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Isomorphism classes of short Gorenstein local rings via Macaulay's inverse system


Authors: J. Elias and M. E. Rossi
Journal: Trans. Amer. Math. Soc. 364 (2012), 4589-4604
MSC (2010): Primary 13H10; Secondary 13H15, 14C05
DOI: https://doi.org/10.1090/S0002-9947-2012-05430-4
Published electronically: April 11, 2012
MathSciNet review: 2922602
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Abstract: Let $ K$ be an algebraically closed field of characteristc zero. In this paper we study the isomorphism classes of Artinian Gorenstein local $ K$-algebras with socle degree three by means of Macaulay's inverse system. We prove that their classification is equivalent to the projective classification of cubic hypersurfaces in $ \mathbb{P}_K ^{n }$. This is an unexpected result because it reduces the study of this class of local rings to the graded case. The result has applications in problems concerning the punctual Hilbert scheme $ \operatorname {Hilb}_d (\mathbb{P}_K^n) $ and in relation to the problem of the rationality of the Poincaré series of local rings.


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

  • 1. L. L. Avramov, S. Iyengar, L. M. Şega, Free resolutions over short local rings, J. Lond. Math. Soc. 78 (2008), 459-476. MR 2439635 (2009h:13011)
  • 2. R. Bøgvad, Gorenstein rings with transcendental Poincaré series, Math. Scand. 53 (1983), no. 1, 5-15. MR 733933 (85c:13012)
  • 3. M. Brundu and A. Logar, Parametrization of the orbits of cubic surfaces, Transform. Groups 3 (1998), 209-239. MR 1640742 (99m:14114)
  • 4. D. A. Cartwright, D. Erman, M. Velasco, and B. Viray, Hilbert schemes of 8 points, arXiv:0803.0341 (2008).
  • 5. G. Casnati and R. Notari, On some Gorenstein loci in $ \textup {Hilb}_6(P^4),$ J. of Algebra 308 (2007). 2055-2074. MR 2295072 (2008d:14006)
  • 6. -,On the Gorenstein locus of some punctual Hilbert schemes, J. Pure and Appl. Alg. 213 (2009), 2055-2074. MR 2533305 (2010g:14003)
  • 7. G. Casnati, J. Elias, M.E.  Rossi, R. Notari, Poincaré series and deformations of Gorenstein local algebras with low socle degree, preprint (2010).
  • 8. A. Conca, M.E.  Rossi, G. Valla, Gröbner flags and Gorenstein Artin rings, Compositio Math. 129 (2001), 95-121. MR 1856025 (2002i:13014)
  • 9. J.  Elias, G. Valla, Structure theorems for certain Gorenstein ideals. Special volume in honor of Melvin Hochster. Michigan Math. J. 57 (2008), 269-292. MR 2492453 (2010i:13020)
  • 10. -, Isomorphism classes of certain Artinian Gorenstein algebras, Algebras and Representation Theory, DOI 10.1007/s10468-009-9196-8 (2009).
  • 11. J. Emsalem, Géométrie des points épais, Bull. Soc. Math. France 106 (1978), no. 4, 399-416. MR 518046 (80j:14008)
  • 12. R. Hartshorne, Algebraic geometry, Graduate Texts in Math., 52, Springer-Verlag, 1977. MR 0463157 (57:3116)
  • 13. I. B. Henriques and L. M. Şega, Free resolutions over short local rings, arXiv: 0904.3510v2 (2009), to appear in Math. Z.
  • 14. A. Iarrobino, Reducibility of the families of 0-dimensional schemes on a variety. Invent. Math. 15 (1972), 72-77. MR 0301010 (46:170)
  • 15. -, Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc. 285 (1984), no. 1, 337-378. MR 748843 (85j:13030)
  • 16. -, Associated graded algebra of a Gorenstein Artin algebra, Mem. Amer. Math. Soc. 107 (1994), no. 514, viii+115. MR 1184062 (94f:13009)
  • 17. A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999, Appendix C by Iarrobino and Steven L. Kleiman. MR 1735271 (2001d:14056)
  • 18. I. Kaplanski, Fields and rings, Univ. of Chicago Press (1970).
  • 19. F.S. Macaulay, The algebraic theory of modular systems, Cambridge University, 1916. MR 1281612 (95i:13001)
  • 20. D.G. Northcott, Injective envelopes and inverse polynomials, J. London Math. Soc. 8 (1972), 290-296. MR 0360555 (50:13003)
  • 21. B. Poonen, Isomorphism types of commutative algebras of finite rank over an algebraically closed field. Computational arithmetic geometry, 111-120, Contemp. Math., 463, Amer. Math. Soc., Providence, RI, 2008. MR 2459993 (2010c:13018)
  • 22. D.W. Sharpe and P. Vamos, Injective modules, Cambridge Univ. Press (1972). MR 0360706 (50:13153)
  • 23. J. Watanabe, The Dilworth number of Artin Gorenstein rings, Advances in Math. 76 (1989), 194-199. MR 1013668 (90j:13023)

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Additional Information

J. Elias
Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
Email: elias@ub.edu

M. E. Rossi
Affiliation: Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Email: rossim@dima.unige.it

DOI: https://doi.org/10.1090/S0002-9947-2012-05430-4
Keywords: Artinian Gorenstein local rings, inverse system, Hilbert functions, isomorphism classes.
Received by editor(s): November 18, 2009
Received by editor(s) in revised form: March 22, 2010, June 29, 2010, and July 16, 2010
Published electronically: April 11, 2012
Additional Notes: The first author was partially supported by MTM2010-20279-C02, Acción Integrada España-Italia 07-09
The second author was partially supported by M.I.U.R.: PRIN 07-09, Azione Integrata Italia-Spagna 07-09
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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