Variation of Hilbert coefficients
Authors:
Laura Ghezzi, Shiro Goto, Jooyoun Hong and Wolmer V. Vasconcelos
Journal:
Proc. Amer. Math. Soc. 141 (2013), 30373048
MSC (2010):
Primary 13A30; Secondary 13B22, 13H10, 13H15
Published electronically:
June 3, 2013
MathSciNet review:
3068957
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Similar Articles 
Additional Information
Abstract: For a Noetherian local ring , the first two Hilbert coefficients, and , of the adic filtration of an primary ideal are known to code for properties of , of the blowup of along , and even of their normalizations. We give estimations for these coefficients when is enlarged (in the case of in the same integral closure class) for general Noetherian local rings.
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T. Phuong, and W.
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Laura
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Hong, and Wolmer
V. Vasconcelos, The signature of the Chern coefficients of local
rings, Math. Res. Lett. 16 (2009), no. 2,
279–289. MR 2496744
(2010b:13003), 10.4310/MRL.2009.v16.n2.a6
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Shiro
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Hong, and Mousumi
Mandal, The positivity of the first
coefficients of normal Hilbert polynomials, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2399–2406. MR 2784804
(2012e:13031), 10.1090/S000299392010107104
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S. Goto, N. Matsuoka and T. T. Phuong, Almost Gorenstein rings, preprint. arXiv:1106.1301v2 [Math.AC].
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Shiro
Goto and Kazuho
Ozeki, Buchsbaumness in local rings possessing constant first
Hilbert coefficients of parameters, Nagoya Math. J.
199 (2010), 95–105. MR 2730412
(2011m:13048)
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Hanumanthu and Craig
Huneke, Bounding the first Hilbert
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(2012e:13032), 10.1090/S000299392011110219
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Mousumi
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Claudia
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(2006m:13006), 10.4310/MRL.2005.v12.n6.a5
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Univ. Padova 121 (2009), 201–222. MR 2542142
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Judith
D. Sally, Numbers of generators of ideals in local rings,
Marcel Dekker, Inc., New YorkBasel, 1978. MR 0485852
(58 #5654)
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Judith
D. Sally, Hilbert coefficients and reduction number 2, J.
Algebraic Geom. 1 (1992), no. 2, 325–333. MR 1144442
(93b:13026)
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Judith
D. Sally and Wolmer
V. Vasconcelos, Stable rings, J. Pure Appl. Algebra
4 (1974), 319–336. MR 0409430
(53 #13185)
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Wolmer
Vasconcelos, Integral closure, Springer Monographs in
Mathematics, SpringerVerlag, Berlin, 2005. Rees algebras, multiplicities,
algorithms. MR
2153889 (2006m:13007)
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Wolmer
V. Vasconcelos, The Chern coefficients of local rings,
Michigan Math. J. 57 (2008), 725–743. Special volume
in honor of Melvin Hochster. MR 2492478
(2009m:13005), 10.1307/mmj/1220879434
 [C]
 A. Corso, Sally modules of primary ideals in local rings, Comm. Algebra 37 (2009), 45034515. MR 2588863 (2011e:13007)
 [ES]
 P. Eakin and A. Sathaye, Prestable ideals, J. Algebra 41 (1976), 439454. MR 0419428 (54:7449)
 [E1]
 J. Elias, On the first normalized Hilbert coefficient, J. Pure and Applied Algebra 201 (2005), 116125. MR 2158750 (2006d:13017)
 [E2]
 J. Elias, Upper bounds of Hilbert coefficients and Hilbert functions, Math. Proc. Camb. Phil. Soc. 145 (2008), 8794. MR 2431640 (2009d:13020)
 [GhGHOPV]
 L. Ghezzi, S. Goto, J. Hong, K. Ozeki, T.T. Phuong and W. V. Vasconcelos, CohenMacaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals, J. London Math. Soc. 81 (2010), 679695. MR 2650791 (2011k:13040)
 [GhHV]
 L. Ghezzi, J. Hong and W. V. Vasconcelos, The signature of the Chern coefficients of local rings, Math. Research Letters 16 (2009), 279289. MR 2496744 (2010b:13003)
 [GHM]
 S. Goto, J. Hong and M. Mandal, The positivity of the first normalized Hilbert coefficients, Proc. Amer. Math. Soc. 139 (2011), 23992406. MR 2784804 (2012e:13031)
 [GMP]
