Multiplicities of weakly graded families of ideals

Abstract In this article, we extend the notion of multiplicity for weakly graded families of ideals which are bounded below linearly. In particular, we show that the limit exists where is a bounded below linearly weakly graded family of ideals in a Noetherian local ring of dimension with . Furthermore, we prove that “volume = multiplicity” formula and Minkowski inequality hold for such families of ideals. We explore some properties of for weakly graded families of ideals of the form where is an ‐primary graded family of ideals. We provide a necessary and sufficient condition for the equality in Minkowski inequality for the weakly graded families of ideals of the form where is a bounded filtration. Moreover, we generalize a result of Rees characterizing the inclusion of ideals with the same multiplicities for the above families of ideals. Finally, we investigate the asymptotic behavior of the length function where is a filtration of ideals (not necessarily ‐primary).