# Cohen–Macaulay ring
A **Cohen–Macaulay ring** (also called a Macaulay ring) is a Noetherian commutative ring in which depth equals [Krull dimension](https://www.edgepedia.org/krull-dimension) locally, or equivalently, a Noetherian commutative unit ring in which every proper ideal of height n contains a regular sequence of length n, that is, a sequence x₁, ..., xₙ such that for each i the residue class of xᵢ in R/(x₁, ..., x_{i−1}) is a non-zero divisor[1](https://mathworld.wolfram.com/Cohen-MacaulayRing.html). The condition is a standard measure of how well behaved a ring is, and it underlies the corresponding notion for schemes, where a scheme is Cohen–Macaulay when its local rings are[2](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf).
| Key fact | Detail |
|---|---|
| Local definition | A Noetherian local ring R is Cohen–Macaulay when its depth equals its Krull dimension; equivalently the maximal ideal contains a regular sequence of length dim(R)[3](https://www.mathematik.uni-muenchen.de/~morel/Teaching/ExSheetAlg/commalg.pdf) • [4](https://math.stackexchange.com/questions/4116019/local-cohen-macaulay-rings) |
| Global definition | A Noetherian ring is Cohen–Macaulay if all of its local rings are Cohen–Macaulay[5](https://stacks.math.columbia.edu/download/book.pdf) |
| Homological test | For a local ring A with maximal ideal m and residue field k, A is Cohen–Macaulay exactly when Extⁱ_A(k, A), or the local cohomology H^i_m(A), vanishes for all i < dim A[6](https://encyclopediaofmath.org/wiki/Cohen%E2%80%93Macaulay_ring) |
| Hierarchy | The class of Cohen–Macaulay rings contains the Gorenstein rings, which include all regular local rings[1](https://mathworld.wolfram.com/Cohen-MacaulayRing.html) |
| Stability | Localizations of a Cohen–Macaulay local ring at every prime are again Cohen–Macaulay[7](https://stacks.math.columbia.edu/tag/00NB) |
| Origin | The property was first proved by F. S. Macaulay in 1916 for every ideal in a polynomial ring over the complex field, and studied by I. S. Cohen in 1946 for complete local rings[1](https://mathworld.wolfram.com/Cohen-MacaulayRing.html) |
## Depth and the local definition
For a nonzero finitely generated module M over a local ring (A, m), the depth of M is the supremum of the lengths of M-regular sequences, sequences f₁, ..., f_r in which each fᵢ is a nonzerodivisor on M/(f₁, ..., f_{i−1})M[8](https://www.cambridge.org/core/books/abs/multiplicities-and-chern-classes-in-local-algebra/homological-properties-of-rings-and-modules/1A20604097CA2D055D4B37056C657525). Depth never exceeds dimension, and a Noetherian local ring R is defined to be Cohen–Macaulay when the two are equal[3](https://www.mathematik.uni-muenchen.de/~morel/Teaching/ExSheetAlg/commalg.pdf). Concretely, one can iteratively pick nonzerodivisors from the maximal ideal until the dimension drops to zero, obtaining a regular sequence of length dim(R)[3](https://www.mathematik.uni-muenchen.de/~morel/Teaching/ExSheetAlg/commalg.pdf).
The definition behaves well only in the Noetherian local setting: the Stacks project records examples showing that depth misbehaves for non-Noetherian rings even when local, and for Noetherian rings that are not local[9](https://stacks.math.columbia.edu/tag/00LE). For a nonlocal [Noetherian ring](https://www.edgepedia.org/noetherian-ring), the convention is that the ring is Cohen–Macaulay when its localization at every maximal ideal is[10](https://sites.math.duke.edu/~ezra/358/regSeq.pdf).
The property descends well along quotients by regular elements: if x is a nonzerodivisor on a finite module M over a Noetherian local ring, then M is Cohen–Macaulay if and only if M/xM is, since the quotient removes exactly one from both depth and dimension[11](https://stacks.math.columbia.edu/tag/0C6G).
