Definition
A Cohen–Macaulay ring is a commutative Noetherian ring $R$ (typically assumed to have unity) such that for every (or equivalently, for some) maximal ideal $\mathfrak{m}$ of $R$, the depth of the local ring $R_{\mathfrak{m}}$ equals its Krull dimension:
$$
\operatorname{depth} R_{\mathfrak{m}} = \dim R_{\mathfrak{m}}.
$$
The notion is often extended to finitely generated $R$-modules: a module $M$ is called Cohen–Macaulay if $\operatorname{depth} M = \dim M$.
Overview
Cohen–Macaulay rings occupy a central position in commutative algebra and algebraic geometry. They generalize regular local rings—where the equality of depth and dimension is automatic—while retaining many favorable homological properties. In algebraic geometry, the coordinate rings of varieties that are Cohen–Macaulay exhibit well‑behaved dimension theory, possess no embedded components, and allow the use of duality theorems analogous to those for smooth varieties. The class includes important families such as polynomial rings over fields, complete intersections, and many rings arising as quotients by homogeneous ideals generated by regular sequences.
Etymology / Origin
The terminology honors two mathematicians:
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Francis S. Macaulay (1862–1937), who developed early work on Hilbert functions and introduced what are now called Macaulay’s theorem and Macaulay’s inverse systems. His contributions laid groundwork for the concept of unmixedness and the depth of ideals.
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Irving S. Cohen (1914–2001), who, in the 1950s, formalized the notion of depth for modules over Noetherian rings and proved fundamental results relating depth, dimension, and regular sequences.
The combined term “Cohen–Macaulay” appeared in the literature during the 1960s, notably in works on local cohomology and the theory of regular sequences.
Characteristics
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Depth–Dimension Equality – By definition, the equality $\operatorname{depth}= \dim$ holds for all localizations at maximal ideals.
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Unmixedness – In a Cohen–Macaulay ring, all associated primes of an ideal have the same height; the ring has no embedded components.
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Serre’s Condition $S_{k}$ – A Cohen–Macaulay ring satisfies Serre’s condition $S_{k}$ for every $k\ge 0$; in particular it is $S_{2}$.
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Homological Regularity – Such rings have finite projective dimension for modules of finite length, and their local cohomology modules vanish below the Krull dimension.
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Behaviour under Standard Operations –
- Localization: Any localization of a Cohen–Macaulay ring at a prime ideal remains Cohen–Macaulay.
- Completion: Completion with respect to an ideal preserves the Cohen–Macaulay property.
- Polynomial Extension: If $R$ is Cohen–Macaulay, then the polynomial ring $R[x_{1},\dots ,x_{n}]$ is also Cohen–Macaulay.
- Passing to Quotients: Quotients by regular sequences of elements produce Cohen–Macaulay rings (e.g., complete intersections).
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Examples –
- Regular local rings and their quotients by regular sequences.
- Polynomial rings $k[x_{1},\dots ,x_{n}]$ over a field $k$.
- Coordinate rings of smooth projective varieties.
- Certain toric varieties and determinantal rings.
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Non‑examples – Rings where depth is strictly smaller than dimension, such as many singular hypersurfaces, fail to be Cohen–Macaulay.
Related Topics
- Noetherian Ring – The ambient class within which the Cohen–Macaulay property is defined.
- Depth – Length of a maximal regular sequence contained in the maximal ideal.
- Krull Dimension – The supremum of lengths of chains of prime ideals.
- Regular Local Ring – A Noetherian local ring with $\operatorname{depth} = \dim$ and regularity of the maximal ideal; all regular local rings are Cohen–Macaulay.
- Gorenstein Ring – A Cohen–Macaulay ring with a self‑dual canonical module; an important subclass.
- Complete Intersection – Quotients of regular rings by ideals generated by regular sequences; they are Cohen–Macaulay.
- Serre’s Conditions $R_{k}$ and $S_{k}$ – Homological criteria closely linked to the Cohen–Macaulay condition.
- Local Cohomology – A tool often used to detect the Cohen–Macaulay property via vanishing theorems.
- Hilbert Function and Hilbert Polynomial – Their regularity properties are well‑understood for Cohen–Macaulay graded algebras.