Graded MCM Modules: Structure & Classification

• Graded maximal Cohen–Macaulay modules are finitely generated modules over graded rings with depth equal to the ring's dimension, encapsulating key singular and geometric properties.
• The associated graded module construction, together with Hilbert functions and h-polynomials, provides sharp depth bounds and insights into deformations and extensions.
• Recent advances enable explicit classification via matrix factorizations, superficial sequences, and combinatorial invariants in both commutative and noncommutative contexts.

Graded maximal Cohen–Macaulay (MCM) modules are a central object of study in commutative algebra, singularity theory, and noncommutative algebraic geometry. These modules, often defined over graded Gorenstein rings, hypersurfaces, or more general singularities, encode deep information about the structure and representation theory of singular spaces, both commutative and noncommutative, and underlie the geometry of cones over projective varieties, quotient singularities, and their invariants.

1. Definition and Basic Properties

Let $A$ be a commutative Noetherian $\mathbb{N}_0$-graded ring (often local or standard graded) over a field $k$. A finitely generated graded $A$-module $M$ is called graded maximal Cohen–Macaulay (MCM) if

$\operatorname{depth}_A M = \operatorname{dim} A.$

This property is required to hold at each homogeneous (prime) ideal or at the unique graded maximal ideal in the local case. The grading structure refines classical homological and geometric invariants, enabling analysis via Hilbert series, local cohomology, and graded extensions.

A central grading-compatible construction is the associated graded module with respect to a relevant filtration (typically the filtration by powers of the homogeneous maximal ideal $\mathfrak{m}$): $$G(M) = \bigoplus_{n \geq 0} \mathfrak{m}^n M / \mathfrak{m}^{n+1} M.$$ This module reflects the tangent cone to the support of $M$ and governs deformation-theoretic and singularity-theoretic properties.

2. Depth, Hilbert Functions, and Associated Graded Modules

The depth of the associated graded module $G(M)$ is a sensitive invariant linking the module-theoretic structure of $M$ to the geometry of the tangent cone. Recent work has led to the development of precise bounds:

• On a hypersurface $A=Q/(f)$ of dimension $d$ with multiplicity $e(A) = 3$, for $M$ an MCM $A$-module with minimal number of generators $\mu(M) = 4$,

$\operatorname{depth}\, G(M) \geq d - 3,$

with this lower bound being sharp and equality attained for suitable $M$ (Mishra et al., 2023).

The proof involves reduction to lower-dimensional cases via $\sigma$-superficial sequences, case analysis on the possible structure of the module after dimension reduction, and detailed computation of the Hilbert series and associated graded invariants, including the use of Sally descent and Ratliff-Rush closure.

Key Technical Ingredients

• Superficial sequences: Systematically used to reduce dimension while preserving depth relations.
• Hilbert and $h$-polynomials: The shape of the $h$-polynomial of $M$ determines whether $G(M)$ is Cohen–Macaulay and provides sharp depth bounds.
• Minimal presentations and resolutions: The grading and valuation of entries in minimal presentations over regular local rings or their completions are closely tied to module invariants such as $i(M)$, the highest power in which presentation entries appear.

These techniques have been systematically extended to different classes of MCM modules and to various settings with multiple numbers of generators or different ring multiplicities (Mishra et al., 2022, Mishra et al., 2021).

3. Classification, Variety Structure, and Moduli

The set of isomorphism classes of graded MCM modules over a fixed ring $A$ often displays rich geometric and combinatorial structure.

• Varieties of graded MCM modules: Generalizing classical module varieties, for a fixed presentation type $V$ (i.e., a graded $k$-vector space structure corresponding to a given Hilbert series), the scheme

$$\mathrm{Reps}(A, V)(k) = \operatorname{Hom}_{S\text{-alg}}(A, \mathrm{End}_S(S \otimes_k V))_0$$

parameterizes graded MCM $A$-modules of type $V$ as orbits under an automorphism group $G_V$ (Hiramatsu, 2020, Dao et al., 2015).

• Finiteness: For commutative graded Cohen–Macaulay algebras $A$ of countable CM representation type (e.g., simple hypersurfaces, cones over smooth projective varieties), for each fixed $V$ there are only finitely many isomorphism classes of graded MCM modules with that Hilbert series (Hiramatsu, 2020), and only finitely many indecomposable rigid MCM modules of a given multiplicity or rank (Dao et al., 2015). This is proved using geometric properties of parameter spaces and the action of automorphism groups, often leveraging the uncountability of the ground field.
• Rigidity and moduli: Rigid MCM modules (those with vanishing self-extensions in degree zero) play a special role in the representation theory and the study of (maximal) modifying modules, especially relevant for 3-dimensional compound Du Val (cDV) singularities.

4. Extensions, Homological Functors, and Bounded Multiplicity

Extensions between graded MCM modules and their behavior under operations like construction of associated graded modules have received attention due to their impact on the possible families of MCM modules.

