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Free resolutions of parameter ideals for some rings with finite local cohomology

Author: Hamidreza Rahmati
Journal: Proc. Amer. Math. Soc. 136 (2008), 467-474
MSC (2000): Primary 13D02, 13D40; Secondary 13H10
Published electronically: November 3, 2007
MathSciNet review: 2358485
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Abstract: Let $ R$ be a $ d$-dimensional local ring, with maximal ideal $ \mathfrak{m}$, containing a field and let $ x_1, \dots , x_d$ be a system of parameters for $ R$. If $ \operatorname{depth}\,R \geq d - 1$ and the local cohomology module $ \operatorname{H}_{\mathit{m}}^{d-1}(R)$ is finitely generated, then there exists an integer $ n$ such that the modules $ R/(x_1^i,\dots,x_d^i)$ have the same Betti numbers, for all $ i\geq n$.

References [Enhancements On Off] (What's this?)

  • 1. L. L. Avramov and S. Iyengar, Gaps in Hochschild cohomology imply smoothness for commutative algebra, Math. Res. Letters 12 (2005), 789-804. MR 2189239 (2006i:13028)
  • 2. W. Bruns and J. Herzog, Cohen-Macaulay rings, vol. 39, Cambridge Stud. Adv. Math., Cambridge Univ. Press, 1998. MR 1251956 (95h:13020)
  • 3. R. C. Heitmann, The direct summand conjecture in dimension 3, Ann. of Math. (2) 156 (2002), no. 2, 695-712. MR 1933722 (2003m:13008)
  • 4. L. T. Hoa, Koszul homology and generalized Cohen-Macaulay modules, Acta Math. Vietnam 18 (1993), no. 1, 91-98. MR 1248885 (95c:13007)
  • 5. M. Hochster, Canonical elements in local cohomology and the direct summand conjecture, J. Algebra 84 (1983), 503-553. MR 723406 (85j:13021)
  • 6. C. Huneke and J. Koh, Some dimension $ 3$ cases of the canonical element conjecture, Proc. Amer. Math. Soc. 98 (1986), no. 3, 394-398. MR 857928 (87m:13030)
  • 7. Y. H. Lai, On the relation type of system of parameters and on the Poincaré series of system of parameters., Ph.D. Thesis Purdue Unviversity, 1995. MR 1338982 (96i:13026)
  • 8. P. Roberts, The equivalence of two forms of the Canonical Element Conjecture, undated manuscript.
  • 9. P. Schenzel, N. V. Trung and N. T. Coung, Verallgemeinerte Cohen-Macaulay Moduln, Math. Nachr., 85 (1978), 57-73. MR 517641 (80i:13008)
  • 10. J. Stückard and W. Vogel, Buchsbaum rings and applications, Berlin-Heidelberg-New York, Springer-Verlag, 1986. MR 881220 (88h:13011a)
  • 11. N. V. Trung, Toward a theory of generalized Cohen-Macaulay modules, Nagoya Math. J. 102 (1986), 1-49. MR 846128 (87h:13018)

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Additional Information

Hamidreza Rahmati
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588

Received by editor(s): April 12, 2006
Received by editor(s) in revised form: January 12, 2007
Published electronically: November 3, 2007
Additional Notes: This research was partly supported by NSF grant DMS-0201904
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society

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