Numerical characterizations for integral dependence of graded ideals

TL;DR

This paper provides numerical criteria for determining the integral dependence of graded ideals in standard graded rings, using multiplicities and invariants that do not require localization, thus offering computationally accessible characterizations.

Contribution

It introduces new numerical characterizations of integral dependence for graded ideals using multiplicities, extending classical results and avoiding localization.

Findings

Characterizations in terms of Hilbert-Samuel multiplicities.

Criteria involving epsilon and mixed multiplicities.

Connections to polar multiplicities of ideals.

Abstract

Let $R=\oplus_{m\geq 0}R_m$ be a standard graded equidimensional ring over a field $R_0$, and $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Then we give a set of numerical characterizations of the integral dependence of $I$ and $J$ in terms of certain multiplicities. A novelty of this approach is that it does not involve localization and only requires checking computable and well-studied invariants. In particular, we show the following: let $S=R[y]$, $\mathsf{I} = IS$ and $\mathsf{J} = JS$ and $\bf d$ be the maximum of the generating degrees of both $I$ and $J$. Let $c>{\bf d}$ be any given integer. Then $$\overline{I} = \overline{J}\iff e\big(S[\mathsf{I}t]_{\Delta_{(c,1)}}\big) = e\big(S[\mathsf{J}t]_{\Delta_{(c,1)}}\big),$$ where $e\big(S[\mathsf{I}t]_{\Delta_{(c,1)}}\big)$ denotes the Hilbert-Samuel multiplicity of the standard graded domain $S[\mathsf{I}t]_{\Delta_{(c,1)}} = \oplus_{n\geq 0}(\mathsf{I}^n)_{cn}t^n$. Further, if $I$ is of finite colength in $R$ then $e\big(S[\mathsf{I}t]_{\Delta_{(c,1)}}\big) = c^de(R) - e(I,R)$. If $R$ is also a domain, then other numerical criteria are the following: \begin{align*} \overline{I} = \overline{J} & \iff \varepsilon(I)=\varepsilon(J)\;\;\mbox{and}\;\; e_i(R[It]) = e_i(R[Jt])\;\;\mbox{for all}\;\; 0\leq i <\dim(R/I), \end{align*} where $\varepsilon(I)$ denotes the epsilon multiplicity of $I$, and $e_i(R[It])$'s are the mixed multiplicities of the Rees algebra $R[It]$. The relation between $e_i(S[\mathsf{I}t])$ and the polar multiplicities of $\mathsf{I}_{\geq {\bf d}}$ provides another criterion in terms of polar multiplicities of $\mathsf{I}_{\geq {\bf d}}$. The first two characterizations generalize Rees's classical result for ideals of finite colengths. Apart from several well-established results, the proofs of these results use the theory of density functions, which was developed in arXiv:2311.17679.