Non-Noetherian Cohen–Macaulay Rings

• Non-Noetherian Cohen–Macaulay rings are commutative rings that exhibit regularity properties analogous to classical Cohen–Macaulay rings without requiring the Noetherian condition.
• They encompass various generalizations, including n‑subperfect, extended Cohen–Macaulay, and Hamilton–Marley approaches, each emphasizing different aspects of regular sequences and homological invariants.
• These rings show stability under operations such as invariant subring formation, flat direct limits, and polynomial extensions, offering deeper insights into non-Noetherian algebraic structures.

A non-Noetherian Cohen–Macaulay ring is a commutative ring exhibiting regularity properties analogous to those of Cohen–Macaulay rings, yet without requiring the Noetherian condition. Multiple generalizations of the Cohen–Macaulay property have been studied in the non-Noetherian context, reflecting both local and global homological features, grade/height correspondences, parameter sequence regularity, and various module-theoretic or combinatorial invariants. Contemporary developments have produced several frameworks, such as $n$-subperfect rings, extended Cohen–Macaulay rings, Cohen–Macaulayness “in the sense of ideals,” the Hamilton–Marley approach via parameter sequences, and variants relying on flat direct limits and Bourbaki unmixedness.

1. Background: Classical and Non-Noetherian Notions

Classically, a Noetherian local ring $(R, \mathfrak{m})$ is Cohen–Macaulay (CM) if the depth and Krull dimension coincide, i.e., $\operatorname{depth} R = \operatorname{dim} R$. Equivalent formulations rely on the existence of maximal regular sequences or on the regularity of all systems of parameters.

In the non-Noetherian setting, the equivalence of depth/dimension, maximal regular sequences, and regularity of parameter sequences splits, leading to a proliferation of possible generalizations. Central among these are:

• Cohen–Macaulay in the sense of ideals: $R$ is called Cohen–Macaulay (ideals) if for every finitely generated ideal $I \subseteq R$, the maximal length of an $R$-regular sequence contained in $I$ equals the height of $I$ (Chlopecki et al., 6 Jan 2026, Asgharzadeh et al., 2008).
• Hamilton–Marley definition: $R$ is Cohen–Macaulay (Hamilton–Marley) if every strong parameter sequence (weakly proregular, cohomologically nonvanishing) is a regular sequence (Kim et al., 2018, Asgharzadeh et al., 2008).
• $n$-subperfect and extended CM rings: These are constructed by imposing Bass–perfectness conditions on total quotient rings of regular sequence quotients, or by requiring subperfect behavior at localizations with Noetherian spectrum (Fuchs et al., 2017).
• Weak Bourbaki unmixed: Requires all minimal primes over a finitely generated ideal to have equal height and for an appropriate unmixed decomposition to hold (Chlopecki et al., 6 Jan 2026, Asgharzadeh et al., 2008).

2. $n$-Subperfect and Extended Cohen–Macaulay Rings

$n$-Subperfect Rings

Following (Fuchs et al., 2017), a commutative ring $R$ is $n$-subperfect (for $n \ge 0$) if:

• Every maximal regular sequence in $R$ has length $n$,
• For each regular sequence $(x_1, \dots, x_i)$, the total ring of quotients $Q(R/(x_1,\dots,x_i))$ is perfect (i.e., every flat module is projective; Bass).

This construction generalizes almost perfect rings ($n=1$) and perfect rings ($n=0$). The definition is recursive: $R$ is $n$-subperfect if and only if $R$ is subperfect and for each non-zero-divisor $x$, $R/xR$ is $(n-1)$-subperfect.

Key properties of $n$-subperfect rings include:

• $\operatorname{dim} R = n$,
• Catenarity (all maximal prime chains have length $n$),
• For any ideal $I \subset R$, $$\operatorname{ht}(I) = \max\{\text{length of regular sequences in }I\} = \min\{t : \operatorname{Ext}^t_R(R/I, R) \neq 0\}$$.

Extended Cohen–Macaulay Rings

A ring $R$ is extended Cohen–Macaulay if:

• $\operatorname{Spec} R$ is a Noetherian topological space,
• For each maximal ideal $M$, the localization $R_M$ is $\operatorname{ht}(M)$-subperfect.

Noetherian Cohen–Macaulay rings are precisely the extended CM rings in this framework. These rings satisfy localization, invariance under finite group actions (order invertible), and direct summand descent (subrings integrally closed and direct S-module summand inherit $n$-subperfectness).

Non-Noetherian examples include $R[X_1,\dots,X_n]$ for perfect $R$, and $R \ltimes N$ where $R$ is local CM and $N$ is a big Cohen–Macaulay module (Fuchs et al., 2017).

3. Homological and Combinatorial Generalizations

Hamilton–Marley Approach

The Hamilton–Marley theory develops a definition wherein a ring $R$ is CM if every strong parameter sequence is regular (Kim et al., 2018, Asgharzadeh et al., 2008). A strong parameter sequence is weakly proregular, not a unit-ideal, and has cohomological nonvanishing at all primes above the relevant ideal.

