Cohen–Macaulay ring

In mathematics,[] a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.
They are named for, who proved the unmixedness theorem for polynomial rings, and for, who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the unmixedness property.
For Noetherian local rings, there is the following chain of inclusions.

Definition

For a commutative Noetherian local ring R, a finite R-module is a Cohen–Macaulay module if . On the other hand, is a module on itself, so we call a Cohen–Macaulay ring if it is a Cohen–Macaulay module as an -module. A maximal Cohen–Macaulay module is a Cohen–Macaulay module M such that.
The above definition was for a Noetherian local ring. But we can expand the definition for a more general Noetherian ring: If is a commutative Noetherian ring, then an R-module M is called Cohen–Macaulay module if is a Cohen–Macaulay module for all maximal ideals. Now, to define maximal Cohen–Macaulay modules for these rings, we require that to be such an -module for each maximal ideal of R. As in the local case, R is a Cohen–Macaulay ring if it is a Cohen–Macaulay module.

Examples

Noetherian rings of the following types are Cohen–Macaulay.

Any regular local ring. This leads to various examples of Cohen–Macaulay rings, such as the integers, or a polynomial ring over a field K, or a power series ring . In geometric terms, every regular scheme, for example a smooth variety over a field, is Cohen–Macaulay.
Any 0-dimensional ring.
Any 1-dimensional reduced ring, for example any 1-dimensional domain.
Any 2-dimensional normal ring.
Any Gorenstein ring. In particular, any complete intersection ring.
The ring of invariants when R is a Cohen–Macaulay algebra over a field of characteristic zero and G is a finite group. This is the Hochster–Roberts theorem.
Any determinantal ring. That is, let R be the quotient of a regular local ring S by the ideal I generated by the r × r minors of some p × q matrix of elements of S. If the codimension of I is equal to the "expected" codimension, R is called a determinantal ring. In that case, R is Cohen−Macaulay. Similarly, coordinate rings of determinantal varieties are Cohen–Macaulay.

Some more examples:

The ring K/ has dimension 0 and hence is Cohen–Macaulay, but it is not reduced and therefore not regular.
The subring K of the polynomial ring K, or its localization or completion at t=0, is a 1-dimensional domain which is Gorenstein, and hence Cohen–Macaulay, but not regular. This ring can also be described as the coordinate ring of the cuspidal cubic curve y² = x³ over K.
The subring K of the polynomial ring K, or its localization or completion at t=0, is a 1-dimensional domain which is Cohen–Macaulay but not Gorenstein.

Rational singularities over a field of characteristic zero are Cohen–Macaulay. Toric varieties over any field are Cohen–Macaulay. The minimal model program makes prominent use of varieties with klt singularities; in characteristic zero, these are rational singularities and hence are Cohen–Macaulay, One successful analog of rational singularities in positive characteristic is the notion of F-rational singularities; again, such singularities are Cohen–Macaulay.
Let X be a projective variety of dimension n ≥ 1 over a field, and let L be an ample line bundle on X. Then the section ring of L
is Cohen–Macaulay if and only if the cohomology group Hⁱ is zero for all 1 ≤ i ≤ n−1 and all integers j. It follows, for example, that the affine cone Spec R over an abelian variety X is Cohen–Macaulay when X has dimension 1, but not when X has dimension at least 2. See also Generalized Cohen–Macaulay ring.

Cohen–Macaulay schemes

We say that a locally Noetherian scheme is Cohen–Macaulay if at each point the local ring is Cohen–Macaulay.

Cohen–Macaulay curves

Cohen–Macaulay curves are a special case of Cohen–Macaulay schemes, but are useful for compactifying moduli spaces of curves where the boundary of the smooth locus is of Cohen–Macaulay curves. There is a useful criterion for deciding whether or not curves are Cohen–Macaulay. Schemes of dimension are Cohen–Macaulay if and only if they have no embedded primes. The singularities present in Cohen–Macaulay curves can be classified completely by looking at the plane curve case.

