Vol. 9, No. 8, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Noetherianity for infinite-dimensional toric varieties

Jan Draisma, Rob Eggermont, Robert Krone and Anton Leykin

Vol. 9 (2015), No. 8, 1857–1880
DOI: 10.2140/ant.2015.9.1857
Abstract

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
Mathematical Subject Classification 2010
Primary: 13E05
Secondary: 13P10, 14M25
Milestones
Received: 6 November 2014
Revised: 6 March 2015
Accepted: 12 June 2015
Published: 29 October 2015
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
Rob Eggermont
Department of Mathematics and Computer Science
Technische Universiteit Eindhoven
P.O. Box 513, 5600 MB Eindhoven
Netherlands
Robert Krone
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
United States
Anton Leykin
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
United States