On Zanello's lower bound for level algebras
Author:
Dan Laksov
Journal:
Proc. Amer. Math. Soc. 141 (2013), 15191527
MSC (2010):
Primary 13E10, 13D40, 05E40, 14A05, 14M05, 14M07
Published electronically:
October 11, 2012
MathSciNet review:
3020839
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Abstract: We consider the proof of Söderberg of Zanello's lower bound for the Hilbert function of level algebras from the point of view of vector spaces. Our results, when specialised to level algebras, generalise those of Zanello and Söderberg to the case when the modules involved may have nontrivial annihilators. In the process we clarify why the methods of Zanello and Söderberg consist of two distinct parts. As a contrast we show that for polynomial rings, Zanello's bound, in the generic case, can be obtained by simple manipulations of numbers without dividing into two separate cases. We also consider the inclusionexclusion principle of dimensions of vector spaces used by Zanello in special cases. It turns out that the resulting alternating sums are extremely difficult to handle and have many unexpected properties. This we illustrate by a couple of results and examples. The examples show that the inclusionexclusion principle does not hold for vector spaces in the way it is used by Zanello.
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 A. Iarrobino, Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc. 285 (1984), 337378. MR 748843 (85j:13030)
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 A. Iarrobino, Hilbert functions of Gorenstein algebras associated to a pencil of forms, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 273286. MR 2202259 (2006i:13030)
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 A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci. Appendix A, Lecture Notes in Math., 1721, SpringerVerlag, Berlin, 1999. MR 1735271 (2001d:14056)
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 R. Fröberg and D. Laksov, Compressed algebras, Complete intersections (Acireale, 1983), Lecture Notes in Math., 1092, SpringerVerlag, Berlin, 1984, pp. 121151. MR 775880 (86f:13012)
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 J. Söderberg, On Zanello's lower bound for generic Gorenstein quotients of level algebras, Thesis, KTH, 2007.
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 F. Zanello, Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type, Trans. Amer. Math. Soc. 359 (2007), 26752686. MR 2286051 (2007k:13034)
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Additional Information
Dan Laksov
Affiliation:
Department of Mathematics, Royal Institute of Technology, KTH, S100 44 Stockholm, Sweden
Email:
laksov@math.kth.se
DOI:
http://dx.doi.org/10.1090/S000299392012114273
Keywords:
Graded modules,
level algebras,
Hilbert functions,
inclusionexclusion
Received by editor(s):
May 30, 2011
Received by editor(s) in revised form:
August 30, 2011
Published electronically:
October 11, 2012
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
