TY - JOUR

T1 - Noetherianity for infinite-dimensional toric varieties

AU - Draisma, J.

AU - Eggermont, R.H.

AU - Krone, R.

AU - Leykin, A.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

U2 - 10.2140/ant.2015.9.1857

DO - 10.2140/ant.2015.9.1857

M3 - Article

VL - 9

SP - 1857

EP - 1880

JO - Algebra & Number Theory

JF - Algebra & Number Theory

SN - 1937-0652

IS - 8

ER -