Multiplicities of weakly graded families of ideals (2411.04831v4)

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 $e_W(\mathfrak{I}):=\lim\limits_{n\to\infty}d!\frac{\ell_R(R/I_n)}{n^d}$ exists where $\mathfrak I={I_n}$ is a bounded below linearly weakly graded families of ideals in a Noetherian local ring $(R,\mathfrak m)$ of dimension $d\geq 1$ with $\dim(N(\hat{R}))<d$. Furthermore, we prove that ``volume=multiplicity" formula and Minkowski inequality hold for such families of ideals. We explore some properties of $e_W(\mathfrak J)$ for weakly graded families of ideals of the form $\mathfrak J={(I_n:K)}$ where ${I_n}$ is an $\mathfrak m$-primary graded family of ideals. We provide a necessary and sufficient condition for the equality in Minkowski inequality for the weakly graded family of ideals of the form $\mathfrak J={(I_n:K)}$ where ${I_n}$ 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 behaviour of the length function $\ell_R(H_{\mathfrak m}^0(R/(I_n:K)))$ where ${I_n}$ is a filtration of ideals (not necessarily $\mathfrak m$-primary).