Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Louiza Fouli, Claudia Polini and Bernd Ulrich
Journal: Trans. Amer. Math. Soc. 362 (2010), 6183-6203
MSC (2010): Primary 13B21; Secondary 13A30, 13B22, 13C40
DOI: https://doi.org/10.1090/S0002-9947-2010-05056-1
Published electronically: July 22, 2010
MathSciNet review: 2678970
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. W. Bruns and J. Herzog, Cohen-Macaulay Rings, revised edition, Cambridge University Press, Cambridge, 1998. MR 1251956 (95h:13020)
  • 2. M. Chardin and B. Ulrich, Liaison and Castelnuovo-Mumford regularity, Amer. J. Math. 124 (2002), 1103-1124. MR 1939782 (2004c:14095)
  • 3. A. Corso, C. Polini, and B. Ulrich, The structure of the core of ideals, Math. Ann. 321 (2001), 89-105. MR 1857370 (2002j:13005)
  • 4. C. Cumming, The canonical module of the extended Rees ring, preprint.
  • 5. F. El Zein, Complexe dualisant et applications à la classe fondamentale d'un cycle, Bull. Soc. Math. France 58 (1978). MR 0518299 (80h:14009)
  • 6. E. G. Evans and P. Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), 323-333. MR 632842 (83i:13006)
  • 7. L. Fouli, C. Polini, and B. Ulrich, The core of ideals in arbitrary characteristic, Michigan Math. J. 57 (2008), 305-319.
  • 8. A. Geramita, M. Kreuzer, and L. Robbiano, Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339 (1993), 163-189. MR 1102886 (93k:14065)
  • 9. J. Harris, Curves in projective space, With the collaboration of David Eisenbud, Séminaire de Mathématiques Supérieures 85, Presses de l'Université de Montréal (1982). MR 685427 (84g:14024)
  • 10. E. Hyry and K. E. Smith, On a non-vanishing conjecture of Kawamata and the core of an ideal, Amer. J. Math. 125 (2003), 1349-1410. MR 2018664 (2006c:13036)
  • 11. E. Hyry and K. E. Smith, Core versus graded core, and global sections of line bundles, Trans. Amer. Math. Soc. 356 (2004), 3143-3166. MR 2052944 (2005g:13007)
  • 12. E. Kunz and R. Waldi, Regular differential forms, Contemp. Math. 79 (1988). MR 971502 (90a:14021)
  • 13. J. Lipman, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque 117 (1984). MR 759943 (86g:14008)
  • 14. C. Polini and B. Ulrich, A formula for the core of an ideal, Math. Ann. 331 (2005), 487-503. MR 2122537 (2006k:13020)
  • 15. C. Polini, B. Ulrich, and M. A. Vitulli, The core of zero-dimensional monomial ideals, Adv. Math. 211 (2007), 72-93. MR 2313528 (2008b:13033)
  • 16. B. Ulrich, Artin-Nagata properties and reductions of ideals, Contemp. Math. 159 (1994), 373-400. MR 1266194 (95a:13017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 13B21, 13A30, 13B22, 13C40

Retrieve articles in all journals with MSC (2010): 13B21, 13A30, 13B22, 13C40


Additional Information

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

Claudia Polini
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: cpolini@nd.edu

Bernd Ulrich
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: ulrich@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05056-1
Received by editor(s): August 14, 2007
Published electronically: July 22, 2010
Additional Notes: The second and third author gratefully acknowledge partial support from the NSF. The second author was also supported in part by the NSA. The first and second author thank the Department of Mathematics of Purdue University for its hospitality
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society