Algebra & Number Theory
- Algebra Number Theory
- Volume 7, Number 9 (2013), 2059-2083.
Multiplicities associated to graded families of ideals
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 “” 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.
Algebra Number Theory, Volume 7, Number 9 (2013), 2059-2083.
Received: 20 July 2012
Revised: 11 October 2012
Accepted: 17 November 2012
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Cutkosky, Steven. Multiplicities associated to graded families of ideals. Algebra Number Theory 7 (2013), no. 9, 2059--2083. doi:10.2140/ant.2013.7.2059. https://projecteuclid.org/euclid.ant/1513730087