[Paper Review] Castelnuovo-Mumford regularity and related invariants

Ngô Viêt Trung|arXiv (Cornell University)|Jul 26, 2019

Commutative Algebra and Its Applications18 references4 citations

TL;DR

This paper provides a comprehensive introduction to Castelnuovo-Mumford regularity and related invariants—weak regularity, $a^*$-invariant, and partial regularities—in the context of graded modules over standard graded algebras. It establishes equivalent characterizations using local cohomology, filter-regular sequences, and Gröbner bases, and proves that in characteristic zero, the regularity and $a^*$-invariant of an ideal are determined by the minimal generators of its generic initial ideal with respect to reverse lexicographic order.

ABSTRACT

These notes are an introduction to some basic aspects of the Castelnuovo-Mumford regularity and related topics such as weak regularity, a*-invariant and partial regularities.

Motivation & Objective

• To provide a unified and accessible introduction to Castelnuovo-Mumford regularity and related invariants in commutative algebra and algebraic geometry.
• To establish equivalent definitions of regularity using local cohomology, Tor/Ext modules, and filter-regular sequences.
• To introduce and analyze weak regularity and partial regularities as computationally more tractable alternatives to full regularity.
• To develop effective computational methods for regularity and related invariants using Gröbner bases and generic initial ideals.
• To characterize the regularity and $a^*$-invariant of ideals in polynomial rings over fields of characteristic zero via the minimal generators of their generic initial ideals.

Proposed method

• Define regularity via local cohomology modules: $\operatorname{reg}(M) = \max\{a_i(M) + i \mid i \geq 0\}$, where $a_i(M)$ is the largest non-vanishing degree of the $i$-th local cohomology module.
• Introduce filter-regular sequences to reduce regularity computation to finite-length quotient modules, especially useful over Artinian base rings.
• Define weak regularity as $\operatorname{g-reg}(M) = \max\{a_i(M) + i \mid i \geq 1\}$, which controls the vanishing of local cohomology in shifted degrees.
• Introduce the $a^*$-invariant as $a^*(M) = \max\{a_i(M) \mid i \geq 0\}$, which captures the maximal degree of local cohomology without shift.
• Define partial regularities $\operatorname{reg}_t(M) = \max\{a_i(M) + i \mid i \leq t\}$ and $a_t^*(M) = \max\{a_i(M) \mid i \leq t\}$, allowing finer control over graded structure.
• Use Gröbner basis techniques, particularly the reverse lexicographic order and generic initial ideals (Gin), to compute regularity and related invariants effectively.

Experimental results

Research questions

• RQ1 How are the different definitions of Castelnuovo-Mumford regularity—via resolution degrees, local cohomology, and Tor/Ext modules—equivalent?
• RQ2 In what way does the filter-regular sequence simplify the computation of regularity for graded modules?
• RQ3 How does weak regularity provide a finite criterion for regularity, and how does it relate to geometric regularity in algebraic geometry?
• RQ4 What is the precise relationship between the $a^*$-invariant and the regularity, especially in polynomial rings?
• RQ5 Can the regularity and $a^*$-invariant of an ideal be computed effectively from its generic initial ideal in characteristic zero?

Key findings

• The regularity of a graded module $M$ is equivalent to $\max\{b_i(M) - i \mid i = 0,\dots,s\}$, where $b_i(M)$ is the maximal degree of generators of the $i$-th syzygy module.
• For a polynomial ring in $n$ variables, $a^*(M) = \max\{b_i(M) - i \mid i = 0,\dots,s\} - n$, showing that $a^*$ is the unshifted version of regularity.
• The partial regularity $\operatorname{reg}_t(M)$ equals $\max\{b_i(M) - i \mid i \geq n - t\}$, which corresponds to the regularity of the $t$-th syzygy module.
• In characteristic zero, $\operatorname{reg}(I) = \max\{\deg(x^A) \mid x^A \in \operatorname{Min}(\operatorname{Gin}(I))\}$, so regularity is determined by the highest degree of minimal generators of the generic initial ideal.
• In characteristic zero, $a^*(I) = \max\{\deg(x^A) + m(x^A) \mid x^A \in \operatorname{Min}(\operatorname{Gin}(I))\} - n - 1$, where $m(x^A)$ is the largest index with non-zero exponent.
• The invariants $c_i(I)$, defined via quotients $\tilde{J}_i / J_i$, satisfy $\operatorname{reg}(R/I) = \max\{c_i(I) \mid i = 0,\dots,n\}$, and can be computed combinatorially from the lattice vectors of $\operatorname{in}(I)$.