Cohen–Macaulay Ring¶

A commutative Noetherian ring whose localizations all have depth equal to Krull dimension.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13067
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Commutative Algebra → Mathematics
Aliases: Cohen-Macaulay ring

Core Idea¶

A Cohen–Macaulay ring is a commutative Noetherian ring $R$ for which every localization $R_{\mathfrak p}$ at a prime ideal has depth equal to Krull dimension. For a Noetherian local ring $(R,\mathfrak m)$, this is the single test $\operatorname{depth}_{\mathfrak m}R=\dim R$: a regular sequence in $\mathfrak m$ reaches the full dimension. In a nonlocal ring, one favorable localization is not enough; the definition quantifies over all local rings. Localization of an already Cohen–Macaulay local ring retains the property.^[1]

This equality says that a local ring has a full-length succession of nonzerodivisors, not that it is nonsingular. In a Cohen–Macaulay local ring, a sequence in $\mathfrak m$ is regular exactly when the quotient dimension drops by its length; thus a full system of parameters is regular. Regular local rings are Cohen–Macaulay, but singular quotients can be Cohen–Macaulay too. The named class sits between all Noetherian commutative rings and the stricter regular-local condition, without reducing to a numerical coincidence divorced from regular sequences.^[1]^[2]

Structural Signature¶

Sig role-phrases: commutative Noetherian carrier → prime localization frame → depth via regular sequences → Krull dimension → local equality test.

Commutative Noetherian carrier. The ring's commutativity and Noetherian hypothesis type its prime spectrum, regular sequences and finite-dimensional local tests. An arbitrary non-Noetherian or noncommutative variant requires a separate definition, not an unannounced extension.^[1]
Prime localization frame. A local ring is checked at its maximal ideal; a general Noetherian ring is checked through every $R_{\mathfrak p}$. This all-primes quantifier prevents a good generic point from concealing a bad local depth defect.^[1]
Depth via regular sequences. At each local ring, depth is the maximal length of a sequence from its maximal ideal that remains a nonzerodivisor successively on earlier quotients. Counting arbitrary generators without the nonzerodivisor condition does not compute depth.^[1]
Krull dimension. The comparison value is the dimension of that same localization, not a dimension imported from another point or a global ambient variety.^[1]
Local equality test. Require $\operatorname{depth}R_{\mathfrak p}=\dim R_{\mathfrak p}$ at each prime. When it holds locally, every full system of parameters is regular by the Stacks dimension-drop criterion. Failing at one prime defeats the global identity.^[1]

What It Is Not¶

It is not synonymous with regularity. Stacks proves regular local rings are Cohen–Macaulay, but the converse fails: $k[[x,y]]/(xy)$ has dimension and depth one while its maximal ideal needs two generators. The intersecting-branches singularity therefore passes the full-depth test without being regular.^[1]^[2]^[3]

It is not merely unmixedness or absence of embedded primes. Those are related consequences under appropriately stated local or geometric hypotheses, not the constitutive equality itself; in particular, “every associated prime has the same global dimension” should not be asserted for arbitrary nonlocal Cohen–Macaulay rings without equidimensionality assumptions. Nor does the definition say that every Cohen–Macaulay ring is finite free over a regular subring: the Cohen structure theorem and finite regular-subring results in Stacks have complete-local and, in a cited lemma, domain hypotheses. Those cannot be promoted into an unrestricted equivalence.^[1]^[3]

It is not every Stanley–Reisner ring. The face ring of a constructible simplicial complex is Cohen–Macaulay by Stanley's theorem, but the face-ring construction itself does not force constructibility or full depth. It is also not a Cohen–Macaulay scheme as an exact alias: the scheme property tests local rings on a geometric carrier and is a related local-ring realization with its own scope.^[4]^[5]

Scope of Application¶

In local commutative algebra, the equality controls parameter sequences and dimension drop in singular as well as regular rings. If $(R,\mathfrak m)$ has dimension $d$ and is Cohen–Macaulay, a length-$c$ sequence in $\mathfrak m$ is regular exactly when the quotient has dimension $d-c$; full parameter systems have length $d$. These are theorem-backed uses, not an alternate definition by a convenient single parameter choice.^[1]

