Mixed Multiplicities of Filtrations

2018/05/03 by Cutkosky, Steven Dale, Sarkar, Parangama, Srinivasan, Hema
#13D40 #13H15 #14B99 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.01440

Abstract

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of m_R-primary ideals in a Noetherian local ring R, generalizing the classical theory for m_R-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of R is less than the dimension of R, which holds for instance if R is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of m_R-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp.

Related

• Multiplicities and Mixed Multiplicities of arbitrary Filtrations
• An inequality in mixed multiplicities
• Mixed multiplicities of Divisorial Filtrations
• The Minkowski Equality for Filtrations
• Positivity of Mixed Multiplicities of Filtrations