Cohen–Macaulay Ring¶
A commutative Noetherian ring whose localizations all have depth equal to Krull dimension.
Core Idea¶
A Cohen–Macaulay ring is a commutative Noetherian ring whose localization \(R_{\mathfrak p}\) at every prime ideal has depth equal to its Krull dimension. For a Noetherian local ring \((R,\mathfrak m)\), the test is \(\operatorname{depth}_{\mathfrak m}R=\dim R\): a regular sequence in the maximal ideal reaches full dimension. The local property persists under localization; a nonlocal ring requires the all-prime test.[^ref-b6c5168ce611]
Scope of Application¶
In local algebra, full depth means systems of parameters are regular sequences and dimension drops by their length. It includes regular local rings but also some singular rings. In combinatorics, Stanley's original theorem makes face rings of constructible simplicial complexes Cohen–Macaulay, including boundaries of simplicial convex polytopes; not every face ring qualifies. A Cohen–Macaulay scheme is related through its local rings but is a separately typed geometric object.[ref-b6c5168ce611][ref-20677f1002b0][ref-78dd93c01c37][ref-3a8890fbb0fa]
Clarity¶
Do not replace the definition by global unmixedness, absence of embedded components, or being finite free over a regular subring: such statements require additional scope or hypotheses. Also distinguish the depth invariant from the ring class, and regularity from full depth. One favorable localization does not establish the property of a general ring.[ref-b6c5168ce611][ref-a2a0aaa59d36]
Manages Complexity¶
The equality supplies one sharp test for a family of local parameter-sequence behaviors. Rather than checking every parameter system separately, a full-depth local ring has Stacks's dimension-drop criterion for regular sequences. The compression keeps singular but well-controlled rings in view, while the global definition preserves the obligation to cover all prime localizations.[^ref-b6c5168ce611]
Abstract Reasoning¶
The singular local ring \(k[[x,y]]/(xy)\) is Cohen–Macaulay of dimension one: \(k[[x,y]]\) is regular of dimension two and \(xy\) is a nonzerodivisor, so the quotient retains full depth. It is not regular because its maximal ideal needs two generators. In a different graded setting, the boundary of a square has face ring \(k[x_1,x_2,x_3,x_4]/(x_1x_3,x_2x_4)\); constructibility of the simplicial-polytope boundary and Stanley's theorem make that two-dimensional face ring Cohen–Macaulay.[ref-a2a0aaa59d36][ref-b6c5168ce611][ref-20677f1002b0][ref-78dd93c01c37]
Knowledge Transfer¶
The full-depth test transfers literally from local singular quotients to graded Stanley–Reisner rings, though their proofs differ. It extends to schemes by evaluating their local rings, not by treating “scheme” as an alias of “ring.” The proposed strict DAG parent is live Commutative Ring; Depth is a component invariant and Buchsbaum is a weaker nearby local class, not synonyms. A broader cross-domain balance metaphor remains only a future-prime question.[ref-3a8890fbb0fa][ref-b6c5168ce611]
[^ref-b6c5168ce611]: The Stacks Project, §10.104 “Cohen-Macaulay rings”, Definitions 10.104.1 and 10.104.6 and Lemmas 10.104.2 and 10.104.5. [^ref-20677f1002b0]: The Stacks Project, Lemma 10.106.3, regular local implies Cohen–Macaulay. [^ref-a2a0aaa59d36]: The Stacks Project, §10.160 “The Cohen structure theorem”, Lemma 10.160.2 and Remark 10.160.9; singular-node calculations are elementary deductions. [^ref-78dd93c01c37]: Richard P. Stanley, “Cohen-Macaulay Rings and Constructible Polytopes”, Bulletin of the American Mathematical Society 81 (1975), pp.133–135. [^ref-3a8890fbb0fa]: The Stacks Project, §28.8 “Cohen-Macaulay schemes”, Definition 28.8.1 and Lemma 28.8.2.
Relationships to Other Abstractions¶
Current abstraction Cohen–Macaulay Ring Domain-specific
Parents (1) — more general patterns this builds on
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Cohen–Macaulay Ring is a kind of Commutative ring Domain-specific
A Cohen–Macaulay ring is a commutative ring constrained by Noetherian and full-local-depth conditions.
Hierarchy paths (2) — routes to 2 parentless roots
- Cohen–Macaulay Ring → Commutative ring → Commutativity → Invariance
- Cohen–Macaulay Ring → Commutative ring → Commutativity → Symmetry
Neighborhood in Abstraction Space¶
Cohen–Macaulay Ring sits in a sparse region of the domain-specific corpus (63rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Ring & Module Structure Theory (14 abstractions)
Nearest neighbors
- Buchsbaum ring — 0.85
- Depth (ring theory) — 0.85
- Ringed Space — 0.85
- Regular scheme — 0.84
- Multiplicatively closed set — 0.84
Computed from structural-signature embeddings · 2026-10-08