| Title:
|
Some results on the cofiniteness of local cohomology modules (English) |
| Author:
|
Laleh, Sohrab Sohrabi |
| Author:
|
Sadeghi, Mir Yousef |
| Author:
|
Mostaghim, Mahdi Hanifi |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
105-110 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative Noetherian ring, $\mathfrak {a}$ an ideal of $R$, $M$ an $R$-module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $\mathcal {AF}$ modules. The main result is that if the $R$-module ${\rm Ext}^t_R(R/\mathfrak {a},M)$ is finite (finitely generated), $H^i_\mathfrak {a}(M)$ is $\mathfrak {a} $-cofinite for all $i<t$ and $H^t_\mathfrak {a}(M)$ is minimax then $H^t_\mathfrak {a}(M)$ is $\mathfrak {a} $-cofinite. As a consequence we show that if $M$ and $N$ are finite $R$-modules and $H^i_\mathfrak {a}(N)$ is minimax for all $i<t$ then the set of associated prime ideals of the generalized local cohomology module $H^t_\mathfrak {a}(M,N)$ is finite. (English) |
| Keyword:
|
local cohomology |
| Keyword:
|
cofinite modules |
| Keyword:
|
mimimax modules |
| Keyword:
|
AF modules |
| Keyword:
|
associated primes |
| MSC:
|
13D45 |
| MSC:
|
13E05 |
| MSC:
|
14B15 |
| idZBL:
|
Zbl 1249.13012 |
| idMR:
|
MR2899737 |
| DOI:
|
10.1007/s10587-012-0019-4 |
| . |
| Date available:
|
2012-03-05T07:14:51Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142043 |
| . |
| Reference:
|
[1] Bahmanpour, K., Naghipour, R.: On the cofiniteness of local cohomology modules.Proc. Am. Math. Soc. 136 2359-2363 (2008). Zbl 1141.13014, MR 2390502, 10.1090/S0002-9939-08-09260-5 |
| Reference:
|
[2] Brodmann, M. P., Sharp, R. Y.: Local Cohomology.An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge (2008). Zbl 1133.13017, MR 1613627 |
| Reference:
|
[3] Enochs, E.: Flat covers and flat cotorsion modules.Proc. Am. Math. Soc. 92 179-184 (1984). Zbl 0522.13008, MR 0754698, 10.1090/S0002-9939-1984-0754698-X |
| Reference:
|
[4] Hartshorne, R.: Affine duality and cofiniteness.Invent. Math. 9 145-164 (1970). Zbl 0196.24301, MR 0257096, 10.1007/BF01404554 |
| Reference:
|
[5] Herzog, J.: Komplexe Auflösungen und Dualität in der lokalen Algebra.Habilitationsschrift Universität Regensburg, Regensburg (1970), German. |
| Reference:
|
[6] Huneke, C., Koh, J.: Cofiniteness and vanishing of local cohomology modules.Math. Proc. Camb. Philos. Soc. 110 421-429 (1991). Zbl 0749.13007, MR 1120477, 10.1017/S0305004100070493 |
| Reference:
|
[7] Mafi, A.: Some results on local cohomology modules.Arch. Math. 87 211-216 (2006). Zbl 1102.13018, MR 2258920, 10.1007/s00013-006-1674-1 |
| Reference:
|
[8] Mafi, A.: On the associated primes of generalized local cohomology modules.Commun. Algebra. 34 2489-2494 (2006). Zbl 1097.13023, MR 2240388, 10.1080/00927870600650739 |
| Reference:
|
[9] Melkersson, L.: Modules cofinite with respect to an ideal.J. Algebra 285 649-668 (2005). Zbl 1093.13012, MR 2125457, 10.1016/j.jalgebra.2004.08.037 |
| Reference:
|
[10] Melkersson, L.: Problems and Results on Cofiniteness: A Survey.IPM Proceedings Series No. II, IPM (2004). |
| Reference:
|
[11] Vasconcelos, W. V.: Divisor Theory in Module Categories, North-Holland Mathematics Studies.14. Notas de Matematica, North-Holland Publishing Company, Amsterdam (1974). MR 0498530 |
| Reference:
|
[12] Yassemi, S.: Cofinite modules.Commun. Algebra 29 2333-2340 (2001). Zbl 1023.13013, MR 1845114, 10.1081/AGB-100002392 |
| Reference:
|
[13] Zink, T.: Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring.German Math. Nachr. 64 239-252 (1974). Zbl 0297.13015, MR 0364223, 10.1002/mana.19740640114 |
| Reference:
|
[14] Zöschinger, H.: Minimax-moduln.German J. Algebra 102 1-32 (1986). Zbl 0593.13012, MR 0853228, 10.1016/0021-8693(86)90125-0 |
| Reference:
|
[15] Zöschinger, H.: Über die Maximalbedingung für radikalvolle Untermoduln.German Hokkaido Math. J. 17 101-116 (1988). Zbl 0653.13011, MR 0928469, 10.14492/hokmj/1381517790 |
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