dimension, cofiniteness and abelian categories of cofinite modules. (English) Zbl 1451.13060
Let $$R$$ be a commutative Noetherian ring, $$I$$ be an ideal of $$R$$, and $$M$$ be an $$R$$-module. Recall that, the notion of the cohomological dimension of $$M$$ relative to $$I$$, denoted by $$\mathrm{cd}(I,M)$$, is defined to be the greatest integer $$i$$ such that $$H_I^i(M)\neq 0$$ if there exist such $$i$$’s and $$-infty$$ otherwise. Also, $$q(I,M)$$ denotes the greatest integer $$i$$ such that $$H_I^i(M)$$ is not Artian if there exist such $$i$$’s and $$-\infty$$ otherwise. In the paper under review, it is proved that, if $$J$$ is an ideal of $$R$$ with $$I\subseteq J$$, and $$M$$ is finitely generated, then $$q(J,M)\leq q(I,M)+\mathrm{cd}(J,M/IM)$$.
Recall that an $$R$$-module $$M$$ is called $$I$$-cofinite, if $$\mathrm{Supp}_RM\subseteq\mathrm{V}(I)$$ and $$\mathrm{Ext}^i_R(R/I,M)$$ is a finitely generated module for all $$i$$. In the paper under review, the author defined $$\tilde{q}(I,M)$$ as the greatest integer $$i$$ such that $$H_I^i(M)$$ is not Artian $$I$$-cofinite if there exist such $$i$$’s and $$-infty$$ otherwise. It is shown that if $$M$$ is finitely generated and $$q(I,M)\leq 1$$, then $$\tilde{q}(I,M)=q(I,M)$$, and $$H_I^i(M)$$ is $$I$$-cofinite for any $$i\geq 0$$. As a consequence, it is proved that, if $$M$$ is finitely generated and $$q(I,R)\leq 1$$, then $$H_I^i(M)$$ is $$I$$-cofinite for any $$i\geq 0$$.
Let $$R,\mathfrak{m}$$ be a complete Noetherian local ring with $$q(I,R)\leq 1$$. It is shown that the category of all $$I$$-cofinite $$R$$-modules is an abelian subcategory of the category of all $$R$$-modules; that is, if $$f:M\rightarrow N$$ is an $$R$$-homomorphism of $$I$$-cofinite modules, then $$\ker f$$ and $$\mathrm{coker }f$$ are $$I$$-cofinite $$R$$-modules.

MSC:
 13D45 Local cohomology and commutative rings 14B15 Local cohomology and algebraic geometry 13E05 Commutative Noetherian rings and modules
Full Text:
References:
 [1] Abazari, N.; Bahmanpour, K., A note on the Artinian cofinite modules, Comm. Algebra, 42, 1270-1275, (2014) · Zbl 1291.13028 [2] Abazari, N.; Bahmanpour, K., Extension functors of local cohomology modules and Serre categories of modules, Taiwanese J. Math., 19, 211-220, (2015) · Zbl 1357.13020 [3] Aghapournahr, M.; Melkersson, L., Local cohomology and Serre subcategories, J. Algebra, 320, 1275-1287, (2008) · Zbl 1153.13014 [4] Asgharzadeh, M.; Divaani-Aazar, K.; Tousi, M., The finiteness dimension of local cohomology modules and its dual notion, J. Pure Appl. Algebra, 213, 321-328, (2009) · Zbl 1156.13007 [5] Asgharzadeh, M.; Tousi, M., A unified approach to local cohomology modules using Serre classes, Canad. Math. Bull., 53, 577-586, (2010) · Zbl 1205.13023 [6] Bahmanpour, K., A note on Lynch’s conjecture, Comm. Algebra, 45, 2738-2745, (2017) · Zbl 1375.13022 [7] Bahmanpour, K.; Naghipour, R., Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra, 321, 1997-2011, (2009) · Zbl 1168.13016 [8] Bahmanpour, K.; Naghipour, R.; Sedghi, M., On the category of cofinite modules which is abelian, Proc. Amer. Math. Soc., 142, 1101-1107, (2014) · Zbl 1286.13017 [9] Brodmann, M. P.; Sharp, R. Y., Local cohomology; an algebraic introduction with geometric applications, (1998), Cambridge University Press Cambridge · Zbl 0903.13006 [10] Chiriacescu, G., Cofiniteness of local cohomology modules, Bull. Lond. Math. Soc., 32, 1-7, (2000) · Zbl 1018.13009 [11] Delfino, D.; Marley, T., Cofinite modules and local cohomology, J. Pure Appl. Algebra, 121, 45-52, (1997) · Zbl 0893.13005 [12] Dibaei, M. T.; Yassemi, S., Associated primes and cofiniteness of local cohomology modules, Manuscripta Math., 117, 199-205, (2005) · Zbl 1105.13016 [13] Divaani-Aazar, K.; Naghipour, R.; Tousi, M., Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc., 130, 3537-3544, (2002) · Zbl 0998.13007 [14] Ghasemi, G.; Bahmanpour, K.; A’zami, J., Upper bounds for the cohomological dimensions of finitely generated modules over a commutative Noetherian ring, Colloq. Math., 137, 263-270, (2014) · Zbl 1314.13034 [15] Grothendieck, A., Local cohomology, Lecture Notes in Math., vol. 862, (1966), Springer New York, Notes by R. Hartshorne · Zbl 0145.17602 [16] Hartshorne, R., Affine duality and cofiniteness, Invent. Math., 9, 145-164, (1970) · Zbl 0196.24301 [17] Hartshorne, R., Cohomological dimension of algebraic varieties, Ann. of Math., 88, 403-450, (1968) · Zbl 0169.23302 [18] Kawasaki, K.-I., On a category of cofinite modules for principal ideals, Nihonkai Math. J., 22, 67-71, (2011) · Zbl 1247.14003 [19] Kawasaki, K.-I., On a category of cofinite modules which is abelian, Math. Z., 269, 587-608, (2011) · Zbl 1228.13020 [20] Kubik, B.; Leamerb, M. J.; Sather-Wagstaff, S., Homology of Artinian and matlis reflexive modules, J. Pure Appl. Algebra, 215, 2486-2503, (2011) · Zbl 1232.13008 [21] Lynch, L. R., Annihilators of top local cohomology, Comm. Algebra, 40, 542-551, (2012) · Zbl 1251.13015 [22] Matsumura, H., Commutative ring theory, (1986), Cambridge University Press Cambridge, UK [23] Melkersson, L., Cofiniteness with respect to ideals of dimension one, J. Algebra, 372, 459-462, (2012) · Zbl 1273.13029 [24] Melkersson, L., Modules cofinite with respect to an ideal, J. Algebra, 285, 649-668, (2005) · Zbl 1093.13012 [25] Pirmohammadi, G.; Ahmadi Amoli, K.; Bahmanpour, K., Some homological properties of ideals with cohomological dimension one, Colloq. Math., (2017), in press · Zbl 1390.13055 [26] Yoshida, K. I., Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J., 147, 179-191, (1997) · Zbl 0899.13018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.