On (FC)-sequences and Mixed Multiplicities of Multi-graded Algebras

Abstract

Let $S=\bigoplus_{n_1,\ldots, n_s \geq 0}S_{(n_1,\ldots, n_s)}$ be a finitely generated standard multi-graded algebra over a Noetherian local ring $A$. This paper investigates the positivity of mixed multiplicities of $S$ and characterizes them in terms of Hilbert-Samuel multiplicities. As an application, we get some results on the mixed multiplicities of ideals that covers the main results in [Vi] and [TV].