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Keywords:
local cohomology module; minimax module; coatomic module; Artinian module; local-global principle
local cohomology module; minimax module; coatomic module; Artinian module; local-global principle
Summary:
Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $\mathcal {C}$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that $(H^i_I(M))_{\frak m}$ is a minimax $R_{\frak m}$-module for all $\frak m \in {\rm Max} (R)$ and for all $i < n$, then the set ${\rm Ass}_R(H^n_I(M))$ is finite. Also, if $H^i_I(M)$ is minimax for all $i \geq n \geq 1$, then $H^i_I(M)$ is Artinian for $i \geq n$. It is shown that if $M$ is a $\mathcal {C}$-minimax module over a local ring such that $H^i_I(M)$ are $\mathcal {C}$-minimax modules for all $i < n$ (or $i\geq n$), where $n\geq 1$, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules.
References:
[1] Abbasi, A., Roshan-Shekalgourabi, H., Hassanzadeh-Lelekaami, D.: Artinianness of local cohomology modules. Honam Math. J. 38 (2016), 295-304. DOI 10.5831/HMJ.2016.38.2.295 | MR 3526773 | Zbl 1346.13030
[2] Aghapournahr, M., Melkersson, L.: Finiteness properties of minimax and coatomic local cohomology modules. Arch. Math. 94 (2010), 519-528. DOI 10.1007/s00013-010-0127-z | MR 2653668 | Zbl 1196.13011
[3] Bahmanpour, K., Naghipour, R.: On the cofiniteness of local cohomology modules. Proc. Am. Math. Soc. 136 (2008), 2359-2363. DOI 10.1090/S0002-9939-08-09260-5 | MR 2390502 | Zbl 1141.13014
[4] Brodmann, M. P., Lashgari, F. A.: A finiteness result for associated primes of local cohomology modules. Proc. Am. Math. Soc. 128 (2000), 2851-2853. DOI 10.1090/S0002-9939-00-05328-4 | MR 1664309 | Zbl 0955.13007
[5] Brodmann, M. P., Sharp, R. Y.: Local Cohomology: An Algebraic Introduction With Geometric Applications. Cambridge Studies in Advanced Mathematics 60. Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511629204 | MR 1613627 | Zbl 0903.13006
[6] Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511608681 | MR 1251956 | Zbl 0909.13005
[7] Dibaei, M. T., Yassemi, S.: Cohomological dimension of complexes. Commun. Algebra 32 (2004), 4375-4386. DOI 10.1081/AGB-200034165 | MR 2102455 | Zbl 1093.13011
[8] Divaani-Aazar, K., Naghipour, R., Tousi, M.: Cohomological dimension of certain algebraic varieties. Proc. Am. Math. Soc. 130 (2002), 3537-3544. DOI 10.1090/S0002-9939-02-06500-0 | MR 1918830 | Zbl 0998.13007
[9] Hartshorne, R.: Affine duality and cofiniteness. Invent. Math. 9 (1970), 145-164. DOI 10.1007/BF01404554 | MR 0257096 | Zbl 0196.24301
[10] Huneke, C.: Problems on local cohomology. Free Resolutions in Commutative Algebra and Algebraic Geometry Research Notes in Mathematics. Jones and Bartlett, Boston (1992), 93-108. MR 1165320 | Zbl 0782.13015
[11] Lorestani, K. B., Sahandi, P., Yassemi, S.: Artinian local cohomology modules. Can. Math. Bull. 50 (2007), 598-602. DOI 10.4153/CMB-2007-058-8 | MR 2364209 | Zbl 1140.13016
[12] Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge (1989). DOI 10.1017/CBO9781139171762 | MR 1011461 | Zbl 0666.13002
[13] Melkersson, L.: Modules cofinite with respect to an ideal. J. Algebra 285 (2005), 649-668. DOI 10.1016/j.jalgebra.2004.08.037 | MR 2125457 | Zbl 1093.13012
[14] Nam, T. T.: Minimax modules, local homology and local cohomology. Int. J. Math. 26 (2015), Article ID 1550102, 16 pages. DOI 10.1142/S0129167X15501025 | MR 3432533 | Zbl 1349.13037
[15] Nam, T. T., Nguyen, M. T.: On coatomic modules and local cohomology modules with respect to a pair of ideals. J. Korean Math. Soc. 54 (2017), 1829-1839. DOI 10.4134/JKMS.j160712 | MR 3718427 | Zbl 1401.13052
[16] Payrovi, S., Parsa, M. Lotfi: Finiteness of local cohomology modules defined by a pair of ideals. Commun. Algebra 41 (2013), 627-637. DOI 10.1080/00927872.2011.631206 | MR 3011786 | Zbl 1263.13016
[17] Rezaei, S.: Minimaxness and finiteness properties of local homology and local cohomology modules. Indian J. Pure Appl. Math. 49 (2018), 383-396. DOI 10.1007/s13226-018-0275-6 | MR 3854443
[18] Rudlof, P.: On the structure of couniform and complemented modules. J. Pure Appl. Algebra 74 (1991), 281-305. DOI 10.1016/0022-4049(91)90118-L | MR 1135033 | Zbl 0754.13010
[19] Rudlof, P.: On minimax and related modules. Can. J. Math. 44 (1992), 154-166. DOI 10.4153/CJM-1992-009-7 | MR 1152672 | Zbl 0762.13003
[20] Yoshida, K.-I.: Cofiniteness of local cohomology modules for ideals of dimension one. Nagoya Math. J. 147 (1997), 179-191. DOI 10.1017/S0027763000006371 | MR 1475172 | Zbl 0899.13018
[21] Yoshizawa, T.: Subcategories of extension modules by Serre subcategories. Proc. Am. Math. Soc. 140 (2012), 2293-2305. DOI 10.1090/S0002-9939-2011-11108-0 | MR 2898693 | Zbl 1273.13018
[22] Zöschinger, H.: Koatomare Moduln. Math. Z. 170 (1980), 221-232 German. DOI 10.1007/BF01214862 | MR 0564202 | Zbl 0411.13009
[23] Zöschinger, H.: Minimax-Moduln. J. Algebra 102 (1986), 1-32 German. DOI 10.1016/0021-8693(86)90125-0 | MR 0853228 | Zbl 0593.13012
[24] Zöschinger, H.: Über die Maximalbedingung für radikalvolle Untermoduln. Hokkaido Math. J. 17 (1988), 101-116 German. DOI 10.14492/hokmj/1381517790 | MR 0928469 | Zbl 0653.13011
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