A bound on the reduction number

of a primary ideal

Author:
M. E. Rossi

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1325-1332

MSC (1991):
Primary 14M05; Secondary 13H10

Published electronically:
October 5, 1999

MathSciNet review:
1670423

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a local ring of positive dimension and let be an -primary ideal. We denote the reduction number of by , which is the smallest integer such that for some reduction of In this paper we give an upper bound on in terms of numerical invariants which are related with the Hilbert coefficients of when is Cohen-Macaulay. If , it is known that where denotes the multiplicity of If in Corollary 1.5 we prove where is the first Hilbert coefficient of From this bound several results follow. Theorem 1.3 gives an upper bound on in a more general setting.

**[B]**Cristina Blancafort,*Hilbert functions of graded algebras over Artinian rings*, J. Pure Appl. Algebra**125**(1998), no. 1-3, 55–78. MR**1600008**, 10.1016/S0022-4049(96)00124-7**[CPV]**A. Corso, C. Polini, M. Vaz Pinto, Sally Modules and associated graded rings,*Communications in Algebra***26**(8) (1998), 2689-2708. CMP**98:15****[E]**J. Elias, On the depth of the tangent cone and the growth of the Hilbert function, to appear in*Trans. Amer. Math. Soc.*CMP**98:07****[GR]**A. Guerrieri and M. E. Rossi,*Hilbert coefficients of Hilbert filtrations*, J. Algebra**199**(1998), no. 1, 40–61. MR**1489353**, 10.1006/jabr.1997.7194**[HM]**Sam Huckaba and Thomas Marley,*Hilbert coefficients and the depths of associated graded rings*, J. London Math. Soc. (2)**56**(1997), no. 1, 64–76. MR**1462826**, 10.1112/S0024610797005206**[H]**Craig Huneke,*Hilbert functions and symbolic powers*, Michigan Math. J.**34**(1987), no. 2, 293–318. MR**894879**, 10.1307/mmj/1029003560**[I]**Shiroh Itoh,*Hilbert coefficients of integrally closed ideals*, J. Algebra**176**(1995), no. 2, 638–652. MR**1351629**, 10.1006/jabr.1995.1264**[N]**D. G. Northcott,*A note on the coefficients of the abstract Hilbet function*, J. London Math. Soc.**35**(1960), 209–214. MR**0110731****[O]**Akira Ooishi,*Δ-genera and sectional genera of commutative rings*, Hiroshima Math. J.**17**(1987), no. 2, 361–372. MR**909621****[RR]**L. J. Ratliff Jr. and David E. Rush,*Two notes on reductions of ideals*, Indiana Univ. Math. J.**27**(1978), no. 6, 929–934. MR**0506202****[R]**M. E. Rossi, Primary ideals with good associated graded ring, to appear in*J. of Pure and Applied Algebra*.**[RV]**M. E. Rossi and G. Valla,*A conjecture of J. Sally*, Comm. Algebra**24**(1996), no. 13, 4249–4261. MR**1414582**, 10.1080/00927879608825812**[S]**Irena Swanson,*A note on analytic spread*, Comm. Algebra**22**(1994), no. 2, 407–411. MR**1255875**, 10.1080/00927879408824857**[S1]**Judith D. Sally,*Bounds for numbers of generators of Cohen-Macaulay ideals*, Pacific J. Math.**63**(1976), no. 2, 517–520. MR**0409453****[S2]**Judith D. Sally,*Cohen-Macaulay local rings of embedding dimension 𝑒+𝑑-2*, J. Algebra**83**(1983), no. 2, 393–408. MR**714252**, 10.1016/0021-8693(83)90226-0**[S3]**Judith D. Sally,*Hilbert coefficients and reduction number 2*, J. Algebraic Geom.**1**(1992), no. 2, 325–333. MR**1144442****[V]**Giuseppe Valla,*On form rings which are Cohen-Macaulay*, J. Algebra**58**(1979), no. 2, 247–250. MR**540637**, 10.1016/0021-8693(79)90159-5**[VV]**Paolo Valabrega and Giuseppe Valla,*Form rings and regular sequences*, Nagoya Math. J.**72**(1978), 93–101. MR**514892****[VW]**W. Vasconcelos, Cohomological Degrees of graded modules,*Six Lectures on Commutative algebra*, Progress in Math.**166**, Birkhauser, Boston (1998), 345-392. CMP**99:02****[W]**Hsin-Ju Wang,*On Cohen-Macaulay local rings with embedding dimension 𝑒+𝑑-2*, J. Algebra**190**(1997), no. 1, 226–240. MR**1442154**, 10.1006/jabr.1996.6894

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
14M05,
13H10

Retrieve articles in all journals with MSC (1991): 14M05, 13H10

Additional Information

**M. E. Rossi**

Affiliation:
Dipartimento di Matematica, Universita’ di Genova, Via Dodecaneso 35, 16146- Genova, Italy

Email:
rossim@dima.unige.it

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05393-9

Keywords:
Cohen-Macaulay local ring,
primary ideals,
reduction number,
minimal reduction,
associated graded ring,
Hilbert function,
Hilbert coefficients

Received by editor(s):
July 6, 1998

Published electronically:
October 5, 1999

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 2000
American Mathematical Society