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2013 Multiplicities associated to graded families of ideals
Steven Cutkosky
Algebra Number Theory 7(9): 2059-2083 (2013). DOI: 10.2140/ant.2013.7.2059


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.


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Steven Cutkosky. "Multiplicities associated to graded families of ideals." Algebra Number Theory 7 (9) 2059 - 2083, 2013.


Received: 20 July 2012; Revised: 11 October 2012; Accepted: 17 November 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1315.13040
MathSciNet: MR3152008
Digital Object Identifier: 10.2140/ant.2013.7.2059

Primary: 13H15
Secondary: 14B05

Keywords: graded family of ideals , multiplicity , Okounkov body

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.7 • No. 9 • 2013
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