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Keywords:
Bass numbers; generalized local cohomology modules; Matlis reflexive
Summary:
Let $(R,\mathfrak m )$ be a complete local ring, $\mathfrak a $ an ideal of $R$ and $N$ and $L$ two Matlis reflexive $R$-modules with $\mathop{{\rm Supp}} (L)\subseteq V(\mathfrak a )$. We prove that if $M$ is a finitely generated $R$-module, then $\mathop{{\rm Ext}}\nolimits_R^i(L,H_{\mathfrak a }^j(M,N))$ is Matlis reflexive for all $i$ and $j$ in the following cases: (a) $\mathop{{\rm dim}} R/{\mathfrak a }=1$; (b) $\mathop{{\rm cd}} (\mathfrak a )=1$; where $\mathop{{\rm cd}} $ is the cohomological dimension of $\mathfrak a $ in $R$; (c) $\mathop{{\rm dim}} R\leq 2$. In these cases we also prove that the Bass numbers of $H_{\mathfrak a }^j(M,N)$ are finite.
References:
[1] Brodmann, M. P., Sharp, R. Y.: Local Cohomology. An Algebraic Introduction with Geometric Applications. Cambridge University Press Cambridge (1998). MR 1613627 | Zbl 0903.13006
[2] Belshoff, R., Slattery, S. P., Wickham, C.: Finiteness properties for Matlis reflexive modules. Commun. Algebra 24 (1996), 1371-1376. DOI 10.1080/00927879608825640 | MR 1380599 | Zbl 0873.13012
[3] Belshoff, R., Slattery, S. P., Wickham, C.: The local cohomology modules of Matlis reflexive modules are almost cofinite. Proc. Am. Math. Soc. 124 (1996), 2649-2654. DOI 10.1090/S0002-9939-96-03326-6 | MR 1326995 | Zbl 0863.13005
[4] Belshoff, R., Wickham, C.: A note on local duality. Bull. Lond. Math. Soc. 29 (1997), 25-31. DOI 10.1112/S0024609396001713 | MR 1416402 | Zbl 0891.13005
[5] Delfino, D.: On the cofiniteness of local cohomology modules. Math. Proc. Camb. Philos. Soc. 115 (1994), 79-84. DOI 10.1017/S0305004100071929 | MR 1253283 | Zbl 0806.13005
[6] Delfino, D., Marley, T.: Cofinite modules and local cohomology. J. Pure Appl. Algebra 121 (1997), 45-52. DOI 10.1016/S0022-4049(96)00044-8 | MR 1471123 | Zbl 0893.13005
[7] Divaani-Aazar, K., Sazeedeh, R.: Cofiniteness of generalized local cohomology modules. Colloq. Math. 99 (2004), 283-290. DOI 10.4064/cm99-2-12 | MR 2079733 | Zbl 1072.13011
[8] Divaani-Aazar, K., Sazeedeh, R., Tousi, M.: On vanishing of generalized local cohomology modules. Algebra Colloq. 12 (2005), 213-218. DOI 10.1142/S1005386705000209 | MR 2127246 | Zbl 1065.13007
[9] Hartshorne, R.: Affine duality and cofiniteness. Invent. Math. 9 (1970), 145-164. DOI 10.1007/BF01404554 | MR 0257096 | Zbl 0196.24301
[10] Herzog, J.: Komplexe Auflösungen und Dualitat in der lokalen Algebra. Habilitationsschrift Universität Regensburg Regensburg (1970), German.
[11] Huneke, C., Koh, J.: Cofiniteness and vanishing of local cohomology modules. Math. Proc. Camb. Philos. Soc. 110 (1991), 421-429. DOI 10.1017/S0305004100070493 | MR 1120477 | Zbl 0749.13007
[12] Kawakami, S., Kawasaki, K.-I.: On the finiteness of Bass numbers of generalized local cohomology modules. Toyama Math. J. 29 (2006), 59-64. MR 2333640 | Zbl 1141.13307
[13] Kawasaki, K.-I.: Cofiniteness of local cohomology modules for principal ideals. Bull. Lond. Math. Soc. 30 (1998), 241-246. DOI 10.1112/S0024609397004347 | MR 1608094 | Zbl 0930.13013
[14] Khashyarmanesh, K., Khosh-Ahang, F.: On the local cohomology of Matlis reflexive modules. Commun. Algebra 36 (2008), 665-669. DOI 10.1080/00927870701724102 | MR 2388029 | Zbl 1133.13018
[15] Mafi, A.: A generalization of the finiteness problem in local cohomology modules. Proc. Indian Acad. Sci. (Math. Sci.) 119 (2009), 159-164. DOI 10.1007/s12044-009-0016-1 | MR 2526419 | Zbl 1171.13011
[16] Mafi, A., Saremi, H.: Cofinite modules and generalized local cohomology. Houston J. Math (to appear). MR 2577138 | Zbl 1185.13019
[17] Melkersson, L.: Properties of cofinite modules and applications to local cohomology. Math. Proc. Camb. Philos. Soc. 125 (1999), 417-423. DOI 10.1017/S0305004198003041 | MR 1656785 | Zbl 0921.13009
[18] 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
[19] Ooishi, A.: Matlis duality and width of a module. Hiroshima Math. J. 6 (1976), 573-587. MR 0422243
[20] Strooker, J.: Homological Questions in Local Algebra. Lecture Notes Series 145. Cambridge University Press Cambridge (1990). MR 1074178
[21] Yassemi, S.: Generalized section functors. J. Pure Appl. Algebra 95 (1994), 103-119. DOI 10.1016/0022-4049(94)90121-X | MR 1289122 | Zbl 0843.13005
[22] Yoshida, K. I.: Cofiniteness of local cohomology modules for ideals of dimension one. Nagoya Math. J. 147 (1997), 179-191. MR 1475172 | Zbl 0899.13018
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