Multiplicities of weakly graded families of ideals
Parangama Sarkar
TL;DR
The paper addresses extending multiplicity theory to weakly graded families of ideals, notably bounded below linearly weakly graded families in Noetherian local rings with $\dim(N(\hat{R}))<d$. It establishes the existence of the invariant $e_W(\mathcal I)=\lim_{n\to\infty} d!\ell_R(R/I_n)/n^d$, proves a volume-multiplicity formula $e_W(\mathcal I)=\lim e(I_n)/n^d$, and proves Minkowski inequalities for such families, including equality criteria. It also analyzes weakly graded families of the form $\{(I_n:K)\}$, showing $e_W(\{(I_n:K)\})$ is bounded above by $e(\mathcal I)$ and identifying conditions for equality, along with a generalized Rees-characterization of equal multiplicities. The work further connects these asymptotics to local cohomology lengths and the epsilon multiplicity framework, broadening multiplicity theory beyond Noetherian filtrations and offering tools for valuations and singularity theory.
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).