On Burch's inequality and a reduction system of a filtration

Authors:
Y. Kinoshita, K. Nishida, Y. Yamanaka and A. Yoneda

Journal:
Proc. Amer. Math. Soc. **134** (2006), 3437-3444

MSC (2000):
Primary 13A02, 13A30

DOI:
https://doi.org/10.1090/S0002-9939-06-08429-2

Published electronically:
June 9, 2006

MathSciNet review:
2240653

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a multiplicative filtration of a local ring such that the Rees algebra is Noetherian. We recall Burch's inequality for and give an upper bound of the a-invariant of the associated graded ring using a reduction system of . Applying those results, we study the symbolic Rees algebra of certain ideals of dimension .

**1.**Aberbach, I., Huneke, C. and Trung, N. V.,*Reduction numbers, Briançon-Skoda theorem and the depth of Rees rings*, Compositio Math.,**97**(1995), 403-434. MR**1353282 (96g:13002)****2.**Bruns, W. and Vetter, U.,*Determinantal rings*, Lecture Notes in Math.,**1327**, Springer, 1988. MR**0953963 (89i:13001)****3.**Burch, L.,*Codimension and analytic spread*, Proc. Camb. Philos. Soc.,**72**(1972), 369-373.MR**0304377 (46:3512)****4.**Conca, A.,*Straightening law and powers of determinantal ideals of Hankel matrices*, Adv. Math.,**138**(1998), 263-292.MR**1645574 (99i:13020)****5.**Goto, S. and Nishida, K.,*The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations*, Mem. Amer. Math. Soc.,**526**(1994).MR**1287443 (95b:13001)****6.**Goto, S., Nishida, K. and Shimoda, Y.,*Topics on symbolic Rees algebras for space monomial curves*, Nagoya Math. J.,**124**(1991), 99-132. MR**1142978 (93e:13002)****7.**Goto, S. and Watanabe, K.,*On graded rings I*, J. Math. Soc. Japan,**30**(1978), 179-213. MR**0494707 (81m:13021)****8.**Johnston, B. and Katz, D.,*Castelnuovo regularity and graded rings associated to an ideal*, Proc. Amer. Math. Soc.,**123**(1995), 727-734. MR**1231300 (95d:13005)****9.**Nishida, K.,*On filtrations having small analytic deviation*, Comm. Algebra,**29**(2001), 2711-2729. MR**1845138 (2002k:13006)****10.**Nishida, K.,*On the depth of the associated graded ring of a filtration*, J. Algebra,**285**(2005), 182-195. MR**2119110****11.**Peskine, C. and Szpiro, L.,*Liaison des variétés algébriques I*, Invent. Math.**26**(1974), 271-302. MR**0364271 (51:526)****12.**Valabrega, P. and Valla, G.,*Form rings and regular sequences*, Nagoya Math. J.,**72**(1978), 93-101. MR**0514892 (80d:14010)**

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Additional Information

**Y. Kinoshita**

Affiliation:
Division of Mathematical Sciences and Physics, School of Science and Technology, Chiba University, 263-8522, Japan

**K. Nishida**

Affiliation:
Division of Mathematical Sciences and Physics, School of Science and Technology, Chiba University, 263-8522, Japan

Email:
nishida@math.s.chiba-u.ac.jp

**Y. Yamanaka**

Affiliation:
Division of Mathematical Sciences and Physics, School of Science and Technology, Chiba University, 263-8522, Japan

**A. Yoneda**

Affiliation:
Division of Mathematical Sciences and Physics, School of Science and Technology, Chiba University, 263-8522, Japan

DOI:
https://doi.org/10.1090/S0002-9939-06-08429-2

Keywords:
Multiplicative filtration,
Rees algebra,
associated graded ring

Received by editor(s):
April 22, 2004

Received by editor(s) in revised form:
July 1, 2005

Published electronically:
June 9, 2006

Additional Notes:
The second author was supported by the Grant-in-Aid for Scientific Researches in Japan (C) (2) No. 15540009

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2006
American Mathematical Society