• Subfunctor for $\operatorname{Ext}^1$: The functor $T_A(M,N)$, forming a subfunctor of $\operatorname{Ext}^1_A(M,N)$, detects exact sequences for which the associated sequence of graded modules is also exact, with $G(E)$ Cohen–Macaulay given appropriate conditions on $G(M)$ and $G(N)$ (Puthenpurakal, 2018).
  • T-split extensions are characterized by a precise formula for the difference in toric Hilbert polynomials:

  $e^T(s) = e^T(M) + e^T(N) - e^T(E)$

  and such $E$ inherit the Cohen–Macaulay property if $M$, $N$ do.

• Weak Brauer–Thrall II for graded MCM modules: It is possible to have infinite families of indecomposable graded MCM modules, all with Cohen–Macaulay associated graded modules and multiplicities bounded above. Explicit constructions in complete intersections and non-isolated singularities demonstrate this behavior (Puthenpurakal, 2018).

5. Graded MCM Modules in Noncommutative and Geometric Settings

Graded maximal Cohen–Macaulay theory generalizes to noncommutative rings and singularities, with distinctive features:

• Over noncommutative, $\mathbb{N}$-graded, locally finite, Auslander Gorenstein, Cohen–Macaulay algebras of dimension two, MCM modules are classified up to isomorphism and shift by modules over noncommutative quasi-resolutions (NQRs) of ADE type, extending the classical McKay correspondence to a broader, noncommutative context. The underlying quiver (Gabriel quiver) is a pretzelization (twist) of the ADE diagrams (Qin et al., 2019).
• The existence of tilting objects in the stable category of graded MCM modules over noncommutative quotient singularities (e.g., $S^G$ for $S$ AS-regular Koszul and $G$ finite) leads to equivalence of the stable category with the derived category of a finite-dimensional algebra, providing deep links to the representation theory and singularity categories (Mori et al., 2015).

6. Matrix Factorization, Geometry, and Explicit Classification

For isolated hypersurface singularities, especially in low dimensions, the theory of graded MCM modules connects closely to:

• Matrix factorizations: Every indecomposable graded MCM module is realized as a matrix factorization of the defining polynomial (cf. Eisenbud’s theorem). This enables explicit classification, as in the case of singularities defined by elliptic curves, where the classification of graded MCM modules of rank one (and algorithms for higher rank) is constructed concretely via Orlov's equivalence between derived categories and matrix factorization categories (Galinat, 2013).
• Graded MCM modules over the cone of an elliptic curve: Explicit 2-by-2 matrix factorizations parameterized by rational points on the elliptic curve give a complete classification of rank one MCM modules.
• Combinatorial invariants: Recent work has developed polyhedral and combinatorial techniques to describe the graded local cohomology of MCM modules over semigroup rings, yielding characterizations of Cohen–Macaulayness using the homology of certain polyhedral cell complexes (“grains” and “chaff”) (Yu et al., 2022).

7. Connections to Hilbert Coefficients, Quasi-Pure Resolutions, and Depth Stratification

Recent advances clarify the influence of homological properties, such as the pattern of syzygies and resolutions, on the graded structure of MCM modules:

• If the associated graded module $G(M)$ of a Cohen–Macaulay module over a regular local ring has a quasi-pure resolution, then $G(M)$ is Cohen–Macaulay (Puthenpurakal et al., 2023). Quasi-pure resolution is defined by degree inequalities between shifts in the minimal graded resolution:

$y_i > a_{i-1} \text{ for all } i \geq 1.$

• Sharp numerical criteria: Lower bounds for Hilbert coefficients $e_0(M)$ and $e_1(M)$ are derived, and equality cases characterize the Cohen–Macaulayness of $G(M)$. For example, for $M$ with projective dimension $p$, if $G(A)$ is Gorenstein and Cohen–Macaulay,

$$e_0(M) \geq \mu(M) + c, \quad e_1(M) \geq \binom{c+1}{2},$$

where $c$ is the regularity, and equality for $e_1$ implies $G(M)$ is Cohen–Macaulay.

This stratification by depth of $G(M)$, governed by the number of generators, the presentation, and invariants such as multiplicity and reduction number, clarifies the landscape of graded MCM modules over hypersurface and complete intersection rings (Mishra et al., 2023, Mishra et al., 2022).

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Summary Table: Lower Bounds and Depth for $G(M)$ over Hypersurfaces ($e(A) = 3$) (Mishra et al., 2023, Mishra et al., 2022, Mishra et al., 2021)

$\mu(M)$	$e(A)$	Lower Bound $\operatorname{depth} G(M)$	Conditions
$2$	$3$	$d-1$	$\operatorname{red}(M)\le 2$
$3$	$3$	$d-2$	$\operatorname{red}(M)\le 2$
$4$	$3$	$d-3$	always (sharp)

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Graded maximal Cohen–Macaulay modules thus provide a framework not only for classifying high-symmetry objects in singularity theory, but also for analyzing their deformation, extension, and representation-theoretic properties. The interplay between module-theoretic invariants, combinatorial and polyhedral techniques, homological algebra, and geometric representation schemes forms the current landscape of research in this area.