|
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
Posted:
February 1, 2002
MathSciNet review:
1895194
Full-text PDF Free Access
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.
Ian
M. Aberbach and Craig
Huneke, A theorem of Briançon-Skoda
type for regular local rings containing a field, Proc. Amer. Math. Soc. 124 (1996), no. 3, 707–713. MR 1301483
(96f:13039), http://dx.doi.org/10.1090/S0002-9939-96-03058-4
- 2.
Lâcezar
Avramov and Jürgen
Herzog, The Koszul algebra of a codimension 2 embedding, Math.
Z. 175 (1980), no. 3, 249–260. MR 602637
(82g:13011), http://dx.doi.org/10.1007/BF01163026
- 3.
M.
Artin and M.
Nagata, Residual intersections in Cohen-Macaulay rings, J.
Math. Kyoto Univ. 12 (1972), 307–323. MR 0301006
(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.
David
A. Buchsbaum and David
Eisenbud, What annihilates a module?, J. Algebra
47 (1977), no. 2, 231–243. MR 0476736
(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.
Alberto
Corso and Claudia
Polini, Reduction number of links of irreducible varieties, J.
Pure Appl. Algebra 121 (1997), no. 1, 29–43. MR 1471122
(98h:13030), http://dx.doi.org/10.1016/S0022-4049(96)00042-4
- 8.
Alberto
Corso and Claudia
Polini, On residually 𝑆₂ ideals
and projective dimension one modules, Proc.
Amer. Math. Soc. 129 (2001), no. 5, 1309–1315 (electronic). MR 1814157
(2001m:13040), http://dx.doi.org/10.1090/S0002-9939-00-05696-3
- 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.
C. Cowsik and M.
V. Nori, On the fibres of blowing up, J. Indian Math. Soc.
(N.S.) 40 (1976), no. 1-4, 217–222 (1977). MR 0572990
(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,
Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math.,
vol. 84, Dekker, New York, 1983, pp. 79–169. MR 686942
(84k:13015)
- 14.
J.
Herzog, W.
V. Vasconcelos, and R.
Villarreal, Ideals with sliding depth, Nagoya Math. J.
99 (1985), 159–172. MR 805087
(86k:13022)
- 15.
Jürgen
Herzog and Bernd
Ulrich, Self-linked curve singularities, Nagoya Math. J.
120 (1990), 129–153. MR 1086575
(92c:13010)
- 16.
Craig
Huneke, Linkage and the Koszul homology of ideals, Amer. J.
Math. 104 (1982), no. 5, 1043–1062. MR 675309
(84f:13019), http://dx.doi.org/10.2307/2374083
- 17.
Craig
Huneke and Irena
Swanson, Cores of ideals in 2-dimensional regular local rings,
Michigan Math. J. 42 (1995), no. 1, 193–208. MR 1322199
(96j:13021), http://dx.doi.org/10.1307/mmj/1029005163
- 18.
Eero
Hyry, Coefficient ideals and the
Cohen-Macaulay property of Rees algebras, Proc.
Amer. Math. Soc. 129 (2001), no. 5, 1299–1308. MR 1712905
(2001h:13005), http://dx.doi.org/10.1090/S0002-9939-00-05673-2
- 19.
Mark
Johnson and Bernd
Ulrich, Artin-Nagata properties and Cohen-Macaulay associated
graded rings, Compositio Math. 103 (1996),
no. 1, 7–29. MR 1404996
(97f:13006)
- 20.
Steven
Kleiman and Bernd
Ulrich, Gorenstein algebras, symmetric
matrices, self-linked ideals, and symbolic powers, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4973–5000. MR 1422609
(98c:13019), http://dx.doi.org/10.1090/S0002-9947-97-01960-0
- 21.
Joseph
Lipman, Adjoints of ideals in regular local rings, Math. Res.
Lett. 1 (1994), no. 6, 739–755. With an
appendix by Steven Dale Cutkosky. MR 1306018
(95k:13028)
- 22.
Joseph
Lipman and Bernard
Teissier, Pseudorational local rings and a theorem of
Briançon-Skoda about integral closures of ideals, Michigan
Math. J. 28 (1981), no. 1, 97–116. MR 600418
(82f:14004)
- 23.
D.
G. Northcott and D.
Rees, Reductions of ideals in local rings, Proc. Cambridge
Philos. Soc. 50 (1954), 145–158. MR 0059889
(15,596a)
- 24.
Claudia
Polini and Bernd
Ulrich, Linkage and reduction numbers, Math. Ann.
310 (1998), no. 4, 631–651. MR 1619911
(99g:13017), http://dx.doi.org/10.1007/s002080050163
- 25.
D.
Rees and Judith
D. Sally, General elements and joint reductions, Michigan
Math. J. 35 (1988), no. 2, 241–254. MR 959271
(89h:13034), http://dx.doi.org/10.1307/mmj/1029003751
- 26.
Bernd
Ulrich, Artin-Nagata properties and reductions of ideals,
(South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc.,
Providence, RI, 1994, pp. 373–400. MR 1266194
(95a:13017), http://dx.doi.org/10.1090/conm/159/01519
- 27.
Bernd
Ulrich, Ideals having the expected reduction number, Amer. J.
Math. 118 (1996), no. 1, 17–38. MR 1375302
(97b:13003)
- 28.
Paolo
Valabrega and Giuseppe
Valla, Form rings and regular sequences, Nagoya Math. J.
72 (1978), 93–101. MR 514892
(80d:14010)
- 29.
Oscar
Zariski and Pierre
Samuel, Commutative algebra. Vol. II, The University Series in
Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.
J.-Toronto-London-New York, 1960. MR 0120249
(22 #11006)
- 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:
http://dx.doi.org/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
Article copyright:
© Copyright 2002 American Mathematical Society
|
