Numerical characterizations for integral dependence of graded modules

Technical Summary: Numerical Characterizations for Integral Dependence of Graded Modules

Problem Statement
The primary objective of this work is to establish a new numerical criterion for detecting the integral dependence of arbitrary graded modules defined over a standard graded Noetherian domain $A$ over a field $k$. While the characterization of integral dependence via numerical invariants was pioneered by Rees for ideals, and subsequently extended to ideals in graded settings by the authors in previous works [5, 6], this paper addresses the more complex case of finitely generated, torsion-free $N$-graded $A$-modules $M$ of rank $e$. The authors aim to construct specific density functions and utilize them to derive necessary and sufficient conditions for a graded submodule $N \subseteq M$ to be a reduction of $M$.

Methodology and Setup
The paper operates within the framework of a standard graded domain $A = \bigoplus_{m \ge 0} A_m$ of dimension $d \ge 2$ over a field $k$. The module $M$ is a finitely generated, torsion-free $A$-module of rank $e$. To define the Rees algebra $A[Mt]$, the authors fix a graded embedding $M \hookrightarrow F$, where $F$ is a free graded $A$-module of finite rank. The Rees algebra is defined as the image of the symmetric algebra map: $A[Mt] = \bigoplus_{n \ge 0} M_n t^n \subseteq \text{Sym}_A(F)$.

The core methodology involves the construction and analysis of three distinct density functions derived from the asymptotic behavior of the lengths of graded components of $M$, its saturation, and the quotient between them. The authors utilize the natural bigrading on the Rees algebra $A[Mt]$, where generators have bidegrees $(d_i, 1)$ and elements of $A$ have bidegree $(1, 0)$.

Key technical tools employed include:

1. Asymptotic Analysis of Lengths: The authors analyze the length function $\ell_k((M_n)_m)$ as $n \to \infty$, utilizing the structure of the Rees algebra as a quotient of a bigraded polynomial ring.
2. Quasi-polynomials and Cones: Drawing on results from [5], the authors partition the domain of the length function into cones $C_j$ generated by the degrees of the generators of $M$. Within these cones, the length function is shown to behave as a quasi-polynomial, which eventually stabilizes to a polynomial.
3. Saturation and $\epsilon$-functions: The authors define the saturation of $M_n$ as $\widetilde{M}_n = (M_n :_F \mathfrak{m}^\infty)$. They introduce the $\epsilon$-density function to measure the difference between the saturated and adic (original) modules.
4. Geometric Interpretation: The proofs leverage the volume function of divisors on projective varieties (specifically $X = \text{Proj}_V B$ where $B$ is the Rees algebra) and asymptotic Riemann-Roch theorems to establish the continuity and polynomial nature of the density functions.
5. Diagonal Subalgebras: To handle the integral dependence criteria, the authors consider $(c, 1)$-diagonal subalgebras $A[Mt]_{\Delta(c,1)}$ and their extensions $S[Mt]_{\Delta(c,1)}$ where $S = A[y]$, relating their multiplicities to the density functions.

Key Contributions and Results

1. Construction of Density Functions:

   • Adic Density Function ($f_M$): Defined as $f_M(x) = \lim_{n \to \infty} \frac{\ell_k((M_n)_{\lfloor xn \rfloor})}{n^{d+e-2}/(d+e-1)!}$. Theorem 3.2 establishes that $f_M$ is a well-defined, piecewise polynomial function, continuous everywhere except possibly at $x=d_1$ (the minimal degree of a generator). It is supported on $[d_1, \infty)$.
   • Saturated Density Function ($\tilde{f}_M$): Defined as $\tilde{f}_M(x) = \lim_{n \to \infty} \frac{\ell_k((\widetilde{M}_n)_{\lfloor xn \rfloor})}{n^{d+e-2}/(d+e-1)!}$. Theorem 4.3 proves this is a continuous function supported on $[-c_0, \infty)$ for some constant $c_0$. Crucially, for $x \ge d_M$, $\tilde{f}_M(x)$ coincides with $f_M(x)$ and is given by a polynomial involving mixed multiplicities of the Rees algebra.
   • $\epsilon$-Density Function ($f_\epsilon(M)$): Defined as the difference $f_\epsilon(M)(x) = \tilde{f}_M(x) - f_M(x)$. Theorem 5.2 shows this function is compactly supported on $[-c_0, d_M]$ and continuous on $\mathbb{R} \setminus \{d_1\}$. The integral of this function, $\epsilon(M)$, is shown to exist as a limit, providing a numerical invariant measuring the "defect" between the module and its saturation.

2. Numerical Criteria for Integral Dependence:
   The paper culminates in Theorem 6.6, which provides a set of equivalent conditions for a graded submodule $N \subseteq M$ (with $\text{rank } N = \text{rank } M = e$) to be a reduction of $M$. The following statements are equivalent:

   • $N$ is a reduction of $M$.
   • The adic and saturated density functions coincide: $f_M \equiv f_N$ and $\tilde{f}_M \equiv \tilde{f}_N$.
   • The $\epsilon$-invariants are equal ($\epsilon(M) = \epsilon(N)$) and the multiplicities of the $(c, 1)$-diagonal subalgebras are equal for some $c > d_{N,M}$.
   • The mixed multiplicities of the Rees algebras $S[Mt]$ and $S[Nt]$ coincide for all $0 \le i \le d$.

Significance and Claims
The paper claims to extend the numerical characterization of integral dependence from the realm of graded ideals (as treated in [5] and [6]) to the more general context of graded modules. The authors note that this extension is non-trivial because, unlike the ideal case where $A^n = A$, the components $F^n$ of the free module vary with $n$, and the invariants of $F^n$ are not constant.

The significance of the work lies in:

• Generalization: Providing a unified framework for detecting integral dependence in modules using asymptotic invariants.
• New Invariants: Introducing the $\epsilon$-density function and the associated invariant $\epsilon(M)$ as a tool for measuring the difference between a module and its saturation in a graded context.
• Rigorous Limits: Establishing the existence of limits for these density functions and the $\epsilon$-invariant in the graded setup, complementing previous results in local settings [3, 20].
• Computational Criteria: Offering concrete, checkable numerical criteria (involving multiplicities of diagonal subalgebras and mixed multiplicities) that are necessary and sufficient for reduction, which are essential for algorithmic applications in commutative algebra and singularity theory.

The authors maintain a modest tone, framing the results as an extension of existing theories for ideals to modules, utilizing similar but adapted strategies (such as linear regularity bounds for powers of modules) to overcome the structural differences between ideals and modules.