## Equivalent characterizations
Several formulations are available. In homological terms, a local Cohen–Macaulay ring A is characterized by the vanishing of the groups Extⁱ_A(k, A), or equivalently the local cohomology groups H^i_m(A), for all i below dim A, where k is the residue field[6](https://encyclopediaofmath.org/wiki/Cohen%E2%80%93Macaulay_ring). Another equivalent local definition asks for a regular sequence a₁, ..., a_k whose quotient A/(a₁, ..., a_k) is Artinian[6](https://encyclopediaofmath.org/wiki/Cohen%E2%80%93Macaulay_ring).
A fundamental result is the unmixedness theorem: in a Cohen–Macaulay ring, ideals of a given height generated by the right number of elements have no embedded components. A common geometric use of unmixedness is to verify that a given set of polynomials generates the homogeneous coordinate ring of a projective variety[12](https://www.math.ens.psl.eu/~benoist/refs/Eisenbud.pdf).
## Where the property comes from
The notion was inspired by polynomial rings. If x₁, ..., xₙ are indeterminates over a field K, the maximal ideal (x₁, ..., xₙ) satisfies the regular-sequence condition, and Macaulay proved in 1916 that every ideal in a polynomial ring over the complex field has the property[1](https://mathworld.wolfram.com/Cohen-MacaulayRing.html). Cohen extended the picture to formal power series rings[13](https://pbelmans.ncag.info/notes/depth.pdf), and his 1946 paper on the structure and ideal theory of complete local rings (Transactions of the American Mathematical Society 59, pp. 54–106) is the second name on the concept[1](https://mathworld.wolfram.com/Cohen-MacaulayRing.html). These properties predate homological algebra by several decades, though homological reformulations came later[8](https://www.cambridge.org/core/books/abs/multiplicities-and-chern-classes-in-local-algebra/homological-properties-of-rings-and-modules/1A20604097CA2D055D4B37056C657525).
**Classes of rings that are Cohen–Macaulay.** Regular local rings are Cohen–Macaulay, as are regular embeddings in smooth varieties[2](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf). More generally, any local complete intersection is a Cohen–Macaulay ring: writing S = k[x₁, ..., xₙ]/(f₁, ..., f_c) with dim S = n − c, the Stacks project shows that f₁, ..., f_c is a regular sequence in the regular local ring k[x₁, ..., xₙ]_{m′}, so S_m is Cohen–Macaulay[14](https://stacks.math.columbia.edu/tag/00S8). Gorenstein rings are Cohen–Macaulay as well[13](https://pbelmans.ncag.info/notes/depth.pdf), giving the chain regular local ⇒ Gorenstein ⇒ Cohen–Macaulay[1](https://mathworld.wolfram.com/Cohen-MacaulayRing.html).
**A standard non-example.** The ring A = k[X, Y, Z]/(ZX, ZY) of global functions on the union of a plane and a transverse line in affine 3-space is not Cohen–Macaulay; its local ring at the intersection point has depth 1: it has a nonzerodivisor in the maximal ideal, but no regular sequence of length 2[15](https://math.stackexchange.com/a/751843) • [21](https://math.uchicago.edu/~amathew/chhomologicallocal.pdf). This reflects the geometric meaning: the plane and line are minimal components of different dimensions at their intersection, and this failure of equidimensionality is what makes the ring non-Cohen–Macaulay[13](https://pbelmans.ncag.info/notes/depth.pdf).
## Geometric meaning and applications
When the depth of a local ring equals its dimension, the ring is Cohen–Macaulay, and this is an important way in which schemes can be nice[2](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf). Cohen–Macaulayness is key to the proof of [Serre duality](https://www.edgepedia.org/serre-duality) through the Miracle Flatness theorem, and it enters Serre's R₁ + S₂ criterion for normality[2](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf). On a regular scheme, the condition has a strong consequence: a coherent sheaf is Cohen–Macaulay with full support exactly when it is finite locally free of positive rank[16](https://stacks.math.columbia.edu/tag/0340).
The property also constrains how schemes can degenerate. The reduced structure of a Cohen–Macaulay scheme may itself fail to be Cohen–Macaulay, though recent work shows this cannot happen under liftable local cohomology hypotheses for equidimensional rings that are homomorphic images of Gorenstein rings[17](https://arxiv.org/abs/2607.08995).