 S. Goto, N. Matsuoka and T. T. Phuong, Almost Gorenstein rings, preprint. arXiv:1106.1301v2 [Math.AC].
 [GO]
 S. Goto and K. Ozeki, Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters, Nagoya Math. J. 199 (2010), 95105. MR 2730412 (2011m:13048)
 [HH]
 K. Hanumanthu and C. Huneke, Bounding the first Hilbert coefficient, Proc. Amer. Math. Soc. 140 (2012), 109117. MR 2833522 (2012e:13032)
 [Hu]
 C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293318. MR 894879 (89b:13037)
 [K]
 D. Kirby, A note on superficial elements of an ideal of a local ring, Q. J. Math. Oxford 14 (1963), 2128. MR 0143780 (26:1332)
 [MV]
 M. Mandal and J. K. Verma, On the Chern number of an ideal, Proc. Amer. Math. Soc. 138 (2010), 19951999. MR 2596035 (2011c:13029)
 [MSV]
 M. Mandal, B. Singh and J. K. Verma, On some conjectures about the Chern numbers of filtrations, J. Algebra 325 (2011), 147162. MR 2745533 (2012a:13028)
 [PUV]
 C. Polini, B. Ulrich and W. V. Vasconcelos, Normalization of ideals and BriançonSkoda numbers, Math. Research Letters 12 (2005), 827842. MR 2189243 (2006m:13006)
 [R]
 M. E. Rossi, A bound on the reduction number of a primary ideal, Proc. Amer. Math. Soc. 128 (2000), 13251332. MR 1670423 (2000j:13004)
 [RV1]
 M. E. Rossi and G. Valla, The Hilbert function of the RattliffRush filtration, J. Pure and Applied Algebra 201 (2005), 2441. MR 2158745 (2006g:13008)
 [RV2]
 M. E. Rossi and G. Valla, On the Chern number of a filtration, Rendiconti Seminario Matematico Padova 121 (2009), 201222. MR 2542142 (2010h:13041)
 [S1]
 J. D. Sally, Numbers of Generators of Ideals in Local Rings, Lecture Notes in Pure and Applied Mathematics 36, Marcel Dekker, New York, 1978. MR 0485852 (58:5654)
 [S2]
 J. D. Sally, Hilbert coefficients and reduction number , J. Algebraic Geometry 1 (1992), 325333. MR 1144442 (93b:13026)
 [SV]
 J. D. Sally and W. V. Vasconcelos, Stable rings, J. Pure and Applied Algebra 4 (1974), 319336. MR 0409430 (53:13185)
 [V1]
 W. V. Vasconcelos, Integral Closure, Springer Monographs in Mathematics, Springer, Heidelberg, 2005. MR 2153889 (2006m:13007)
 [V2]
 W. V. Vasconcelos, The Chern coefficients of local rings, Michigan Math. J. 57 (2008), 725743. MR 2492478 (2009m:13005)
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Additional Information
Laura Ghezzi
Affiliation:
Department of Mathematics, New York City College of TechnologyCUNY, 300 Jay Street, Brooklyn, New York 11201
Email:
lghezzi@citytech.cuny.edu
Shiro Goto
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, 111 Higashimita, Tamaku, Kawasaki 2148571, Japan
Email:
goto@math.meiji.ac.jp
Jooyoun Hong
Affiliation:
Department of Mathematics, Southern Connecticut State University, 501 Crescent Street, New Haven, Connecticut 065151533
Email:
hongj2@southernct.edu
Wolmer V. Vasconcelos
Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 088548019
Email:
vasconce@math.rutgers.edu
DOI:
http://dx.doi.org/10.1090/S000299392013117740
Received by editor(s):
August 22, 2011
Received by editor(s) in revised form:
December 3, 2011
Published electronically:
June 3, 2013
Additional Notes:
The first author was partially supported by a grant from the City University of New York PSCCUNY Research Award Program41
The second author was partially supported by GrantinAid for Scientific Researches (C) in Japan (19540054) and by a grant from MIMS (Meiji Institute for Advanced Study of Mathematical Sciences)
The fourth author was partially supported by the NSF
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
