## On the depth of the associated graded ring

HTML articles powered by AMS MathViewer

- by Anna Guerrieri PDF
- Proc. Amer. Math. Soc.
**123**(1995), 11-20 Request permission

## Abstract:

Let (*R, m*) be a Cohen-Macaulay local ring of positive dimension

*d*, let

*I*be an $m -$ primary ideal of

*R*. In this paper we individuate some conditions on

*I*that allow us to determine a lower bound for depth ${\text {gr}_I}(R)$. It is proved that if $J \subseteq I$ is a minimal reduction of

*I*such that $\lambda ({I^2} \cap J/IJ) = 2$ and ${I^n} \cap J = {I^{n - 1}}J$ for all $n \geq 3$, then depth ${\text {gr}_I}(R) \geq d - 2$; let us remark that $\lambda$ denotes the length function.

## References

- Anna Guerrieri,
*On the depth of the associated graded ring of an $m$-primary ideal of a Cohen-Macaulay local ring*, J. Algebra**167**(1994), no. 3, 745–757. MR**1287068**, DOI 10.1006/jabr.1994.1210 - Sam Huckaba,
*Reduction numbers for ideals of higher analytic spread*, Math. Proc. Cambridge Philos. Soc.**102**(1987), no. 1, 49–57. MR**886434**, DOI 10.1017/S0305004100067037 - Craig Huneke,
*Hilbert functions and symbolic powers*, Michigan Math. J.**34**(1987), no. 2, 293–318. MR**894879**, DOI 10.1307/mmj/1029003560 - Irving Kaplansky,
*$R$-sequences and homological dimension*, Nagoya Math. J.**20**(1962), 195–199. MR**175955**, DOI 10.1017/S0027763000023710 - Thomas Marley,
*The coefficients of the Hilbert polynomial and the reduction number of an ideal*, J. London Math. Soc. (2)**40**(1989), no. 1, 1–8. MR**1028910**, DOI 10.1112/jlms/s2-40.1.1 - Hideyuki Matsumura,
*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR**879273** - D. G. Northcott and D. Rees,
*Reductions of ideals in local rings*, Proc. Cambridge Philos. Soc.**50**(1954), 145–158. MR**59889**, DOI 10.1017/s0305004100029194 - Judith D. Sally,
*Reductions, local cohomology and Hilbert functions of local rings*, Commutative algebra: Durham 1981 (Durham, 1981) London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 231–241. MR**693638** - Judith D. Sally,
*Hilbert coefficients and reduction number $2$*, J. Algebraic Geom.**1**(1992), no. 2, 325–333. MR**1144442**
—, - Judith D. Sally,
*Numbers of generators of ideals in local rings*, Marcel Dekker, Inc., New York-Basel, 1978. MR**0485852** - Ngô Việt Trung,
*Reduction exponent and degree bound for the defining equations of graded rings*, Proc. Amer. Math. Soc.**101**(1987), no. 2, 229–236. MR**902533**, DOI 10.1090/S0002-9939-1987-0902533-1 - Giuseppe Valla,
*On form rings which are Cohen-Macaulay*, J. Algebra**58**(1979), no. 2, 247–250. MR**540637**, DOI 10.1016/0021-8693(79)90159-5 - Paolo Valabrega and Giuseppe Valla,
*Form rings and regular sequences*, Nagoya Math. J.**72**(1978), 93–101. MR**514892**, DOI 10.1017/S0027763000018225 - Wolmer V. Vasconcelos,
*Hilbert functions, analytic spread, and Koszul homology*, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 401–422. MR**1266195**, DOI 10.1090/conm/159/01520 - Yinghwa Wu,
*Reduction numbers and Hilbert polynomials of ideals in higher-dimensional Cohen-Macaulay local rings*, Math. Proc. Cambridge Philos. Soc.**111**(1992), no. 1, 47–56. MR**1131477**, DOI 10.1017/S0305004100075149

*Ideals whose Hilbert function and Hilbert polynomial agree at*$n = 1$, preprint.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 11-20 - MSC: Primary 13A30; Secondary 13C15, 13H10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1211580-9
- MathSciNet review: 1211580