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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Zanello's lower bound for level algebras


Author: Dan Laksov
Journal: Proc. Amer. Math. Soc. 141 (2013), 1519-1527
MSC (2010): Primary 13E10, 13D40, 05E40, 14A05, 14M05, 14M07
Published electronically: October 11, 2012
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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 inclusion-exclusion 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 inclusion-exclusion principle does not hold for vector spaces in the way it is used by Zanello.


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Additional Information

Dan Laksov
Affiliation: Department of Mathematics, Royal Institute of Technology, KTH, S-100 44 Stockholm, Sweden
Email: laksov@math.kth.se

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11427-3
PII: S 0002-9939(2012)11427-3
Keywords: Graded modules, level algebras, Hilbert functions, inclusion-exclusion
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.