Castelnuovo–Mumford Regularity

• Castelnuovo–Mumford regularity is a homological invariant that quantifies the complexity of graded modules and coherent sheaves by analyzing syzygy degrees and local cohomology.
• It provides concrete bounds on generator degrees and syzygy locations, enabling effective computations in algebraic geometry and commutative algebra.
• The concept is pivotal for understanding designer ideals, asymptotic growth of powers, and sharp geometric bounds in both theoretical research and computational applications.

Castelnuovo-Mumford Regularity

Castelnuovo–Mumford regularity (“regularity”, often abbreviated reg) is a homological invariant that quantitatively measures the complexity of a finitely generated graded module or a sheaf, particularly over a standard graded polynomial ring. It plays a central role in algebraic geometry and commutative algebra, controlling the degrees of generators for syzygies, the vanishing of higher cohomology groups, and the effective bounds on algorithms for objects such as ideals, sheaves, or blowup algebras. Originating in the work of Castelnuovo and adapted into homological form by Mumford, the concept has become foundational for both theoretical research and computational applications.

1. Core Definitions and Characterizations

Let $S = k[x_0, \dots, x_n]$ be a standard graded polynomial ring over a field $k$, and $M$ a finitely generated graded $S$-module. The minimal graded free resolution of $M$ has the form

$$\cdots \to \bigoplus_j S(-j)^{\beta_{i,j}(M)} \to \cdots \to M \to 0$$

where $$\beta_{i,j}(M) = \dim_k \operatorname{Tor}_i^S(M, k)_j$$ are the graded Betti numbers. The Castelnuovo–Mumford regularity is defined by

$$\operatorname{reg}(M) = \max \{ j - i \mid \beta_{i,j}(M) \neq 0 \}$$

or, equivalently, via local cohomology,

$$\operatorname{reg}(M) = \max_{i \geq 0} \{ a_i(M) + i \}$$

where $a_i(M) = \sup \{ t \mid H^i_{S_+}(M)_t \neq 0 \}$ and $S_+ = (x_0, \dots, x_n)$ denotes the homogeneous maximal ideal (Ullery, 2013, Trung, 2019, Bruns et al., 2021). For a homogeneous ideal $I \subset S$, often $$\operatorname{reg}(I) = \operatorname{reg}(S/I) + 1$$ is quoted, but it is common to work with either convention.

For a coherent sheaf $\mathcal{F}$ on projective space $\mathbb{P}^n$, $\mathcal{F}$ is said to be $m$-regular if $H^i(\mathbb{P}^n, \mathcal{F}(m-i)) = 0$ for all $i > 0$, and

$$\operatorname{reg}(\mathcal{F}) = \min \{ m \mid \mathcal{F} \text{ is } m\text{-regular} \}$$

(Kwak et al., 2014, Niu, 2013, Bruns et al., 2021, Lee et al., 2024).

The regularity provides a tight bound on the “spread” of syzygy degrees and controls vanishing needed for effective results in projective geometry.

2. Properties, Behaviors, and Related Invariants

Regularity is sensitive to both algebraic and geometric structure:

• Generator degree bound: For a graded $S$-module $M$, the largest degree of a minimal generator $t_0(M)$ satisfies $t_0(M) \leq \operatorname{reg}(M)$ (Bruns et al., 2021).
• Exact sequences: $$\operatorname{reg}(N) \leq \max\{\operatorname{reg}(M), \operatorname{reg}(L)+1\}$$ in any short exact sequence $0 \to N \to M \to L \to 0$.
• Change of grading: Regularity is preserved under scalar extension and quotient by surjective maps from graded polynomial rings (Bruns et al., 2021).
• Syzygy location: For a module with a minimal free resolution, $$\operatorname{reg}(M) = \max_i \left( \max_j \{ j - i \mid \beta_{i,j}(M) \neq 0 \} \right)$$.
• Weak and geometric regularity, $a^*$-invariant: Variants measure vanishing in shifted or partial degrees (Trung, 2019).

There exist further generalized notions in multigraded and non-graded settings (Botbol et al., 2011).

3. Asymptotic and Extremal Regularity: Powers and Designer Ideals

Growth for Powers

For any homogeneous ideal $I$ in a standard graded Noetherian ring $R$, and finitely generated graded module $M$, the asymptotic linearity theorem of Cutkosky–Herzog–Trung and Kodiyalam establishes that

$$\operatorname{reg}(I^v M) = a \cdot v + c \qquad \text{for } v \gg 0$$

where $a$ is the degree of some generator of $I$, and $c$ is an integer depending on $I$ and $M$ (Bruns et al., 2021).

