In mathematics, a CohenâÂÂMacaulay 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 CohenâÂÂMacaulay exactly when it is a finitely generated free module over a regular local subring. CohenâÂÂMacaulay 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 CohenâÂÂMacaulay rings have the unmixedness property.
For Noetherian local rings, there is the following chain of inclusions.
For a commutative Noetherian local ring R, a finite (i.e. finitely generated) R-module is a CohenâÂÂMacaulay module if (in general we have: , see AuslanderâÂÂBuchsbaum formula for the relation between depth and dim of a certain kind of modules). On the other hand, is a module on itself, so we call a CohenâÂÂMacaulay ring if it is a CohenâÂÂMacaulay module as an -module. A maximal CohenâÂÂMacaulay module is a CohenâÂÂMacaulay 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 CohenâÂÂMacaulay module if is a CohenâÂÂMacaulay module for all maximal ideals . (This is a kind of circular definition unless we define zero modules as CohenâÂÂMacaulay. So we define zero modules as CohenâÂÂMacaulay modules in this definition.) Now, to define maximal CohenâÂÂMacaulay 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 CohenâÂÂMacaulay ring if it is a CohenâÂÂMacaulay module (as an -module on itself).
Noetherian rings of the following types are CohenâÂÂMacaulay.
Some more examples:
Rational singularities over a field of characteristic zero are CohenâÂÂMacaulay. Toric varieties over any field are CohenâÂÂMacaulay. The minimal model program makes prominent use of varieties with klt (Kawamata log terminal) singularities; in characteristic zero, these are rational singularities and hence are CohenâÂÂMacaulay, One successful analog of rational singularities in positive characteristic is the notion of F-rational singularities; again, such singularities are CohenâÂÂMacaulay.
Let X be a projective variety of dimension n âÂÂ¥ 1 over a field, and let L be an ample line bundle on X. Then the section ring of L
is CohenâÂÂMacaulay if and only if the cohomology group H<sup>i</sup>(X, L<sup>j</sup>) is zero for all 1 ⤠i ⤠nâÂÂ1 and all integers j. It follows, for example, that the affine cone Spec R over an abelian variety X is CohenâÂÂMacaulay when X has dimension 1, but not when X has dimension at least 2 (because H<sup>1</sup>(X, O) is not zero). See also Generalized CohenâÂÂMacaulay ring.
We say that a locally Noetherian scheme is CohenâÂÂMacaulay if at each point the local ring is CohenâÂÂMacaulay.
CohenâÂÂMacaulay curves are a special case of CohenâÂÂMacaulay schemes, but are useful for compactifying moduli spaces of curves where the boundary of the smooth locus is of CohenâÂÂMacaulay curves. There is a useful criterion for deciding whether or not curves are CohenâÂÂMacaulay. Schemes of dimension are CohenâÂÂMacaulay if and only if they have no embedded primes. The singularities present in CohenâÂÂMacaulay curves can be classified completely by looking at the plane curve case.
Using the criterion, there are easy examples of non-CohenâÂÂMacaulay 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 .
CohenâÂÂMacaulay 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 CohenâÂÂMacaulay, 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 CohenâÂÂMacaulay ring; see #Properties. Multiplicity one criterion, on the other hand, roughly characterizes a regular local ring as a local ring of multiplicity one.
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 CohenâÂÂMacaulay of length two, hence the intersection multiplicity is two, as expected.
There is a remarkable characterization of CohenâÂÂMacaulay 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 CohenâÂÂMacaulay 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 (for example, an affine variety). Let n be the dimension of X. By Noether normalization, there is a finite morphism f from X to affine space A<sup>n</sup> over K. Then X is CohenâÂÂMacaulay 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 â R (with generators in various degrees) such that R is finitely generated as an A-module. Then R is CohenâÂÂMacaulay 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.
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. (This is stronger than saying that A/I is equidimensional; see below.)
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 CohenâÂÂMacaulay if and only if the unmixedness theorem holds for it.
The unmixed theorem applies in particular to the zero ideal (an ideal generated by zero elements) and thus it says a CohenâÂÂMacaulay 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 (a ring in which the unmixed theorem holds for integral closure of an ideal).
The Segre product of two CohenâÂÂMacaulay rings need not be CohenâÂÂMacaulay.
One meaning of the CohenâÂÂMacaulay condition can be seen in coherent duality theory. A variety or scheme X is CohenâÂÂMacaulay if the "dualizing complex", which a priori lies in the derived category of sheaves on X, is represented by a single sheaf. The stronger property of being Gorenstein means that this sheaf is a line bundle. In particular, every regular scheme is Gorenstein. Thus the statements of duality theorems such as Serre duality or Grothendieck local duality for Gorenstein or CohenâÂÂMacaulay schemes retain some of the simplicity of what happens for regular schemes or smooth varieties.