Core and residual intersections of ideals

Authors:
Alberto Corso, Claudia Polini and Bernd Ulrich

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2579-2594

MSC (2000):
Primary 13H10; Secondary 13A30, 13B22, 13C40, 13D45

DOI:
https://doi.org/10.1090/S0002-9947-02-02908-2

Published electronically:
February 1, 2002

MathSciNet review:
1895194

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: D. Rees and J. Sally defined the core of an -ideal as the intersection of all (minimal) reductions of . However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of integrally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.

**1.**I.M. Aberbach and C. Huneke, A theorem of Briançon-Skoda type for regular local rings containing a field, Proc. Amer. Math. Soc.**124**(1996), 707-713. MR**96f:13039****2.**L. Avramov and J. Herzog, The Koszul algebra of a codimension embedding, Math. Z.**175**(1980), 249-260. MR**82g:13011****3.**M. Artin and M. Nagata, Residual intersections in Cohen-Macaulay rings, J. Math. Kyoto Univ.**12**(1972), 307-323. MR**46:166****4.**D. Bayer and M.E. Stillman,`Macaulay`, A computer algebra system for computing in Algebraic Geometry and Commutative Algebra, 1990. Available via anonymous ftp from zariski.harvard.edu.**5.**D. Buchsbaum and D. Eisenbud, What annihilates a module?, J. Algebra**47**(1977), 231-243. MR**57:16293****6.**M. Chardin, D. Eisenbud and B. Ulrich, Hilbert functions, residual intersections, and residually -ideals, Compositio Math.**125**(2001), 193-219. CMP**2001:09****7.**A. Corso and C. Polini, Reduction number of links of irreducible varieties, J. Pure Appl. Algebra**121**(1997), 29-43. MR**98h:13030****8.**A. Corso and C. Polini, On residually ideals and projective dimension one modules, Proc. Amer. Math. Soc.**129**(2001), 1309-1315. MR**2001m:13040****9.**A. Corso, C. Polini and B. Ulrich, The structure of the core of ideals, Math. Ann.**321**(2001), 89-105. CMP**2002:02****10.**R. Cowsik and M. Nori, On the fibers of blowing-up, J. Indian Math. Soc.**40**(1976), 217-222. MR**58:28011****11.**S. Goto, S.-I. Iai and K.-I. Watanabe, Good ideals in Gorenstein local rings, Trans. Amer. Math. Soc.**353**(2001), 2309-2346. CMP**2001:08****12.**R. Hübl and C. Huneke, Fiber cones and the integral closure of ideals, Collect. Math.**52**(2001), 85-100.**13.**J. Herzog, A. Simis and W.V. Vasconcelos, Koszul homology and blowing-up rings, in*Commutative Algebra, Proceedings: Trento 1981*(Greco/Valla eds.), Lecture Notes in Pure and Applied Mathematics**84**, Marcel Dekker, New York, 1983, 79-169. MR**84k:13015****14.**J. Herzog, W.V. Vasconcelos and R.H. Villarreal, Ideals with sliding depth, Nagoya Math. J.**99**(1985), 159-172. MR**86k:13022****15.**J. Herzog and B. Ulrich, Self-linked curve singularities, Nagoya Math. J.**120**(1990), 129-153. MR**92c:13010****16.**C. Huneke, Linkage and Koszul homology of ideals, Amer. J. Math.**104**(1982), 1043-1062. MR**84f:13019****17.**C. Huneke and I. Swanson, Cores of ideals in -dimensional regular local rings, Michigan Math. J.**42**(1995), 193-208. MR**96j:13021****18.**E. Hyry, Coefficient ideals and the Cohen-Macaulay property of Rees algebras, Proc. Amer. Math. Soc.**129**(2001), 1299-1308. MR**2001h:13005****19.**M. Johnson and B. Ulrich, Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compositio Math.**103**(1996), 7-29. MR**97f:13006****20.**S. Kleiman and B. Ulrich, Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers, Trans. Amer. Math. Soc.**349**(1997), 4973-5000. MR**98c:13019****21.**J. Lipman, Adjoints of ideals in regular local rings, Math. Research Letters**1**(1994), 739-755. MR**95k:13028****22.**J. Lipman and B. Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J.**28**(1981), 97-112. MR**82f:14004****23.**D.G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Camb. Phil. Soc.**50**(1954), 145-158. MR**15:596a****24.**C. Polini and B. Ulrich, Linkage and reduction numbers, Math. Ann.**310**(1998), 631-651. MR**99g:13017****25.**D. Rees and J.D. Sally, General elements and joint reductions, Michigan Math. J.**35**(1988), 241-254. MR**89h:13034****26.**B. Ulrich, Artin-Nagata properties and reductions of ideals, Contemp. Math.**159**(1994), 373-400. MR**95a:13017****27.**B. Ulrich, Ideals having the expected reduction number, Amer. J. Math.**118**(1996), 17-38. MR**97b:13003****28.**P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J.**72**(1978), 91-101. MR**80d:14010****29.**O. Zariski and P. Samuel,*Commutative Algebra*, Vol. 2, Van Nostrand, Princeton, 1960. MR**22:11006**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
13H10,
13A30,
13B22,
13C40,
13D45

Retrieve articles in all journals with MSC (2000): 13H10, 13A30, 13B22, 13C40, 13D45

Additional Information

**Alberto Corso**

Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Email:
corso@ms.uky.edu

**Claudia Polini**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Address at time of publication:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Email:
cpolini@nd.edu

**Bernd Ulrich**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Address at time of publication:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Email:
ulrich@math.purdue.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-02908-2

Keywords:
Integral closure,
reductions,
residual intersections of ideals

Received by editor(s):
April 10, 2001

Published electronically:
February 1, 2002

Additional Notes:
The first author was partially supported by the NATO/CNR Advanced Fellowships Programme during an earlier stage of this work. The second and third authors were partially supported by the NSF

Dedicated:
To Professor Craig Huneke on the occasion of his fiftieth birthday

Article copyright:
© Copyright 2002
American Mathematical Society