Noether normalizations, reductions of ideals, and matroids
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- by Joseph P. Brennan and Neil Epstein PDF
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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.References
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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