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On Gorenstein injectivity of top local cohomology modules
Author:
Takeshi Yoshizawa
Journal:
Proc. Amer. Math. Soc. 140 (2012), 1897-1907
MSC (2010):
Primary 13D05, 13D45
Posted:
October 3, 2011
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References |
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Additional Information
Abstract: R. Sazeedeh showed that top local cohomology modules are Gorenstein injective in a Gorenstein local ring with at most two dimensions. In this paper, it is proved that the condition of dimension in his result cannot be relaxed and the conclusion in his result holds for complete local hypersurface rings with any dimension.
References
- 1.
Mohammad
H. Bijan-Zadeh, Torsion theories and local cohomology over
commutative and Noetherian rings, J. London Math. Soc. (2)
19 (1979), no. 3, 402–410. MR 540052
(80h:13013), http://dx.doi.org/10.1112/jlms/s2-19.3.402
- 2.
M.
P. Brodmann and R.
Y. Sharp, Local cohomology: an algebraic introduction with
geometric applications, Cambridge Studies in Advanced Mathematics,
vol. 60, Cambridge University Press, Cambridge, 1998. MR 1613627
(99h:13020)
- 3.
Lars
Winther Christensen, Gorenstein dimensions, Lecture Notes in
Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866
(2002e:13032)
- 4.
Lars
Winther Christensen, Hans-Bjørn
Foxby, and Anders
Frankild, Restricted homological dimensions and
Cohen-Macaulayness, J. Algebra 251 (2002),
no. 1, 479–502. MR 1900297
(2003e:13022), http://dx.doi.org/10.1006/jabr.2001.9115
- 5.
Edgar
E. Enochs and Overtoun
M. G. Jenda, On Gorenstein injective modules, Comm. Algebra
21 (1993), no. 10, 3489–3501. MR 1231612
(94g:13006), http://dx.doi.org/10.1080/00927879308824744
- 6.
Edgar
E. Enochs and Overtoun
M. G. Jenda, Gorenstein injective and projective modules,
Math. Z. 220 (1995), no. 4, 611–633. MR 1363858
(97c:16011), http://dx.doi.org/10.1007/BF02572634
- 7.
Edgar
E. Enochs, Overtoun
M. G. Jenda, and Blas
Torrecillas, Gorenstein flat modules, Nanjing Daxue Xuebao
Shuxue Bannian Kan 10 (1993), no. 1, 1–9
(English, with Chinese summary). MR 1248299
(95a:16004)
- 8.
Edgar
E. Enochs, Overtoun
M. G. Jenda, and Jin
Zhong Xu, Foxby duality and Gorenstein injective
and projective modules, Trans. Amer. Math.
Soc. 348 (1996), no. 8, 3223–3234. MR 1355071
(96k:13010), http://dx.doi.org/10.1090/S0002-9947-96-01624-8
- 9.
A.
A. Gerko, On homological dimensions, Mat. Sb.
192 (2001), no. 8, 79–94 (Russian, with Russian
summary); English transl., Sb. Math. 192 (2001),
no. 7-8, 1165–1179. MR 1862245
(2002h:13024), http://dx.doi.org/10.1070/SM2001v192n08ABEH000587
- 10.
Henrik
Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra
189 (2004), no. 1-3, 167–193. MR 2038564
(2004k:16013), http://dx.doi.org/10.1016/j.jpaa.2003.11.007
- 11.
Henrik
Holm, Rings with finite Gorenstein injective
dimension, Proc. Amer. Math. Soc.
132 (2004), no. 5,
1279–1283 (electronic). MR 2053331
(2005a:13031), http://dx.doi.org/10.1090/S0002-9939-03-07466-5
- 12.
Eben
Matlis, The higher properties of 𝑅-sequences, J.
Algebra 50 (1978), no. 1, 77–112. MR 479882
(80a:13013), http://dx.doi.org/10.1016/0021-8693(78)90176-X
- 13.
Reza
Sazeedeh, Gorenstein injective modules and local
cohomology, Proc. Amer. Math. Soc.
132 (2004), no. 10, 2885–2891 (electronic). MR 2063107
(2005e:13027), http://dx.doi.org/10.1090/S0002-9939-04-07461-1
- 14.
Reza
Sazeedeh, Strongly torsion free, copure flat and Matlis reflexive
modules, J. Pure Appl. Algebra 192 (2004),
no. 1-3, 265–274. MR 2067199
(2005c:13008), http://dx.doi.org/10.1016/j.jpaa.2004.01.010
- 15.
Reza
Sazeedeh, Strongly torsion-free modules and local cohomology over
Cohen-Macaulay rings, Comm. Algebra 33 (2005),
no. 4, 1127–1135. MR 2136689
(2006c:13015), http://dx.doi.org/10.1081/AGB-200053824
- 16.
Peter
Schenzel, Explicit computations around the Lichtenbaum-Hartshorne
vanishing theorem, Manuscripta Math. 78 (1993),
no. 1, 57–68. MR 1201761
(93m:14003), http://dx.doi.org/10.1007/BF02599300
- 17.
Peter
Schenzel, On the use of local cohomology in algebra and
geometry, Six lectures on commutative algebra (Bellaterra, 1996)
Progr. Math., vol. 166, Birkhäuser, Basel, 1998,
pp. 241–292. MR 1648667
(99k:13025)
- 18.
Peter
Schenzel, On connectedness and indecomposibility of local
cohomology modules, Manuscripta Math. 128 (2009),
no. 3, 315–327. MR 2481047
(2010a:13030), http://dx.doi.org/10.1007/s00229-008-0229-0
- 19.
Ryo
Takahashi, Yuji
Yoshino, and Takeshi
Yoshizawa, Local cohomology based on a nonclosed support defined by
a pair of ideals, J. Pure Appl. Algebra 213 (2009),
no. 4, 582–600. MR 2483839
(2009m:13027), http://dx.doi.org/10.1016/j.jpaa.2008.09.008
- 20.
Jin
Zhong Xu, Minimal injective and flat resolutions of modules over
Gorenstein rings, J. Algebra 175 (1995), no. 2,
451–477. MR 1339651
(96h:13025), http://dx.doi.org/10.1006/jabr.1995.1196
- 21.
Okyeon
Yi, On torsion Gorenstein injective modules, Arch. Math.
(Brno) 34 (1998), no. 4, 445–454. MR 1679639
(2000h:16006)
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Additional Information
Takeshi Yoshizawa
Affiliation:
Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan
Email:
tyoshiza@math.okayama-u.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11059-1
PII:
S 0002-9939(2011)11059-1
Keywords:
Local cohomology,
Gorenstein injective module
Received by editor(s):
January 4, 2010
Received by editor(s) in revised form:
October 11, 2010 and February 5, 2011
Posted:
October 3, 2011
Communicated by:
Bernd Ulrich
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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