Multiplicities and mixed multiplicities of arbitrary filtrations
Multiplicities and mixed multiplicities of arbitrary filtrations
复制标题
DOI：
10.1007/s40687-021-00307-x
复制
发表时间：
2022-01
影响因子：
1.2
通讯作者：
S. Cutkosky;Parangama Sarkar
中科院分区：
文献类型：
--
作者：
S. Cutkosky;Parangama Sarkar
我们开发了一个理论的多重性和混合多重性的过滤，扩展理论的过滤的m-主要理想的任意（不一定诺特）过滤。解析非分歧局部环R上的r-滤子的混合重数来自于r-变量的一个适当的齐次多项式的系数，其次数等于环的维数，类似于局部环中m-准素理想的混合重数的经典情况。证明了Minkowski不等式对任意滤子成立。Teissier，Rees和Sharp和Katz在局部环中的m-准素理想的Minkowski不等式中的等式的刻画并不扩展到任意的filtrations，但我们证明了它们在一个大的重要的filtrations子范畴中是真实的。我们定义了可除的和有界的过滤。一个固定理想的幂的滤子是有界滤子，就像一个整除滤子一样。我们表明，在一个优秀的局部区域，平等的Minkowski等式的特征在于条件，适当的Rees像代数的积分闭包是相同的，严格推广定理Teissier，Rees和夏普和Katz。我们还证明了描述具有相同重数的理想包含的里斯定理推广到优秀局部域中的有界过滤。我们给出了一些其他的应用，扩展经典定理的理想。
We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations ofm-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities ofrfiltrations on an analytically unramified local ringRcome from the coefficients of a suitable homogeneous polynomial inrvariables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities ofm-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals.