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), 34373444
MSC (2000):
Primary 13A02, 13A30
Published electronically:
June 9, 2006
MathSciNet review:
2240653
Fulltext PDF Free Access
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 ainvariant 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.
Ian
M. Aberbach, Craig
Huneke, and Ngô
Vi\cfudot{e}t Trung, Reduction numbers, BriançonSkoda
theorems and the depth of Rees rings, Compositio Math.
97 (1995), no. 3, 403–434. MR 1353282
(96g:13002)
 2.
Winfried
Bruns and Udo
Vetter, Determinantal rings, Lecture Notes in Mathematics,
vol. 1327, SpringerVerlag, Berlin, 1988. MR 953963
(89i:13001)
 3.
Lindsay
Burch, Codimension and analytic spread, Proc. Cambridge
Philos. Soc. 72 (1972), 369–373. MR 0304377
(46 #3512)
 4.
Aldo
Conca, Straightening law and powers of determinantal ideals of
Hankel matrices, Adv. Math. 138 (1998), no. 2,
263–292. MR 1645574
(99i:13020), http://dx.doi.org/10.1006/aima.1998.1740
 5.
Shiro
Goto and Koji
Nishida, The CohenMacaulay and Gorenstein Rees algebras associated
to filtrations, American Mathematical Society, Providence, RI, 1994.
Mem. Amer. Math. Soc. 110 (1994), no. 526. MR 1287443
(95b:13001)
 6.
Shiro
Goto, Koji
Nishida, and Yasuhiro
Shimoda, Topics on symbolic Rees algebras for space monomial
curves, Nagoya Math. J. 124 (1991), 99–132. MR 1142978
(93e:13002)
 7.
Shiro
Goto and Keiichi
Watanabe, On graded rings. I, J. Math. Soc. Japan
30 (1978), no. 2, 179–213. MR 494707
(81m:13021), http://dx.doi.org/10.2969/jmsj/03020179
 8.
Bernard
Johnston and Daniel
Katz, Castelnuovo regularity and graded
rings associated to an ideal, Proc. Amer. Math.
Soc. 123 (1995), no. 3, 727–734. MR 1231300
(95d:13005), http://dx.doi.org/10.1090/S00029939199512313001
 9.
Koji
Nishida, On filtrations having small analytic deviation, Comm.
Algebra 29 (2001), no. 6, 2711–2729. MR 1845138
(2002k:13006), http://dx.doi.org/10.1081/AGB100002416
 10.
Koji
Nishida, On the depth of the associated graded ring of a
filtration, J. Algebra 285 (2005), no. 1,
182–195. MR 2119110
(2006b:13011), http://dx.doi.org/10.1016/j.jalgebra.2004.10.026
 11.
C.
Peskine and L.
Szpiro, Liaison des variétés algébriques.
I, Invent. Math. 26 (1974), 271–302 (French).
MR
0364271 (51 #526)
 12.
Paolo
Valabrega and Giuseppe
Valla, Form rings and regular sequences, Nagoya Math. J.
72 (1978), 93–101. MR 514892
(80d:14010)
 1.
 Aberbach, I., Huneke, C. and Trung, N. V., Reduction numbers, BriançonSkoda theorem and the depth of Rees rings, Compositio Math., 97 (1995), 403434. 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), 369373.MR 0304377 (46:3512)
 4.
 Conca, A., Straightening law and powers of determinantal ideals of Hankel matrices, Adv. Math., 138 (1998), 263292.MR 1645574 (99i:13020)
 5.
 Goto, S. and Nishida, K., The CohenMacaulay 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), 99132. MR 1142978 (93e:13002)
 7.
 Goto, S. and Watanabe, K., On graded rings I, J. Math. Soc. Japan, 30 (1978), 179213. 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), 727734. MR 1231300 (95d:13005)
 9.
 Nishida, K., On filtrations having small analytic deviation, Comm. Algebra, 29 (2001), 27112729. MR 1845138 (2002k:13006)
 10.
 Nishida, K., On the depth of the associated graded ring of a filtration, J. Algebra, 285 (2005), 182195. MR 2119110
 11.
 Peskine, C. and Szpiro, L., Liaison des variétés algébriques I, Invent. Math. 26 (1974), 271302. MR 0364271 (51:526)
 12.
 Valabrega, P. and Valla, G., Form rings and regular sequences, Nagoya Math. J., 72 (1978), 93101. MR 0514892 (80d:14010)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
13A02,
13A30
Retrieve articles in all journals
with MSC (2000):
13A02,
13A30
Additional Information
Y. Kinoshita
Affiliation:
Division of Mathematical Sciences and Physics, School of Science and Technology, Chiba University, 2638522, Japan
K. Nishida
Affiliation:
Division of Mathematical Sciences and Physics, School of Science and Technology, Chiba University, 2638522, Japan
Email:
nishida@math.s.chibau.ac.jp
Y. Yamanaka
Affiliation:
Division of Mathematical Sciences and Physics, School of Science and Technology, Chiba University, 2638522, Japan
A. Yoneda
Affiliation:
Division of Mathematical Sciences and Physics, School of Science and Technology, Chiba University, 2638522, Japan
DOI:
http://dx.doi.org/10.1090/S0002993906084292
PII:
S 00029939(06)084292
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 GrantinAid for Scientific Researches in Japan (C) (2) No. 15540009
Communicated by:
Bernd Ulrich
Article copyright:
© Copyright 2006
American Mathematical Society
