Numerical characterizations for integral dependence of graded ideals

TL;DR

The paper develops localization-free numerical criteria for detecting integral dependence of homogeneous ideals in standard graded equidimensional rings, recasting the problem in terms of diagonal Rees algebras and density-function invariants. It proves that equality of integral closures $\overline{I}=\overline{J}$ is equivalent to equalities of diagonal Hilbert–Samuel multiplicities $e(S[\mathsf{I}t]_{\Delta_{(c,1)}})$ and $e(S[\mathsf{J}t]_{\Delta_{(c,1)}})$ for $c>\mathbf{d}$, and to equalities of various multiplicities (RA-, mixed-, polar-, and $\varepsilon$-multiplicities) on suitable truncations, all without localization. The authors provide both algebraic and geometric proofs of finiteness criteria for $\ell_R(\overline{J}/\overline{I})$, along with a Macaulay2-based algorithm to compute these invariants and decide integral dependence in practice. These results generalize Rees’s classical theorem to the graded setting and offer computational pathways for detecting integral dependence via well-studied invariants.

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 $$\overline{I} = \overline{J}\iff e\big(S[\mathsf{I}t]_{Δ_{(c,1)}}\big) = e\big(S[\mathsf{J}t]_{Δ_{(c,1)}}\big),$$ where $e\big(S[\mathsf{I}t]_{Δ_{(c,1)}}\big)$ denotes the Hilbert-Samuel multiplicity of the standard graded domain $S[\mathsf{I}t]_{Δ_{(c,1)}} = \oplus_{n\geq 0}(\mathsf{I}^n)_{cn}t^n$. Further, if $I$ is of finite colength in $R$ then $e\big(S[\mathsf{I}t]_{Δ_{(c,1)}}\big) = c^de(R) - e(I,R)$. 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.

Theorems & Definitions (40)

• Theorem 1.1: Theorem \ref{['equidim']}
• Theorem 1.2: Theorem \ref{['integral1']}
• Theorem 1.3: Theorem \ref{['comp']}
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 3.2
• proof