The approach relies on the Čech complex $\check{C}^x(M)$ and the correspondence between parameter sequences and cohomological vanishing:

• $R$ is Hamilton–Marley CM if, for every strong parameter sequence $x$, the polynomial grade equals the sequence length, i.e., $\operatorname{p-grade}(x_1,\dots,x_e) = e$.

This definition preserves desirable properties such as local-to-global behavior and stability under polynomial rings for valuation and Prüfer domains of finite dimension (Kim et al., 2018).

Flat Direct Limits and Stanley–Reisner Constructions

Direct limits of Noetherian CM rings can be CM in the sense of ideals, provided the system is flat/pure and satisfies grade monotonicity (Asgharzadeh et al., 2013). Infinite Stanley–Reisner rings, such as those arising from initial complexes of Schubert varieties associated to infinite symmetric groups $S_\infty$, provide non-Noetherian but CM examples in the sense of ideals and weak Bourbaki unmixedness (Chlopecki et al., 6 Jan 2026). The construction relies on properties of flat direct limits and combinatorial shellability.

4. Comparisons, Examples, and Counterexamples

The various non-Noetherian CM properties are related by a strict ladder of implications (Asgharzadeh et al., 2008, Kim et al., 2018):

CM(Maximal) ⇒ CM(Spec) ⇒ CM(Ideals) ⇒ CM(Glaz) ⇒ CM(f.g. ideals) ⇒ Hamilton–Marley ⇒ Weak Bourbaki unmixed

These implications are strict; none reverses in general. Examples exhibit the distinction:

• The ring $R[x_1,x_2,\dots]$ over a Noetherian CM $R$ may be CM in all senses.
• Certain valuation domains of dimension $>1$ are CM in the Hamilton–Marley sense but not in the (finitely generated) ideals sense.
• Non-Noetherian normal semigroup rings constructed as $k + Q \subset S$ show the failure of Hochster’s theorem: they may be normal but not CM in any sense currently studied (Kim et al., 2018).

A summary table of these relationships appears below.

Definition (Abbreviation)	Main Criterion	Exemplary Ring Types
$n$-Subperfect	Regular seq. quotients perfect	Polynomial rings over perfect rings, idealizations
Extended Cohen–Macaulay	Local $n$-subperfect + Noeth. Spec	$R[X_1,\dots,X_n]$ for perfect $R$
CM (Ideals)	$\operatorname{p-grade}(I) = \operatorname{ht}(I)$ ∀ $I$	Infinite Stanley–Reisner rings (Chlopecki et al., 6 Jan 2026)
Hamilton–Marley	Strong parameter seq’s regular	Valuation domains, certain semigroup rings
Weak Bourbaki unmixed	All min primes over $I$ same height	Flat limits, invariant subrings

5. Invariant Theory, Direct Limits, and Stability Properties

Non-Noetherian Cohen–Macaulayness exhibits meaningful behavior under several operations:

• Invariant subrings: If $R$ is $n$-subperfect (resp. CM in the sense of ideals or weak Bourbaki unmixed) and $G$ is a finite group of automorphisms with $|G|$ invertible in $R$, then the invariant subring $R^G$ retains the corresponding property (Fuchs et al., 2017, Asgharzadeh et al., 2008).
• Flat direct limits: Flat (equivalently, pure) direct systems of Noetherian CM rings yield direct limits that are CM in the sense of ideals and weak Bourbaki unmixed (Chlopecki et al., 6 Jan 2026, Asgharzadeh et al., 2013).
• Polynomial and Veronese subrings: $R[X_1,\dots,X_n]$ is $n$-subperfect for perfect $R$; more generally, so is any Veronese subring. This recovers the behavior of many infinite Stanley–Reisner rings and face rings (Fuchs et al., 2017, Chlopecki et al., 6 Jan 2026).

However, closure under non-flat direct limits does not hold; counterexamples include valuation domains of infinite dimension and certain non-affine semigroup rings (Asgharzadeh et al., 2013).

6. Open Problems and Future Directions

Several major questions remain open in the theory of non-Noetherian Cohen–Macaulay rings:

• Characterization of which homological invariants generalize as in the Noetherian CM context (e.g., vanishing of local cohomology, spectral sequence degeneration) (Fuchs et al., 2017).
• Classification of extended CM rings in terms of underlying algebraic or geometric data, e.g., for rings of functions on infinite unions of varieties.
• Understanding stability under additional operations (completions, Rees algebras, amalgamations) without recourse to Noetherianity (Fuchs et al., 2017, Kim et al., 2018).
• Developing a unifying definition that:
  • Specializes to the classical CM property in the Noetherian case,
  • Encompasses all coherent regular rings,
  • Is preserved under invariant subring constructions,
  • Recovers desirable theorems such as Hochster’s for semigroup rings (Asgharzadeh et al., 2008, Kim et al., 2018).
• Further study of Gorensteinness, sequential Cohen–Macaulayness, and combinatorial invariants in the infinite and non-Noetherian context (Chlopecki et al., 6 Jan 2026).

These problems drive research at the confluence of commutative algebra, infinite combinatorics, and algebraic geometry, revealing deep structure in the “de-Noetherized” landscape of Cohen–Macaulay theory.