Ideals with linear powers (those for which each $I^v$ has a $d \cdot v$-linear resolution) are characterized by the vanishing of a bigraded regularity of the Rees algebra, $$\operatorname{reg}_{(1,0)}(\operatorname{Rees}(I,M))=0$$. Maximal minors of generic matrices are a central class with linear powers (Bruns et al., 2021).

Ideals with Prescribed High Regularity

Ullery's construction of “designer ideals” provides explicit methods to build ideals with regularity growing superlinearly in the generating degree and with the peak Betti difference $j-i$ forced to occur at an arbitrary syzygy. For example, beginning with an $R$-module $M$ with strictly increasing maximal degree sequence $(k, t_1, \dots, t_r)$ and bounding the number of generators, one embeds $\operatorname{Proj}(R)$ into $\mathbb{P}^{n+N}$ and constructs an ideal $J_M \subset S$ whose regularity is $(t_r - r) + 1$ and whose “spike” can be pushed as far out in the resolution as desired. The growth can be engineered to achieve arbitrarily high polynomial rates in $k$ while keeping generators in a single degree (Ullery, 2013).

The construction demonstrates that regularity is not governed solely by initial syzygy degrees and allows for Betti tables that violate classical “syzygy control” intuitions.

4. Bounds and Sharpness Across Geometric and Combinatorial Settings

Classical and Recent Bounds

• Complete intersections: $\operatorname{reg}(I) \leq N(k-1)+1$ for codimension $N$ and generating degree at most $k$ (Ullery, 2013).
• Eisenbud–Goto conjecture: For a nondegenerate irreducible projective variety $X \subset \mathbb{P}^r$ of dimension $d$ and degree $d$, the conjecture predicts $$\operatorname{reg} X \leq \deg X - \operatorname{codim} X + 1$$ (Niu, 2013, Kwak et al., 2014). The bound is confirmed in numerous special cases (e.g., rational, elliptic, log canonical surfaces (Niu, 2013), seminormal simplicial semigroup rings (Nitsche, 2011), scrolls (Niu et al., 2016), and certain singular surfaces) but is false in the most general singular setting (Kwak et al., 2014).
• Smooth varieties: For a smooth projective variety $X \subset \mathbb{P}^r$ of degree $d$ and codimension $e$, sharp bounds include $\operatorname{reg} (\mathcal{O}_X) \leq d-e$ and $$\operatorname{reg}(X) \leq \max\{d-e+1, \text{normality index}\}$$ (Kwak et al., 2014). Further results, such as $\operatorname{reg}(X) \leq n(d-2)+1$ and variants involving double point divisors and projections, improve classical results (Kwak et al., 2014, Rathmann, 2020).
• Powers of ideal sheaves: For $X \subset \mathbb{P}^r$ a smooth variety scheme-theoretically cut out by hypersurfaces of degrees $d_1 \geq \dots \geq d_m$, the regularity of $\mathcal{I}_X^a$ satisfies $$\operatorname{reg}(\mathcal{I}_X^a) \leq a d_1 + d_2 + \cdots + d_e - e + 1$$ with equality if and only if $X$ is a complete intersection (Shang, 2022).

Monomial Ideals and Combinatorics

For quadratic square-free monomial ideals, identification with graph invariants yields tight combinatorial bounds on regularity via decompositions, genus, and separator theory. For instance, for chordal graphs, regularity is uniformly bounded by 2; more generally, hereditary graph families, separator-based bounds, and genus-based inequalities achieve sharp results, reflecting the tight interaction between algebraic and combinatorial data (Blekherman et al., 2019).

Finite Schemes and Secant Theory

For 0-dimensional subschemes $\Gamma \subset \mathbb{P}^n$ of degree $d$, the secant invariant $t(\Gamma)$ — the minimal $t$ such that $\Gamma$ has a $(t+2)$-secant $t$-plane — governs the regularity bound: $$\operatorname{reg}(\Gamma) \leq \left \lceil \frac{d - n - 1}{t(\Gamma)} \right \rceil + 2$$ with maximal regularity achieved exactly when $\Gamma$ is supported in a unique rational normal curve of degree $t(\Gamma)$ (Lee et al., 2024).

