Noether normalizations, reductions of ideals, and matroids

Authors:
Joseph P. Brennan and Neil Epstein

Journal:
Proc. Amer. Math. Soc. **139** (2011), 2671-2680

MSC (2010):
Primary 13A30; Secondary 05B35, 13B21, 13H15

DOI:
https://doi.org/10.1090/S0002-9939-2011-10719-6

Published electronically:
January 24, 2011

MathSciNet review:
2801606

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that given a finitely generated standard graded algebra of dimension over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded -algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a *generic matroid*, which is a common generalization of a topological space and a matroid.

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Additional Information

**Joseph P. Brennan**

Affiliation:
Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, Orlando, Florida 32816

Email:
jpbrenna@mail.ucf.edu

**Neil Epstein**

Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109

Address at time of publication:
Universität Osnabrück, Institut für Mathematik, 49069 Osnabrück, Germany

Email:
nepstein@uos.de

DOI:
https://doi.org/10.1090/S0002-9939-2011-10719-6

Received by editor(s):
December 13, 2009

Received by editor(s) in revised form:
August 1, 2010

Published electronically:
January 24, 2011

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.