**Invariant theory.** Whether the Cohen–Macaulay property passes from a ring R to a subring of invariants R^G under a finite group G is a classical question. The property is inherited whenever the trace map tr_G: R → R^G, r ↦ Σ_{g∈G} g(r), is surjective; otherwise it usually does not transfer, even for linear actions on polynomial algebras, and the linear case remains without a final solution[18](https://arxiv.org/abs/math/0012201). For Galois actions in the sense of Auslander and Goldman, R^G is Cohen–Macaulay if and only if R is[18](https://arxiv.org/abs/math/0012201).
**Combinatorics.** Assigning to a simplicial complex Δ its face ring K[Δ] = S/I_Δ transfers the algebraic property to combinatorics. By Reisner's criterion, Δ is Cohen–Macaulay over a field k if and only if, for every face A ∈ Δ, the reduced homology of the link of A vanishes in degrees up to dim(link_Δ(A)) − 1[19](https://ndl.ethernet.edu.et/bitstream/123456789/24541/1/Peter%20Orlik.pdf). This criterion is the most widely used test for Cohen–Macaulayness of simplicial complexes[20](https://doi.org/10.1007/s10801-020-00943-6).
## References
1. [Cohen-Macaulay Ring, Wolfram MathWorld](https://mathworld.wolfram.com/Cohen-MacaulayRing.html)
2. [Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry (2024 prepublication)](https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf)
3. [Commalg course notes, Lina Weitzenböck, LMU Munich](https://www.mathematik.uni-muenchen.de/~morel/Teaching/ExSheetAlg/commalg.pdf)
4. [Local Cohen-Macaulay rings, Mathematics Stack Exchange](https://math.stackexchange.com/questions/4116019/local-cohen-macaulay-rings)
5. [The Stacks Project, Chapter 10: Commutative Algebra (Definition 10.104.6)](https://stacks.math.columbia.edu/download/book.pdf)
6. [Cohen–Macaulay ring, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Cohen%E2%80%93Macaulay_ring)
7. [The Stacks Project, Lemma 10.104.5](https://stacks.math.columbia.edu/tag/00NB)
8. [Paul C. Roberts, Homological Properties of Rings and Modules, in Multiplicities and Chern Classes in Local Algebra, Cambridge University Press](https://www.cambridge.org/core/books/abs/multiplicities-and-chern-classes-in-local-algebra/homological-properties-of-rings-and-modules/1A20604097CA2D055D4B37056C657525)
9. [The Stacks Project, Section 10.72: Depth](https://stacks.math.columbia.edu/tag/00LE)
10. [Regular Sequences and Regular Rings, Duke University lecture notes](https://sites.math.duke.edu/~ezra/358/regSeq.pdf)
11. [The Stacks Project, Lemma 10.103.5](https://stacks.math.columbia.edu/tag/0C6G)
12. [David Eisenbud, Commut Algebra reference text](https://www.math.ens.psl.eu/~benoist/refs/Eisenbud.pdf)
13. [Pieter Belmans, Depth notes, Section 2: Properties](https://pbelmans.ncag.info/notes/depth.pdf)
14. [The Stacks Project, Section 10.135: Local complete intersections](https://stacks.math.columbia.edu/tag/00S8)
15. [Examples of Cohen-Macaulay rings, Mathematics Stack Exchange](https://math.stackexchange.com/a/751843)
16. [The Stacks Project, Section 30.11: Depth](https://stacks.math.columbia.edu/tag/0340)
17. [S_r properties of generalized Cohen–Macaulay modules and rings having liftable local cohomology via Serre depth, arXiv:2607.08995](https://arxiv.org/abs/2607.08995)
18. [On Cohen-Macaulay rings of invariants, arXiv:math/0012201](https://arxiv.org/abs/math/0012201)
19. [Peter Orlik, Cohen-Macaulay Simplicial Complexes](https://ndl.ethernet.edu.et/bitstream/123456789/24541/1/Peter%20Orlik.pdf)
20. [Cohen–Macaulayness of two classes of circulant graphs, Journal of Algebraic Combinatorics](https://doi.org/10.1007/s10801-020-00943-6)
21. [math.uchicago.edu](https://math.uchicago.edu/~amathew/chhomologicallocal.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Cohen–Macaulay and Gorenstein rings*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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