5. Multigraded, Symmetric, and Functorial Extensions

Multigraded Regularity

In multigraded or toric settings, regularity generalizes to a region in the grading lattice, defined as the locus where shifted local cohomologies vanish. This Maclagan–Smith regularity controls syzygy degrees and persists under graded truncation (Botbol et al., 2011). Specialized regularity notions for scrolls relate closely to — but are generally weaker than — full multigraded regularity, yet suffice for splitting and generation theorems (Malaspina et al., 10 Jan 2025).

Regularity Under Symmetry and Functoriality

Chains of ideals or modules invariant under symmetric or hierarchical category actions display eventual linear or bounded growth of regularity. For instance, in symmetric group-invariant chains, the regularity is eventually linear in $n$ with explicit combinatorial bounds, often verified for Artinian or square-free monomial cases (Le et al., 2018). For representations of finite products of EI-categories, such as FI-modules and their generalizations, finite regularity is equivalent to finite degree of presentation, stabilizing abelian properties of the module categories (Gan et al., 2019).

6. Applications and New Directions

Regularity remains a methodologically crucial invariant across a range of algebraic and geometric problems:

• Controls the global generation and vanishing of higher cohomology of sheaves, central in effective projective geometry, syzygy theory, and Hilbert scheme computations (Trung, 2019, Cioffi et al., 2013).
• Provides thresholds for the stabilization of Ratliff–Rush closure, reduction numbers, and for equational complexity in Rees, associated graded, and fiber cones; in important classes, regularity of blowup algebras coincides and matches conjectured patterns (Dinh et al., 2015, Jayanthan et al., 2011).
• Underlies explicit splitting criteria for vector bundles on scrolls and toric varieties: regularity conditions force bundles to decompose as direct sums of line bundles in the presence of suitable vanishing (Malaspina et al., 10 Jan 2025).
• Facilitates algorithmic approaches to minimal Hilbert functions and lowest-regularity Hilbert scheme loci, via combinatorial and constructive methods (ideal graft, lifting) (Cioffi et al., 2013).

Open directions include sharpening polynomial bounds in the smooth (or mild singularity) case, generalization to broader ambient spaces (e.g., Segre varieties, higher codimensional intersections), and deeper understanding of regularity behavior in highly structured or symmetric settings (Ullery, 2013, Botbol et al., 2011, Malaspina et al., 10 Jan 2025).

7. Illustrative Special Cases and Examples

• Designer ideals: Construction permits realization of Betti tables with delayed regularity spikes and arbitrarily high polynomial growth relative to the generating degree (Ullery, 2013).
• Matrix Schubert varieties: The regularity is given by the difference between the Rajchgot index (a permutation statistic) and the inversion number, connecting homological and combinatorial data (Pechenik et al., 2021).
• Seminormal simplicial affine semigroup rings: The Eisenbud–Goto bound holds and gives a combinatorial calculation for Veronese subrings, where $$\operatorname{reg}(K[x_1, \dots, x_d]^{(a)}) = \lfloor d - d/a \rfloor$$ (Nitsche, 2011).
• Hilbert schemes: The minimal regularity for given Hilbert polynomial and Hilbert function is achieved by schemes with minimal admissible Hilbert function constructed by growth-height-lex Borel sets (Cioffi et al., 2013).
• Scroll regularity: Generalized regularity defines a splitting theory for vector bundles on scrolls, extending classical projective and biprojective regularity (Malaspina et al., 10 Jan 2025).

Research on Castelnuovo–Mumford regularity thus continues to form a nexus of interaction between resolution theory, geometric syzygies, deformation theory, and computational algebraic geometry.

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Key References

• Designer ideals and delayed regularity: (Ullery, 2013)
• Regularity for representations and FI-modules: (Gan et al., 2019)
• Singular surfaces and the Eisenbud–Goto conjecture: (Niu, 2013)
• Regularity of powers and asymptotic growth: (Bruns et al., 2021, Dinh et al., 2015, Shang, 2022)
• Monomial ideals and combinatorial bounds: (Blekherman et al., 2019, Lee et al., 2024)
• Scrolls and splitting criteria: (Niu et al., 2016, Malaspina et al., 10 Jan 2025)
• Seminormal semigroup rings: (Nitsche, 2011)
• Minimal regularity and Hilbert function methods: (Cioffi et al., 2013)
• Multigraded and toric frameworks: (Botbol et al., 2011)
• Regularity and category invariance: (Le et al., 2018)
• Matrix Schubert varieties and permutation-statistics connections: (Pechenik et al., 2021)