Core of ideals of Noetherian local rings
HTML articles powered by AMS MathViewer
- by Hsin-Ju Wang PDF
- Proc. Amer. Math. Soc. 136 (2008), 801-807 Request permission
Abstract:
The core of an ideal is the intersection of all its reductions. In 2005, Polini and Ulrich explicitly described the core as a colon ideal of a power of a single reduction and a power of $I$ for a broader class of ideals, where $I$ is an ideal in a local Cohen-Macaulay ring. In this paper, we show that if $I$ is an ideal of analytic spread $1$ in a Noetherian local ring with infinite residue field, then with some mild conditions on $I$, we have $\operatorname {core} (I)\supseteq J(J^n: I^n)=I(J^n: I^n)=(J^{n+1}: I^n)\cap I$ for any minimal reduction $J$ of $I$ and for $n\gg 0$.References
- Alberto Corso, Claudia Polini, and Bernd Ulrich, The structure of the core of ideals, Math. Ann. 321 (2001), no. 1, 89–105. MR 1857370, DOI 10.1007/PL00004502
- Alberto Corso, Claudia Polini, and Bernd Ulrich, Core and residual intersections of ideals, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2579–2594. MR 1895194, DOI 10.1090/S0002-9947-02-02908-2
- 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, DOI 10.1307/mmj/1029005163
- Craig Huneke and Ngô Viêt Trung, On the core of ideals, Compos. Math. 141 (2005), no. 1, 1–18. MR 2099767, DOI 10.1112/S0010437X04000910
- Eero Hyry and Karen E. Smith, On a non-vanishing conjecture of Kawamata and the core of an ideal, Amer. J. Math. 125 (2003), no. 6, 1349–1410. MR 2018664, DOI 10.1353/ajm.2003.0041
- Claudia Polini and Bernd Ulrich, A formula for the core of an ideal, Math. Ann. 331 (2005), no. 3, 487–503. MR 2122537, DOI 10.1007/s00208-004-0560-z
- D. Rees and Judith D. Sally, General elements and joint reductions, Michigan Math. J. 35 (1988), no. 2, 241–254. MR 959271, DOI 10.1307/mmj/1029003751
Additional Information
- Hsin-Ju Wang
- Affiliation: Department of Mathematics, National Chung Cheng University, Chiayi 621, Taiwan
- Received by editor(s): November 2, 2004
- Received by editor(s) in revised form: November 27, 2006
- Published electronically: November 23, 2007
- Communicated by: Bernd Ulruch
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 801-807
- MSC (2000): Primary 13H10, 13A15; Secondary 13A30
- DOI: https://doi.org/10.1090/S0002-9939-07-09038-7
- MathSciNet review: 2361851