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A formula for the $ *$-core of an ideal


Authors: Louiza Fouli, Janet C. Vassilev and Adela N. Vraciu
Journal: Proc. Amer. Math. Soc. 139 (2011), 4235-4245
MSC (2010): Primary 13A30, 13A35, 13B22
DOI: https://doi.org/10.1090/S0002-9939-2011-10858-X
Published electronically: April 27, 2011
MathSciNet review: 2823069
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Abstract: Expanding on the 2010 work of Fouli and Vassilev, we determine a formula for the $ *\textrm{-}\textrm{core}$ of an ideal in two different settings: (1) in a Cohen-Macaulay local ring of characteristic $ p>0$, with perfect residue field and test ideal of depth at least two, where the ideal has a minimal $ *$-reduction that is a parameter ideal, and (2) in a normal local domain of characteristic $ p>0$, with perfect residue field and $ \mathfrak{m}$-primary test ideal, where the ideal is a sufficiently high Frobenius power of an ideal. We also exhibit some examples where our formula fails if our hypotheses are not met.


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  • [Ab] Aberbach, I., Extensions of weakly and strongly F-rational rings by flat maps, J. Algebra, 241 (2001), 799-807. MR 1843326 (2002f:13008)
  • [Br] Brenner, H., Computing the tight closure in dimension two, Math. Comp., 74 (2005), 1495-1518. MR 2137014 (2006h:13008)
  • [CEU] Chardin, M., Eisenbud, D., Ulrich, B., Hilbert functions, residual intersections, and residually $ {\rm S}_2$ ideals, Compositio Math., 125 (2001), 193-219. MR 1815393 (2002g:13034)
  • [CPU1] Corso, A., Polini, C., Ulrich, B., The structure of the core of ideals, Math. Ann., 321 (2001), 89-105. MR 1857370 (2002j:13005)
  • [CPU2] Corso, A., Polini, C., Ulrich, B., Core and residual intersections of ideals, Trans. Amer. Math. Soc., 354 (2002), 2579-2594. MR 1895194 (2003b:13035)
  • [Ep] Epstein, N., A tight closure analogue of analytic spread, Math. Proc. Camb. Phil. Soc., 139 (2005), 371-383. MR 2168094 (2006e:13003)
  • [EV] Epstein, N., Vraciu, A., A length characterization of $ *$-spread, Osaka J. Math., 45 (2008), 445-456. MR 2441949 (2009d:13003)
  • [FV] Fouli, L., Vassilev, J., The $ cl$-core of an ideal, Math. Proc. Camb. Phil. Soc., 149 (2010), 247-262. MR 2670215
  • [HH1] Hochster, M., Huneke, C., Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc., 3 (1990), 31-116. MR 1017784 (91g:13010)
  • [HH2] Hochster, M., Huneke, C., F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346 (1994), 1-62. MR 1273534 (95d:13007)
  • [Hu] Huneke, C., Tight Closure and Its Applications, CBMS Regional Conference Series in Mathematics, 88, American Math. Soc., Providence, RI, 1996. MR 1377268 (96m:13001)
  • [HS1] Huneke, C., Swanson, I., Cores of ideals in 2-dimensional regular local rings, Michigan Math. J., 42 (1995), 193-208. MR 1322199 (96j:13021)
  • [HT] Huneke, C., Trung, N., On the core of ideals, Compos. Math., 141 (2005), 1-18. MR 2099767 (2005g:13003)
  • [HV] Huneke, C., Vraciu, A., Special tight closure, Nagoya Math. J., 170 (2003), 175-183. MR 1994893 (2004e:13008)
  • [HySm1] Hyry, E., Smith, K., On a non-vanishing conjecture of Kawamata and the core of an ideal, Amer. J. Math., 125 (2003), 1349-1410. MR 2018664 (2006c:13036)
  • [HySm2] Hyry, E., Smith, K., Core versus graded core, and global sections of line bundles, Trans. Amer. Math. Soc., 356 (2004), 3143-3166. MR 2052944 (2005g:13007)
  • [JU] Johnson, M., Ulrich, B., Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compositio Math., 103 (1996), 7-29. MR 1404996 (97f:13006)
  • [NR] Northcott, D. G. and Rees  D., Reductions of ideals in local rings, Proc. Camb. Phil. Soc., 50 (1954), 145-158. MR 0059889 (15:596a)
  • [PU] Polini, C., Ulrich, B., A formula for the core of an ideal, Math. Ann., 331 (2005), 487-503. MR 2122537 (2006k:13020)
  • [RS] Rees, D., Sally, J., General elements and joint reductions, Michigan Math. J., 35 (1988), 241-254. MR 959271 (89h:13034)
  • [U] Ulrich, B., Artin-Nagata properties and reductions of ideals, Contemp. Math., 159, American Math. Soc., Providence, RI, 1994, 373-400. MR 1266194 (95a:13017)
  • [Va] Vassilev, J., Test ideals in quotients of $ F$-finite regular local rings, Trans. Amer. Math. Soc., 350 (1998), 4041-4051. MR 1458336 (98m:13009)
  • [Vr1] Vraciu, A., $ *$-independence and special tight closure, J. Algebra, 249 (2002), 544-565. MR 1901172 (2003d:13003)
  • [Vr2] Vraciu, A., Chains and families of tightly closed ideals, Bull. London Math. Soc., 38 (2006), 201-208. MR 2214472 (2007e:13009)

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Additional Information

Louiza Fouli
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: lfouli@math.nmsu.edu

Janet C. Vassilev
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131
Email: jvassil@math.unm.edu

Adela N. Vraciu
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: vraciu@math.sc.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10858-X
Keywords: Tight closure, reduction, core, *-independent, spread
Received by editor(s): October 23, 2009
Received by editor(s) in revised form: October 22, 2010
Published electronically: April 27, 2011
Additional Notes: The second author was partly supported by the NSA grant H98230-09-1-0057
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2011 American Mathematical Society

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