On Zanello’s lower bound for level algebras

Laksov, Dan

doi:10.1090/S0002-9939-2012-11427-3

On Zanello’s lower bound for level algebras
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Proc. Amer. Math. Soc. 141 (2013), 1519-1527 Request permission

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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