Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-19T04:13:46.487Z Has data issue: false hasContentIssue false

The cl-core of an ideal

Published online by Cambridge University Press:  03 June 2010

LOUIZA FOULI
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003–8001, U.S.A. e-mail: lfouli@math.nmsu.edu
JANET C. VASSILEV
Affiliation:
Department of Mathematics, University of New Mexico, Albuquerque, NM 87131–0001, U.S.A. e-mail: jvassil@math.unm.edu

Abstract

We expand the notion of core to cl-core for Nakayama closures cl. In the characteristic p > 0 setting, when cl is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that *-core(I) = core(I), if I is an ideal in a one-dimensional domain with infinite residue field or if I is an ideal generated by a system of parameters in any Noetherian ring. More generally, we show the same result in a Cohen–Macaulay normal local domain with infinite perfect residue field, if the analytic spread, ℓ, is equal to the *-spread and I is G and weakly-(ℓ − 1)-residually S2. This last is dependent on our result that generalizes the notion of general minimal reductions to general minimal *-reductions. We also determine that the *-core of a tightly closed ideal in certain one-dimensional semigroup rings is tightly closed and therefore integrally closed.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aberbach, I.Extensions of weakly and strongly F-rational rings by flat maps. J. Algebra 241 (2001), 799807.Google Scholar
[2]Avramov, L. and Herzog, J.The Koszul algebra of a codimension 2 embedding. Math. Z. 175 (1980), 249260.Google Scholar
[3]Chardin, M., Eisenbud, D. and Ulrich, B.Hilbert functions, residual intersections, and residually S 2-ideals. Compositio Math. 125 (2001), 193219.CrossRefGoogle Scholar
[4]Corso, A., Polini, C. and Ulrich, B.The structure of the core of ideals. Math. Ann. 321 (2001), no. 1, 89105.CrossRefGoogle Scholar
[5]Corso, A., Polini, C. and Ulrich, B.Core and residual intersections of ideals. Trans. Amer. Math. Soc. 354 (2002), no. 7, 25792594.Google Scholar
[6]Eisenbud, D.Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150, (Springer-Verlag, New York, 1995).Google Scholar
[7]Epstein, N.A tight closure analogue of analytic spread. Math. Proc. Camb. Phil. Soc. 139 (2005), 371383.Google Scholar
[8]Fouli, L., Polini, C. and Ulrich, B. Annihilators of graded components of the canonical module and the core of standard graded algebras. Preprint, to appear in Trans. Amer. Math. Soc., arXiv:0903.3439 [math.AC].Google Scholar
[9]Herzog, J., Vasconcelos, W. V. and Villarreal, R. H.Ideals with sliding depth. Nagoya Math. J. 99 (1985), 159172.Google Scholar
[10]Hochster, M. and Huneke, C.Tight closure, invariant theory, and the Briançon-Skoda theorem. J. Amer. Math. Soc. 3 (1990), no. 1, 31116.Google Scholar
[11]Huneke, C.Linkage and Koszul homology of ideals. Amer. J. Math. 104 (1982), 10431062.CrossRefGoogle Scholar
[12]Huneke, C.Tight closure and its applications. CBMS Lecture Notes in Math. 88 (Amer. Math. Soc., Providence, 1996).CrossRefGoogle Scholar
[13]Huneke, C. and Swanson, I.Cores of ideals in 2-dimensional regular local rings. Michigan Math. J. 42 (1995), 193208.CrossRefGoogle Scholar
[14]Huneke, C. and Swanson, I. Integral closure of ideals, rings, and modules. London Math. Soc. Lecture Note Series 336, (Cambridge University Press, 2006).Google Scholar
[15]Huneke, C. and Trung, N.On the core of ideals. Compos. Math. 141 (2005), no. 1, 118.CrossRefGoogle Scholar
[16]Huneke, C. and Vraciu, A.Special tight closure. Nagoya Math. J. 170 (2003), 175183.CrossRefGoogle Scholar
[17]Hyry, E. and Smith, K.On a non-vanishing conjecture of Kawamata and the core of an ideal. Amer. J. Math. 125 (2003), no. 6, 13491410.CrossRefGoogle Scholar
[18]Hyry, E. and Smith, K.Core versus graded core, and global sections of line bundles. Trans. Amer. Math. Soc. 356 (2004), no. 8, 31433166.Google Scholar
[19]Kunz, E.Introduction to Commutative Algebra and Algebraic Geometry. (Birkhäuser Boston, 1985).Google Scholar
[20]Lipman, J. and Sathaye, A.Jacobian ideals and a theorem of Briançon-Skoda. Michigan Math. J. 28 (1981), 199222.CrossRefGoogle Scholar
[21]Grayson, D. and Stillman, M. Macaulay 2, A computer algebra system for computing in Algebraic Geometry and Commutative Algebra, available at http://www.math.uiuc.edu/Macaulay2.Google Scholar
[22]Northcott, D. G. and Rees, D.Reductions of ideals in local rings. Proc. Camb. Phil. Soc. 50 (1954), 145158.Google Scholar
[23]Polini, C. and Ulrich, B.A formula for the core of an ideal. Math. Ann. 331 (2005), no. 3, 487503.CrossRefGoogle Scholar
[24]Rees, D. and Sally, J.General elements and joint reductions. Michigan Math. J. 35 (1988), no. 2, 241254.Google Scholar
[25]Smith, K.Test ideals in local rings. Trans. Amer. Math. Soc. 347 (1995), no. 9, 34533472.Google Scholar
[26]Ulrich, B.Artin-Nagata properties and reductions of ideals. Contemp. Math. 159 (1994), 373400.Google Scholar
[27]Vassilev, J. Test ideals in Gorenstein isolated singularities and F-finite reduced rings. Thesis (University of California, Los Angeles, 1997).Google Scholar
[28]Vassilev, J.Structure on the set of closure operations of a commutative ring. J. Algebra 321 (2009), 27372753.Google Scholar
[29]Vassilev, J. and Vraciu, A.When is tight closure determined by the test ideal?. J. Comm. Alg. 1 (2009), 591602.Google Scholar
[30]Vraciu, A.*-independence and special tight closure. J. Algebra 249 (2002), no. 2, 544565.CrossRefGoogle Scholar
[31]Vraciu, A.Chains and families of tightly closed ideals. Bull. London Math. Soc. 38 (2006), no. 2, 201208.Google Scholar