In combinatorial commutative algebra, a simplicial complex $\Delta$ is encoded by its Stanley–Reisner face ring $k[\Delta]$. Stanley's original 1975 paper shows that constructible complexes, including boundaries of simplicial convex polytopes, give Cohen–Macaulay face rings; that algebraic fact constrains possible face-count data. Here full depth belongs to the Ring, while constructibility is a sufficient property of the complex. In algebraic geometry, a locally Noetherian scheme is called Cohen–Macaulay when its local rings satisfy the ring test; the scheme is a related typed object, not the ring itself.^[4]^[5]

Clarity¶

Ask first whether the subject is a local ring, a general ring, a module, or a scheme. The local ring has one maximal-ideal depth/dimension calculation. A general Noetherian ring requires all-prime localization; a finite module has a related but separately typed depth criterion; a scheme inherits the property from its local rings. Reusing the same adjective across those carriers does not collapse them into synonyms.^[1]^[5]

Then compare the same local ring's depth and dimension. A regular sequence is a succession of nonzerodivisors on successive quotients, not simply an algebraically independent list or any collection of dimension-many equations. Stacks's result says full-depth local rings turn dimension-dropping parameter sequences into regular sequences. The implication “regular local ⇒ Cohen–Macaulay” is one-way.^[1]^[2]

Manages Complexity¶

The class compresses many local phenomena into a precise test: does each local ring admit a regular sequence as long as its dimension? Instead of treating every parameter system and every quotient separately, the Cohen–Macaulay condition licenses a uniform dimension-drop rule for sequences in the maximal ideal. It makes singularities more tractable without falsely declaring them regular.^[1]

The compression is not a free pass to global geometric claims. One still has to know which primes or affine charts are being covered; a calculation at one closed point may establish only a local conclusion. In combinatorics, Stanley's constructibility theorem supplies a broad sufficient route to full depth for face rings, but it is not a statement that all simplicial complexes have the property. The explicit local-to-global quantifier is what keeps the shortcut honest.^[4]^[5]

Abstract Reasoning¶

To test a local example, identify its maximal ideal, calculate dimension, then exhibit a regular sequence of that length or invoke a justified permanence theorem. For $A=k[[x,y]]$, Stacks states that $A$ is a regular complete local ring of dimension two. The nonzero element $xy$ is a nonzerodivisor because $A$ is a domain. Quotienting by this one-element regular sequence gives $R=A/(xy)$, Cohen–Macaulay of dimension one by the dimension-drop theorem. The element $x+y$ is outside both minimal primes $(x)$ and $(y)$ of $R$ and therefore is a nonzerodivisor; modding out by it yields an Artinian quotient, concretely witnessing depth one.^[3]^[2]^[1]

To test a nonlocal ring, do not assume the local computation alone answers the global question. Prove that every prime localization is Cohen–Macaulay, perhaps by a theorem that already yields the global ring class. Stanley's constructible-complex theorem does this for the face ring of a square's boundary. The inference is from a source-backed combinatorial condition to the algebraic full-depth property, not from a superficial visual resemblance between a polytope and a regular ring.^[4]^[1]

Knowledge Transfer¶

The same formal test transfers literally from local singular quotients to graded face rings: identify a commutative Noetherian carrier, localize, compare regular-sequence depth with Krull dimension, and require equality everywhere. What changes is the route of proof: a regular nonzerodivisor quotient for the singular node, versus constructibility of a simplicial complex and Stanley's face-ring theorem for the graded example.^[1]^[4]

The ring test can then be used in the geometric scheme setting through local rings, as Stacks explicitly formalizes. That is a transfer of criterion to a different typed carrier, not proof that ring and scheme are aliases. Outside commutative algebra, “capacity equals dimension” is at most an analogy; without regular sequences and prime localization it is not the Cohen–Macaulay condition.^[5]

