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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Noether normalizations, reductions of ideals, and matroids
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by Joseph P. Brennan and Neil Epstein PDF
Proc. Amer. Math. Soc. 139 (2011), 2671-2680 Request permission

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
  • 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
  • MR Author ID: 768826
  • Email: nepstein@uos.de
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2801606