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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Partial derivatives of a generic subspace of a vector space of forms: Quotients of level algebras of arbitrary type


Author: Fabrizio Zanello
Journal: Trans. Amer. Math. Soc. 359 (2007), 2675-2686
MSC (2000): Primary 13E10; Secondary 13H10
Published electronically: January 4, 2007
MathSciNet review: 2286051
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Abstract: Given a vector space $ V$ of homogeneous polynomials of the same degree over an infinite field, consider a generic subspace $ W$ of $ V$. The main result of this paper is a lower-bound (in general sharp) for the dimensions of the spaces spanned in each degree by the partial derivatives of the forms generating $ W$, in terms of the dimensions of the spaces spanned by the partial derivatives of the forms generating the original space $ V$.

Rephrasing our result in the language of commutative algebra (where this result finds its most important applications), we have: let $ A$ be a type $ t$ artinian level algebra with $ h$-vector $ h=(1,h_1,h_2,...,h_e)$, and let, for $ c=1,2,...,t-1$, $ H^{c,gen}=(1,H_1^{c,gen},H_2^{c,gen},...,H_e^{c,gen})$ be the $ h$-vector of the generic type $ c$ level quotient of $ A$ having the same socle degree $ e$. Then we supply a lower-bound (in general sharp) for the $ h$-vector $ H^{c,gen}$. Explicitly, we will show that, for any $ u\in \lbrace 1,...,e\rbrace $,

$\displaystyle H_u^{c,gen}\geq {1\over t^2-1}\left((t-c)h_{e-u}+(ct-1)h_u\right).$

This result generalizes a recent theorem of Iarrobino (which treats the case $ t=2$).

Finally, we begin to obtain, as a consequence, some structure theorems for level $ h$-vectors of type bigger than 2, which is, at this time, a very little explored topic.


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Additional Information

Fabrizio Zanello
Affiliation: Dipartimento di Matematica, Università di Genova, Genova, Italy
Address at time of publication: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: zanello@kth.se

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04015-9
PII: S 0002-9947(07)04015-9
Keywords: Artinian algebra, level algebra, $h$-vector, generic quotient, dimension, partial derivatives.
Received by editor(s): February 22, 2005
Received by editor(s) in revised form: March 17, 2005
Published electronically: January 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.