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

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 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

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.