Hilbert-Samuel functions of modules over Cohen-Macaulay rings
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- by Srikanth Iyengar and Tony J. Puthenpurakal PDF
- Proc. Amer. Math. Soc. 135 (2007), 637-648 Request permission
Abstract:
For a finitely generated, non-free module $M$ over a CM local ring $(R,\mathfrak {m},k)$, it is proved that for $n\gg 0$ the length of $\operatorname {Tor}_1^R(M,R/\mathfrak {m}^{n+1})$ is given by a polynomial of degree $\dim R-1$. The vanishing of $\operatorname {Tor}_ i^R(M,N/\mathfrak {m}^{n+1}N)$ is studied, with a view towards answering the question: If there exists a finitely generated $R$-module $N$ with $\dim N\ge 1$ such that the projective dimension or the injective dimension of $N/\mathfrak {m}^{n+1}N$ is finite, then is $R$ regular? Upper bounds are provided for $n$ beyond which the question has an affirmative answer.References
- Luchezar L. Avramov, Small homomorphisms of local rings, J. Algebra 50 (1978), no. 2, 400–453. MR 485906, DOI 10.1016/0021-8693(78)90163-1
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Songqing Ding, Cohen-Macaulay approximation and multiplicity, J. Algebra 153 (1992), no. 2, 271–288. MR 1198202, DOI 10.1016/0021-8693(92)90156-G
- Juan Elias, On the computation of the Ratliff-Rush closure, J. Symbolic Comput. 37 (2004), no. 6, 717–725. MR 2095368, DOI 10.1016/j.jsc.2003.10.003
- Vijay Kodiyalam, Homological invariants of powers of an ideal, Proc. Amer. Math. Soc. 118 (1993), no. 3, 757–764. MR 1156471, DOI 10.1090/S0002-9939-1993-1156471-5
- G. Levin and W. V. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315–323. MR 230715
- Tony J. Puthenpurakal, Hilbert-coefficients of a Cohen-Macaulay module, J. Algebra 264 (2003), no. 1, 82–97. MR 1980687, DOI 10.1016/S0021-8693(03)00231-X
- —, Ratliff-Rush filtration, regularity and depth of higher associated grade modules, Part I, J. Pure App. Algebra, to appear.
- Judith D. Sally, Numbers of generators of ideals in local rings, Marcel Dekker, Inc., New York-Basel, 1978. MR 0485852
- Liana M. Şega, Homological properties of powers of the maximal ideal of a local ring, J. Algebra 241 (2001), no. 2, 827–858. MR 1843329, DOI 10.1006/jabr.2001.8786
- Jack Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470. MR 241411, DOI 10.1016/0021-8693(69)90023-4
- John Tate, Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–27. MR 86072
- Emanoil Theodorescu, Derived functors and Hilbert polynomials, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 1, 75–88. MR 1866325, DOI 10.1017/S0305004101005412
Additional Information
- Srikanth Iyengar
- Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
- MR Author ID: 616284
- ORCID: 0000-0001-7597-7068
- Email: iyengar@math.unl.edu
- Tony J. Puthenpurakal
- Affiliation: Department of Mathematics, IIT Bombay, Powai, Mumbai 400 076, India
- MR Author ID: 715327
- Email: tputhen@math.iitb.ac.in
- Received by editor(s): November 18, 2004
- Received by editor(s) in revised form: September 22, 2005
- Published electronically: August 28, 2006
- Additional Notes: The first author was partly supported by NSF grant DMS 0442242
- Communicated by: Bernd Ulrich
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 637-648
- MSC (2000): Primary 13D40; Secondary 13D02, 13D07
- DOI: https://doi.org/10.1090/S0002-9939-06-08519-4
- MathSciNet review: 2262858