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Noetherianity for
infinite-dimensional toric varieties
Jan Draisma, Rob Eggermont, Robert Krone and Anton Leykin |
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Vol. 9 (2015), No. 8, 1857–1880
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DOI: 10.2140/ant.2015.9.1857
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Abstract |
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We consider a large class of monomial maps respecting an action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples where Noetherianity was recently proved or conjectured. In particular, our results imply Hillar–Sullivant’s independent set theorem and settle several finiteness conjectures due to Aschenbrenner, Martín del Campo, Hillar, and Sullivant. We introduce a matching monoid and show that its monoid ring is Noetherian up to symmetry. Our approach is then to factorize a more general equivariant monomial map into two parts going through this monoid. The kernels of both parts are finitely generated up to symmetry: recent work by Yamaguchi–Ogawa–Takemura on the (generalized) Birkhoff model provides an explicit degree bound for the kernel of the first part, while for the second part the finiteness follows from the Noetherianity of the matching monoid ring. |
Keywords
Noetherianity up to symmetry, binomial ideals
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Mathematical Subject Classification 2010
Primary: 13E05
Secondary: 13P10, 14M25
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Milestones
Received: 6 November 2014
Revised: 6 March 2015
Accepted: 12 June 2015
Published: 29 October 2015
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Authors |
| Jan Draisma | |
| Department of Mathematics and
Computer Science Technische Universiteit Eindhoven P.O. Box 513, 5600 MB Eindhoven Netherlands Vrije Universiteit and Centrum voor Wiskunde en Informatica Amsterdam Netherlands |
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| Rob Eggermont | |
| Department of Mathematics and
Computer Science Technische Universiteit Eindhoven P.O. Box 513, 5600 MB Eindhoven Netherlands |
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| Robert Krone | |
| School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160 United States |
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| Anton Leykin | |
| School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160 United States |
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