Multiplicities associated to graded families of ideals
Multiplicities associated to graded families of ideals
复制标题
DOI：
10.2140/ant.2013.7.2059
复制
发表时间：
2012-06
影响因子：
1.3
通讯作者：
S. Cutkosky
中科院分区：
文献类型：
--
作者：
S. Cutkosky
我们证明了与分次理想族相关的重数极限在非常一般的条件下存在。我们的大多数结果适用于解析非分歧等特征局部环，具有完美剩余域。我们给出了一些应用程序，包括一个“体积=多重性”公式，推广公式的Lazarsfeld和Mustata，并证明了Ulrich和Validashti的多重性存在作为一个限制理想在相当一般的环，包括分析的局部域。我们还证明了一个渐近的“加法公式”的限制的多重性，和公式限制增长的估值，这回答了一个问题所提出的作者，起亚Dalili和奥尔加Kashcheyeva。我们的证明的灵感来自奥肯科夫的哲学，计算限制的多重性的体积切片的适当的锥所产生的半群确定的适当的过滤上的一个家庭的代数对象。
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 Mustata, 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 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.