Examples¶

Canonical: a singular local ring¶

Let $A=k[[x,y]]$ over a field $k$ and $R=A/(xy)$. The power-series ring is Noetherian, complete, regular local and two-dimensional. Because $xy$ is a nonzero element of the domain $A$, it is regular; Stacks's quotient theorem therefore makes $R$ Cohen–Macaulay of dimension one. Its maximal ideal $\mathfrak m=(x,y)/(xy)$ has two independent classes in $\mathfrak m/\mathfrak m^2$, so its embedding dimension is two, not its Krull dimension one: $R$ is not regular. This is the algebraic node of two branches, and it directly refutes “Cohen–Macaulay means smooth.”^[3]^[2]^[1]

Mapped back: Commutative Noetherian carrier → complete local quotient $k[[x,y]]/(xy)$; prime localization frame → the ring is local at $\mathfrak m$ and its other prime localizations inherit the condition; depth via regular sequences → one nonzerodivisor parameter, such as $x+y$; Krull dimension → $2-1=1$; local equality test → depth and dimension both equal one at $\mathfrak m$, with remaining primes covered by localization stability.

Applied: a square's boundary face ring¶

Let $\Delta$ be the boundary cycle of a square with vertices $1,2,3,4$ in cyclic order. Its minimal nonfaces are the opposite pairs $\{1,3\}$ and $\{2,4\}$, so its Stanley–Reisner ring over a field is $$ k[\Delta]=k[x_1,x_2,x_3,x_4]/(x_1x_3,\;x_2x_4). $$ The square is a simplicial convex polygon: its boundary is a constructible one-dimensional simplicial complex. Stanley's 1975 result therefore makes its face ring Cohen–Macaulay; its Krull dimension is $\dim\Delta+1=2$. The nonface ideal and combinatorial route are unlike the local node's single-hypersurface proof. The property is conditional on this complex, not automatic for any face ring.^[4]

Mapped back: Commutative Noetherian carrier → graded square-face quotient over $k$; prime localization frame → global ring conclusion covers each prime localization; depth via regular sequences → full local parameter sequences exist by the Cohen–Macaulay theorem; Krull dimension → complex dimension one gives ring dimension two; local equality test → depth equals local dimension at each prime by Stanley's constructibility result.

Structural Tensions¶

T1: Regularity's stronger control versus Cohen–Macaulay's wider singular reach. Requiring regularity can support stronger smooth-local conclusions, but it excludes the nodal quotient even though that ring has full depth. Cohen–Macaulayness retains dimension-drop and parameter regularity in singular settings, at the cost of not guaranteeing regular local geometry. Diagnostic: Does a desired theorem actually need a regular local ring, or only the full-depth condition?^[1]^[2]

T2: Economical local check versus justified global claim. Verifying one favorable localization is cheaper and may suffice for a point-specific result; it cannot license “the whole ring is Cohen–Macaulay” if another prime has a depth defect. A global theorem such as Stanley's constructibility result carries the all-localizations burden, but requires its stronger premises. Diagnostic: Is the conclusion about one local ring, an open neighborhood, or every point of a ring or scheme?^[1]^[4]^[5]

Structural–Framed Character¶

Cohen–Macaulay Ring is near the structural end within commutative algebra, while its algebraic frame remains indispensable. Evaluative weight: “well behaved” is an assessment, not membership; a singular ring may satisfy the exact equality. Human-practice dependence: proof strategy and chosen coordinates are human choices, but regular-sequence depth and local dimension decide the mathematical claim. Institutional origin: the name and theorem tradition arose in algebra; neither one author's presentation nor a particular software test defines it. Vocabulary travel: the adjective travels to modules, complexes and schemes only with a declared typed criterion, not by free metaphor. Import versus recognition: recognize a new ring instance by all-localizations depth/dimension equality, not by importing the label because it is equidimensional or has a nice Hilbert series. Its character: a domain-specific local-algebra class with a robust formal test and multiple literal settings, but no license to leave the commutative Noetherian frame.^[1]^[4]^[5]

Structural Core vs. Domain Accent¶

Skeletal core. A property is checked locally by comparing a regularity-like capacity with an intrinsic size, then required across all local views. Whether this local-equality pattern is a genuinely portable future-prime question remains open; no current prime should be selected merely from shared words like depth or dimension.

