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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
Published electronically: January 24, 2011
MathSciNet review: 2801606
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Abstract: We show that given a finitely generated standard graded algebra of dimension $ d$ 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 $ d$ 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 $ k$-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

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

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.

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