Consecutive cancellations in Betti numbers of local rings
Authors:
Maria Evelina Rossi and Leila Sharifan
Journal:
Proc. Amer. Math. Soc. 138 (2010), 6173
MSC (2000):
Primary 13D02
Published electronically:
August 28, 2009
MathSciNet review:
2550170
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Abstract: Let be a homogeneous ideal in a polynomial ring over a field. By Macaulay's Theorem there exists a lexicographic ideal with the same Hilbert function as Peeva has proved that the Betti numbers of can be obtained from the graded Betti numbers of by a suitable sequence of consecutive cancellations. We extend this result to any ideal in a regular local ring by passing through the associated graded ring. To this purpose it will be necessary to enlarge the list of the allowed cancellations. Taking advantage of EliahouKervaire's construction, we present several applications. This connection between the graded perspective and the local one is a new viewpoint, and we hope it will be useful for studying the numerical invariants of classes of local rings.
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 A.M. Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra 21 (7) (1993), 23172334. MR 1218500 (94c:13014)
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 CoCoA Team, CoCoA: a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.
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 S. Eliahou, M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), no. 1, 125. MR 1037391 (91b:13019)
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 J. Elias, G. Valla, Structure theorems for certain Gorenstein ideals, Michigan Math. J. 57 (2008), 269292. MR 2492453
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 T. Hibi, S. Murai, The depth of an ideal with a given Hilbert function, Proc. Amer. Math. Soc. 136 (2008), no. 5, 15331538. MR 2373580 (2009b:13028)
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 J. Herzog, M. E. Rossi, G. Valla, On the depth of the symmetric algebra, Trans. Amer. Math. Soc. 296 (1986), no. 2, 577606. MR 846598 (87f:13009)
 [Hu]
 H. A. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 23352350. MR 1218501 (94c:13015)
 [I]
 A. Iarrobino, Punctual Hilbert schemes, Mem. Amer. Math. Soc. 10, no. 188 (1977). MR 0485867 (58:5667)
 [M]
 F. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531555.
 [P]
 K. Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), no. 4, 564585. MR 1415019 (97g:13029)
 [Pe]
 I. Peeva, Consecutive cancellations in Betti numbers, Proc. Amer. Math. Soc. 132 (2004), no. 12, 35033507. MR 2084070 (2006a:13023)
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 L. Robbiano, Coni tangenti a singolaritá razionali, Curve algebriche, Istituto di Analisi Globale, Firenze, 1981.
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 L. Robbiano, G. Valla, On the equations defining tangent cones, Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 2, 281297. MR 578272 (81i:14004)
 [RSh]
 M. E. Rossi, L. Sharifan, Extremal Betti numbers of filtered modules over a regular local ring, to appear in J. of Algebra, in press.
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 [Sh]
 T. Shibuta, CohenMacaulayness of almost complete intersection tangent cones, J. Algebra 319 (8) (2008), 32223243. MR 2408315 (2009b:13061)
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Additional Information
Maria Evelina Rossi
Affiliation:
Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146 Genoa, Italy
Email:
rossim@dima.unige.it
Leila Sharifan
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Avenue, 15914 Tehran, Iran
Email:
leilasharifan@aut.ac.ir
DOI:
http://dx.doi.org/10.1090/S0002993909100102
PII:
S 00029939(09)100102
Keywords:
Minimal free resolution,
filtered module,
associated graded module,
Betti numbers,
lexicographic ideal,
standard bases.
Received by editor(s):
February 11, 2009
Received by editor(s) in revised form:
April 17, 2009
Published electronically:
August 28, 2009
Communicated by:
Bernd Ulrich
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
