In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Cohen–Macaulay ring is a particular type of commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

, possessing some of the algebraic-geometric

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 properties of a nonsingular

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

 variety, such as local equidimensionality

Equidimensionality

In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p is constant. The Euclidean space is an example of...

.

They are named for Francis Sowerby Macaulay

Francis Sowerby Macaulay

Francis Sowerby Macaulay FRS was an English mathematician who made significant contributions to algebraic geometry. He is most famous for his 1916 book, The Algebraic Theory of Modular Systems, which greatly influenced the later course of algebraic geometry...

, who proved the unmixedness theorem for polynomial rings in Macaulay (1916), and for Irvin S. Cohen, who proved the unmixedness theorem for formal power series rings in Cohen (1946). (All Cohen–Macaulay rings have the unmixedness property.)

Formal definition

A local Cohen–Macaulay ring is defined as a commutative noetherian

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

 local ring

Local ring

In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...

 with Krull dimension

Krull dimension

In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....

 equal to its depth. The depth is always bounded above by the Krull dimension; equality provides some interesting regularity conditions on the ring, enabling some powerful theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s to be proven in this rather general setting.

A non-local ring is called Cohen–Macaulay if all of its localization

Localization of a ring

In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...

s at prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

s are Cohen–Macaulay.

Examples

1. Every regular local ring
   Regular local ring
   In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
   
    is Cohen–Macaulay.
2. A field
   Field (mathematics)
   In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
   
    is a particular example of a regular local ring, so it is Cohen–Macaulay.
3. A local ring is Cohen–Macaulay if and only if its completion
   Completion (ring theory)
   In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
   
    is Cohen–Macaulay.
4. A ring R is Cohen–Macaulay if and only if the polynomial ring
   Polynomial ring
   In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
   
    R[x] is Cohen–Macaulay.
5. If K is a field, then the formal power series ring in one variable Kx is a regular local ring and so is Cohen–Macaulay, but is not a field.
6. Any Gorenstein ring
   Gorenstein ring
   In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition....
   
    is Cohen–Macaulay. In particular, complete intersection rings are Cohen–Macaulay.
7. Rational singularities
   Rational singularity
   In mathematics, more particularly in the field of algebraic geometry, a scheme X has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational mapf \colon Y \rightarrow X...
   
    are Cohen–Macaulay but not necessarily Gorenstein.
8. Any Artinian ring
   Artinian ring
   In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
   
    is Cohen–Macaulay.
9. Following the last idea, if K is a field and X is an indeterminate, the ring K[x]/(x²) is a local Artinian ring and so is Cohen–Macaulay, but it is not regular.
10. If K is a field, then the formal power series ring Kt², t³, where t is an indeterminate, is an example of a 1-dimensional local ring which is not regular but is Gorenstein, so is Cohen–Macaulay.
11. If K is a field, then the formal power series ring Kt³, t⁴, t⁵, where t is an indeterminate, is an example of a 1-dimensional local ring which is not Gorenstein but is Cohen–Macaulay.
12. More generally, any 1-dimensional Noetherian integral domain is Cohen–Macaulay.

Counterexamples

1. If K is a field, then the formal power series ring (the completion of the local ring at the double point of a line with an embedded double point) is not Cohen–Macaulay, because it has depth zero but dimension 1.
2. If K is a field, then the ring (the completion of the local ring at the intersection of a plane and a line) is not Cohen–Macaulay (it is not even equidimensional
   Equidimensionality
   In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p is constant. The Euclidean space is an example of...
   
   ); quotienting by gives the previous example.
3. If K is a field, then the ring (the completion of the local ring at the intersection of two planes meeting in a point) is not Cohen–Macaulay; quotienting by gives the previous example.

Consequences of the condition

One meaning of the Cohen–Macaulay condition is seen in coherent duality

Coherent duality

In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory....

 theory. Here the condition corresponds to case when the dualizing object, which a priori lies in a derived category

Derived category

In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...

, is represented by a single module (coherent sheaf

Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...

). The finer Gorenstein condition is then expressed by this module being projective (an invertible sheaf

Invertible sheaf

In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...

). Non-singularity (regularity) is still stronger— it corresponds to the notion of smoothness of a geometric object at a particular point. Thus, in a geometric sense, the notions of Gorenstein and Cohen–Macaulay capture increasingly larger sets of points than the smooth ones, points which are not necessarily smooth but behave in many ways like smooth points.

The unmixedness theorem

An ideal I of a Noetherian ring A is called unmixed if ht(I)= ht(P) for any associated prime P of A/I. The unmixedness theorem is said to hold for the ring A if every ideal I generated by ht(I) elements is unmixed. A Noetherian ring is Cohen–Macaulay if and only if the unmixedness theorem holds for it.