|
Core and residual intersections of ideals
Author(s):
Alberto
Corso;
Claudia
Polini;
Bernd
Ulrich
Journal:
Trans. Amer. Math. Soc.
354
(2002),
2579-2594.
MSC (2000):
Primary 13H10;
Secondary 13A30, 13B22, 13C40, 13D45
Posted:
February 1, 2002
MathSciNet review:
1895194
Retrieve article in:
PDF
This article is available free of charge
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.
References:
-
- 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
Similar Articles:
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:
10.1090/S0002-9947-02-02908-2
PII:
S 0002-9947(02)02908-2
Keywords:
Integral closure,
reductions,
residual intersections of ideals
Received by editor(s):
April 10, 2001
Posted:
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
Copyright of article:
Copyright
2002,
American Mathematical Society
|
