Multiplicities associated to graded families of ideals

Cutkosky, Steven

doi:10.2140/ant.2013.7.2059

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Multiplicities associated to graded families of ideals

Steven Dale Cutkosky

Vol. 7 (2013), No. 9, 2059–2083

DOI: 10.2140/ant.2013.7.2059

Abstract

We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings with perfect residue fields. We give a number of applications, including a “ $volume = multiplicity$ ” formula, generalizing the formula of Lazarsfeld and Mustaţă, and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic “additivity formula” for limits of multiplicities and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

Keywords

multiplicity, graded family of ideals, Okounkov body

Mathematical Subject Classification 2010

Primary: 13H15

Secondary: 14B05

Milestones

Received: 20 July 2012

Revised: 11 October 2012

Accepted: 17 November 2012

Published: 18 December 2013

Authors

Steven Dale Cutkosky
Department of Mathematics University of Missouri Columbia, MO 65211 United States

Vol. 7, No. 9, 2013