Domain-bound mechanism. Here “capacity” means maximal regular-sequence length, “size” means Krull dimension, and local views are prime localizations of a commutative Noetherian ring. These are constitutive, not decorative. Stanley–Reisner combinatorics and singular local geometry supply different proof routes to the same exact ring property.^[1]^[4]

Why not a prime. Remove local rings, nonzerodivisor sequences and Krull dimension, and one has only a vague balance metaphor. The actual strict broader carrier is the proposed live domain-specific Commutative Ring node, not an unsupported cross-domain prime.

This entry is a kind of Commutative ring.

DAG parent: live domain-specific Commutative Ring. A Cohen–Macaulay ring has the full additive, multiplicative and commutativity structure, narrowed by Noetherianity and full local depth.

Live Depth (ring theory) measures one constituent invariant, not the genus of a ring satisfying an equality. Live Stanley–Reisner Ring supplies a conditional example class, not an upward parent: some face rings fail the depth test. Live Buchsbaum Ring admits weaker local parameter behavior and is not a synonym; a hierarchy relation would need exact scope and convention before adding an edge. Regular local rings are a subclass of Cohen–Macaulay local rings, so regularity is not an upward ring parent here.^[1]^[2]^[4]

Relationships to Other Abstractions¶

Current abstraction Cohen–Macaulay Ring Domain-specific

Parents (1) — more general patterns this builds on

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

Show alternative path (1)

Neighborhood in Abstraction Space¶

Cohen–Macaulay Ring sits in a sparse region of the domain-specific corpus (63^rd 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

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Regular local ring. Its maximal ideal has a minimal number of generators equal to dimension; regular implies Cohen–Macaulay, not conversely. Tell: Does a full-depth ring still have a singular embedding dimension?^[2]
Depth invariant. An integer (under typed local hypotheses), not the class of rings whose depth reaches dimension everywhere. Tell: Is the claim a measured value or a ring property?
Stanley–Reisner ring. Encodes a simplicial complex through nonfaces; only qualified complexes give Cohen–Macaulay face rings. Tell: Has the complex's necessary combinatorial/topological condition been proved?^[4]
Buchsbaum ring. A weaker universal weak-parameter condition in the local setting. Tell: Are full regular parameter sequences required, or only controlled defects?
Cohen–Macaulay module. A finite module can have full depth relative to support; it is not the ring $R$ viewed as a module over itself unless that carrier is specified.^[1]
Cohen–Macaulay scheme. A locally Noetherian geometric object whose local rings satisfy the test. Tell: Is the subject a ring or a scheme assembled from local charts?^[5]
Global equidimensionality or finite free regular subring. Such statements require extra conditions; neither is the unrestricted definition. Tell: Which locality, completeness, domain or equidimensional hypotheses have actually been established?^[3]

References¶

[1] The Stacks Project, §10.104 “Cohen-Macaulay rings”, Definitions 10.104.1 and 10.104.6 and Lemmas 10.104.2, 10.104.5 and 10.104.7. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w

[2] The Stacks Project, Lemma 10.106.3 “Regular local rings”, including regular local implies Cohen–Macaulay. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[3] The Stacks Project, §10.160 “The Cohen structure theorem”, Lemma 10.160.2, Theorem 10.160.8, Remark 10.160.9 and Lemma 10.160.11. The node example additionally uses elementary quotient and tangent-space calculations. registry ↩a ↩b ↩c ↩d ↩e

[4] Richard P. Stanley, “Cohen-Macaulay Rings and Constructible Polytopes”, Bulletin of the American Mathematical Society 81 (1975), pp.133–135; constructible polytope boundaries on p.133 and face-ring Cohen–Macaulay conclusion on p.134. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k

[5] The Stacks Project, §28.8 “Cohen-Macaulay schemes”, Definition 28.8.1 and Lemmas 28.8.2–3. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h