Consecutive cancellations in Betti numbers of local rings

Authors:
Maria Evelina Rossi and Leila Sharifan

Journal:
Proc. Amer. Math. Soc. **138** (2010), 61-73

MSC (2000):
Primary 13D02

DOI:
https://doi.org/10.1090/S0002-9939-09-10010-2

Published electronically:
August 28, 2009

MathSciNet review:
2550170

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 Eliahou-Kervaire'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.

**[B]**V. Bertella, Hilbert function of local Artinian level rings in codimension two, J. Algebra 321 (2009), no. 5, 1429-1442.**[Bi]**Anna Maria Bigatti,*Upper bounds for the Betti numbers of a given Hilbert function*, Comm. Algebra**21**(1993), no. 7, 2317–2334. MR**1218500**, https://doi.org/10.1080/00927879308824679**[C]**CoCoA Team, CoCoA: a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.**[E]**David Eisenbud,*The geometry of syzygies*, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR**2103875****[EK]**Shalom Eliahou and Michel Kervaire,*Minimal resolutions of some monomial ideals*, J. Algebra**129**(1990), no. 1, 1–25. MR**1037391**, https://doi.org/10.1016/0021-8693(90)90237-I**[EV]**Juan Elias and Giuseppe Valla,*Structure theorems for certain Gorenstein ideals*, Michigan Math. J.**57**(2008), 269–292. Special volume in honor of Melvin Hochster. MR**2492453**, https://doi.org/10.1307/mmj/1220879409**[HM]**Satoshi Murai and Takayuki Hibi,*The depth of an ideal with a given Hilbert function*, Proc. Amer. Math. Soc.**136**(2008), no. 5, 1533–1538. MR**2373580**, https://doi.org/10.1090/S0002-9939-08-09067-9**[HRV]**J. Herzog, M. E. Rossi, and G. Valla,*On the depth of the symmetric algebra*, Trans. Amer. Math. Soc.**296**(1986), no. 2, 577–606. MR**846598**, https://doi.org/10.1090/S0002-9947-1986-0846598-8**[Hu]**Heather A. Hulett,*Maximum Betti numbers of homogeneous ideals with a given Hilbert function*, Comm. Algebra**21**(1993), no. 7, 2335–2350. MR**1218501**, https://doi.org/10.1080/00927879308824680**[I]**Anthony A. Iarrobino,*Punctual Hilbert schemes*, Mem. Amer. Math. Soc.**10**(1977), no. 188, viii+112. MR**0485867**, https://doi.org/10.1090/memo/0188**[M]**F. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555.**[P]**Keith Pardue,*Deformation classes of graded modules and maximal Betti numbers*, Illinois J. Math.**40**(1996), no. 4, 564–585. MR**1415019****[Pe]**Irena Peeva,*Consecutive cancellations in Betti numbers*, Proc. Amer. Math. Soc.**132**(2004), no. 12, 3503–3507. MR**2084070**, https://doi.org/10.1090/S0002-9939-04-07517-3**[R]**L. Robbiano, Coni tangenti a singolaritá razionali, Curve algebriche, Istituto di Analisi Globale, Firenze, 1981.**[RoV]**Lorenzo Robbiano and Giuseppe Valla,*On the equations defining tangent cones*, Math. Proc. Cambridge Philos. Soc.**88**(1980), no. 2, 281–297. MR**578272**, https://doi.org/10.1017/S0305004100057583**[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.**[RV]**M. E. Rossi, G. Valla, Hilbert function of filtered modules, arXiv:0710.2346v1 [math.AC].**[Sh]**Takafumi Shibuta,*Cohen-Macaulayness of almost complete intersection tangent cones*, J. Algebra**319**(2008), no. 8, 3222–3243. MR**2408315**, https://doi.org/10.1016/j.jalgebra.2007.11.023

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
13D02

Retrieve articles in all journals with MSC (2000): 13D02

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:
leila-sharifan@aut.ac.ir

DOI:
https://doi.org/10.1090/S0002-9939-09-10010-2

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.