Non-examples

Using the criterion, there are easy examples of non-Cohen–Macaulay curves from constructing curves with embedded points. For example, the scheme
has the decomposition into prime ideals. Geometrically it is the -axis with an embedded point at the origin, which can be thought of as a fat point. Given a smooth projective plane curve, a curve with an embedded point can be constructed using the same technique: find the ideal of a point in and multiply it with the ideal of. Then
is a curve with an embedded point at.

Intersection theory

Cohen–Macaulay schemes have a special relation with intersection theory. Precisely, let X be a smooth variety and V, W closed subschemes of pure dimension. Let Z be a proper component of the scheme-theoretic intersection, that is, an irreducible component of expected dimension. If the local ring A of at the generic point of Z is Cohen–Macaulay, then the intersection multiplicity of V and W along Z is given as the length of A:
In general, that multiplicity is given as a length essentially characterizes Cohen–Macaulay ring; see #Properties. Multiplicity one criterion, on the other hand, roughly characterizes a regular local ring as a local ring of multiplicity one.

Example

For a simple example, if we take the intersection of a parabola with a line tangent to it, the local ring at the intersection point is isomorphic to
which is Cohen–Macaulay of length two, hence the intersection multiplicity is two, as expected.

Miracle flatness or Hironaka's criterion

There is a remarkable characterization of Cohen–Macaulay rings, sometimes called miracle flatness or Hironaka's criterion. Let R be a local ring which is finitely generated as a module over some regular local ring A contained in R. Such a subring exists for any localization R at a prime ideal of a finitely generated algebra over a field, by the Noether normalization lemma; it also exists when R is complete and contains a field, or when R is a complete domain. Then R is Cohen–Macaulay if and only if it is flat as an A-module; it is also equivalent to say that R is free as an A-module.
A geometric reformulation is as follows. Let X be a connected affine scheme of finite type over a field K. Let n be the dimension of X. By Noether normalization, there is a finite morphism f from X to affine space Aⁿ over K. Then X is Cohen–Macaulay if and only if all fibers of f have the same degree. It is striking that this property is independent of the choice of f.
Finally, there is a version of Miracle Flatness for graded rings. Let R be a finitely generated commutative graded algebra over a field K,
There is always a graded polynomial subring A ⊂ R such that R is finitely generated as an A-module. Then R is Cohen–Macaulay if and only if R is free as a graded A-module. Again, it follows that this freeness is independent of the choice of the polynomial subring A.

Properties

A Noetherian local ring is Cohen–Macaulay if and only if its completion is Cohen–Macaulay.
If R is a Cohen–Macaulay ring, then the polynomial ring R and the power series ring Rx are Cohen–Macaulay.
For a non-zero-divisor u in the maximal ideal of a Noetherian local ring R, R is Cohen–Macaulay if and only if R/ is Cohen–Macaulay.
The quotient of a Cohen–Macaulay ring by any ideal is universally catenary.
If R is a quotient of a Cohen–Macaulay ring, then the locus is an open subset of Spec R.
Let be a Noetherian local ring of embedding codimension c, meaning that c = dim_k − dim. In geometric terms, this holds for a local ring of a subscheme of codimension c in a regular scheme. For c=1, R is Cohen–Macaulay if and only if it is a hypersurface ring. There is also a structure theorem for Cohen–Macaulay rings of codimension 2, the Hilbert–Burch theorem: they are all determinantal rings, defined by the r × r minors of an × r matrix for some r.
For a Noetherian local ring, the following are equivalent:
#R is Cohen–Macaulay.
#For every parameter ideal Q,
#: := the Hilbert–Samuel multiplicity of Q.
#For some parameter ideal Q,.
The unmixedness theorem

An ideal I of a Noetherian ring A is called unmixed in height if the height of I is equal to the height of every associated prime P of A/''I.
The unmixedness theorem is said to hold for the ring A'' if every ideal I generated by a number of elements equal to its height is unmixed. A Noetherian ring is Cohen–Macaulay if and only if the unmixedness theorem holds for it.
The unmixed theorem applies in particular to the zero ideal and thus it says a Cohen–Macaulay ring is an equidimensional ring; in fact, in the strong sense: there is no embedded component and each component has the same codimension.
See also: quasi-unmixed ring.