Annihilators of graded components of the canonical module, and the core of standard graded algebras

Fouli, Louiza; Polini, Claudia; Ulrich, Bernd

doi:10.1090/S0002-9947-2010-05056-1

Annihilators of graded components of the canonical module, and the core of standard graded algebras
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by Louiza Fouli, Claudia Polini and Bernd Ulrich

Trans. Amer. Math. Soc. 362 (2010), 6183-6203

DOI: https://doi.org/10.1090/S0002-9947-2010-05056-1

Published electronically: July 22, 2010

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Abstract:

We relate the annihilators of graded components of the canonical module of a graded Cohen-Macaulay ring to colon ideals of powers of the homogeneous maximal ideal. In particular, we connect them to the core of the maximal ideal. An application of our results characterizes Cayley-Bacharach sets of points in terms of the structure of the core of the maximal ideal of their homogeneous coordinate ring. In particular, we show that a scheme is Cayley-Bacharach if and only if the core is a power of the maximal ideal.

References

Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Marc Chardin and Bernd Ulrich, Liaison and Castelnuovo-Mumford regularity, Amer. J. Math. 124 (2002), no. 6, 1103–1124. MR 1939782
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
C. Cumming, The canonical module of the extended Rees ring, preprint.
Fouad El Zein, Complexe dualisant et applications à la classe fondamentale d’un cycle, Bull. Soc. Math. France Mém. 58 (1978), 93 (French, with English summary). MR 518299
E. Graham Evans and Phillip Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333. MR 632842, DOI 10.2307/1971296
L. Fouli, C. Polini, and B. Ulrich, The core of ideals in arbitrary characteristic, Michigan Math. J. 57 (2008), 305–319.
Anthony V. Geramita, Martin Kreuzer, and Lorenzo Robbiano, Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339 (1993), no. 1, 163–189. MR 1102886, DOI 10.1090/S0002-9947-1993-1102886-5
Joe Harris, Curves in projective space, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 85, Presses de l’Université de Montréal, Montreal, Que., 1982. With the collaboration of David Eisenbud. MR 685427
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
Eero Hyry and Karen E. Smith, Core versus graded core, and global sections of line bundles, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3143–3166. MR 2052944, DOI 10.1090/S0002-9947-03-03337-3
Ernst Kunz and Rolf Waldi, Regular differential forms, Contemporary Mathematics, vol. 79, American Mathematical Society, Providence, RI, 1988. MR 971502, DOI 10.1090/conm/079
Joseph Lipman, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque 117 (1984), ii+138 (English, with French summary). MR 759943
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
Claudia Polini, Bernd Ulrich, and Marie A. Vitulli, The core of zero-dimensional monomial ideals, Adv. Math. 211 (2007), no. 1, 72–93. MR 2313528, DOI 10.1016/j.aim.2006.07.020
Bernd Ulrich, Artin-Nagata properties and reductions of ideals, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 373–400. MR 1266194, DOI 10.1090/conm/159/01519