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)

 
 

 

An iterative construction of Gorenstein ideals


Authors: C. Bocci, G. Dalzotto, R. Notari and M. L. Spreafico
Journal: Trans. Amer. Math. Soc. 357 (2005), 1417-1444
MSC (2000): Primary 14M05, 13H10; Secondary 14M06, 13D02, 18G10
DOI: https://doi.org/10.1090/S0002-9947-04-03521-4
Published electronically: July 22, 2004
MathSciNet review: 2115371
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we present a method to inductively construct Gorenstein ideals of any codimension $ c.$ We start from a Gorenstein ideal $ I $ of codimension $ c $ contained in a complete intersection ideal $ J $ of the same codimension, and we prove that under suitable hypotheses there exists a new Gorenstein ideal contained in the residual ideal $ I : J.$ We compare some numerical data of the starting and the resulting Gorenstein ideals of the construction. We compare also the Buchsbaum-Eisenbud matrices of the two ideals, in the codimension three case. Furthermore, we show that this construction is independent from the other known geometrical constructions of Gorenstein ideals, providing examples.


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

  • 1. C. Bocci, G. Dalzotto, Gorenstein points in $ \mathbb{P} ^3 $, Rend. Sem. Mat. Univ. Politec. Torino 59 (2001) n. 1, 155-164.
  • 2. W. Bruns, J. Herzog, ``Cohen-Macaulay Rings'', Cambridge Studies in Adv. Math. 39, Cambridge University Press, 1993. MR 95h:13020
  • 3. D. Buchsbaum, D. Eisenbud, Algebra structures for finite free resolutions and some structure theorems for ideals of codimension three, Amer. J. Math. 99 (1977), 447-485. MR 56:11983
  • 4. A. Capani, G. Niesi, L. Robbiano, Coem CoA, a System for Doing Computations in Commutative Algebra, Version 4.2 is available at http://cocoa.dima.unige.it.
  • 5. M. Casanellas, R.M. Miró-Roig, Gorenstein liaison of curves in $ \mathbb{P} ^4 $, J. Algebra 230 (2000), 656-664. MR 2001g:14082
  • 6. M. Casanellas, R.M. Miró-Roig, Gorenstein liaison of divisors on standard determinantal schemes and on rational normal scrolls, Journal of Pure and Applied Algebra 164 (2001), 325-343. MR 2002g:14071
  • 7. M. Casanellas, R.M. Miró-Roig, Gorenstein liaison and special linear configurations, Illinois Math. J. 46 (2002), 129-143 MR 2003i:14062
  • 8. E. Davis, A.V. Geramita, F. Orecchia, Gorenstein Algebras and the Cayley-Bacharach Theorem, Proceedings AMS 93, n. 4 (1985), 593-597. MR 86k:14034
  • 9. S. Diesel, Irreducibility and Dimension Theorems for Families of Height 3 Gorenstein Algebras, Pacific J. Math. 172 (1996), n. 2, 365-397. MR 99f:13016
  • 10. A.V. Geramita, J.C. Migliore, Reduced Gorenstein codimension three subschemes of projective schemes, Proc. Amer. Math. Soc. 125 (1997), 943-950. MR 97h:14068
  • 11. R. Hartshorne, Some examples of Gorenstein liaison in codimension three, Collectanea Math. 53 (2002), 21-48. MR 2003d:14059
  • 12. J.O. Kleppe, J.C. Migliore, R.M. Miró-Roig, C. Peterson, Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness, Mem. Amer. Math. Soc. 154, n. 732, (2001). MR 2002e:14083
  • 13. J. Lesperance, Gorenstein liaison of some curves in $ \mathbb{P} ^4 $, Collectanea Math. 52 (2001), 219-230. MR 2003g:14065
  • 14. J.C. Migliore, ``Introduction to Liaison Theory and Deficiency Modules'', Progress in Mathematics 165, Birkhäuser, 1998. MR 2000g:14058
  • 15. J.C. Migliore, C. Peterson, A construction of codimension three arithmetically Gorenstein subschemes of projective space, Trans. Amer. Math. Soc. 349 (1997), 3803-3821. MR 98d:14060
  • 16. J.C. Migliore, U. Nagel, C. Peterson, Buchsbaum-Rim sheaves and their multiple sections, J. Algebra 219 (1999), 378-420. MR 2000f:14076
  • 17. J.C. Migliore, U. Nagel, Monomial ideals and the Gorenstein liaison class of a complete intersection, Compositio Math. 133 (2002), n. 1, 25-36. MR 2003g:13010
  • 18. J.C. Migliore, U. Nagel, Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers, Adv. Math. (to appear)
  • 19. C. Peskine, L. Szpiro, Liaison des variétés algébriques I, Inv. Math. 26 (1974), 271-302. MR 51:526
  • 20. A. Ragusa, G. Zappalà, Properties of 3-codimensional Gorenstein schemes, Comm. Alg. 29 (2001), n. 1, 303-318. MR 2002f:14061
  • 21. R. Stanley, Hilbert functions of graded algebras, Adv. in Math. 28 (1978), 57-82. MR 58:5637
  • 22. J. Watanabe, A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227-232. MR 47:8526

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14M05, 13H10, 14M06, 13D02, 18G10

Retrieve articles in all journals with MSC (2000): 14M05, 13H10, 14M06, 13D02, 18G10


Additional Information

C. Bocci
Affiliation: Dipartimento di Matematica, Università di Torino, I-10123 Torino, Italy
Address at time of publication: Dipartimento di Matematica, Università di Milano, I-20133 Milano, Italy
Email: bocci@dm.unito.it, cristiano.bocci@unimi.it

G. Dalzotto
Affiliation: Dipartimento di Matematica, Università di Genova, I-16146 Genova, Italy
Address at time of publication: Dipartimento di Matematica, Università di Pisa, I-56127 Pisa, Italy
Email: dalzotto@module.dima.unige.it, dalzotto@mail.dm.unipi.it

R. Notari
Affiliation: Dipartimento di Matematica, Politecnico di Torino, I-10129 Torino, Italy
Email: roberto.notari@polito.it

M. L. Spreafico
Affiliation: Dipartimento di Matematica, Politecnico di Torino, I-10129 Torino, Italy
Email: maria.spreafico@polito.it

DOI: https://doi.org/10.1090/S0002-9947-04-03521-4
Received by editor(s): February 24, 2003
Received by editor(s) in revised form: September 26, 2003
Published electronically: July 22, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society