In
mathematics, a CohenâMacaulay 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 CohenâMacaulay exactly when it is a finitely generated free module over a regular local subring. CohenâMacaulay rings play a central role in
commutative algebra: they form a very broad class, and yet they are well understood in many ways.
For a
commutativeNoetherianlocal ringR, a finite (i.e.
finitely generated) R-module is a Cohen-Macaulay module if (in general we have: , see
AuslanderâBuchsbaum 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 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 rings. 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 CohenâMacaulay module if is a Cohen-Macaulay module for all
maximal ideals. (This is a kind of
circular definition unless we define zero modules as Cohen-Macaulay. So we define zero modules as Cohen-Macaulay modules in this definition.) 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 (as an -module on itself).[1]
Examples
Noetherian rings of the following types are CohenâMacaulay.
Any determinantal ring. That is, let R be the quotient of a regular local ring S by the ideal I generated by the r Ă rminors of some p Ă qmatrix of elements of S. If the codimension (or
height) of I is equal to the "expected" codimension (pâr+1)(qâr+1), R is called a determinantal ring. In that case, R is CohenâMacaulay.[2] Similarly, coordinate rings of
determinantal varieties are Cohen-Macaulay.
Some more examples:
The ring Kx]/(xÂČ) has dimension 0 and hence is CohenâMacaulay, but it is not reduced and therefore not regular.
The subring Kt2, t3] of the polynomial ring Kt], or its localization or
completion at t=0, is a 1-dimensional domain which is Gorenstein, and hence CohenâMacaulay, but not regular. This ring can also be described as the coordinate ring of the
cuspidal cubic curve y2 = x3 over K.
The subring Kt3, t4, t5] of the polynomial ring Kt], or its localization or completion at t=0, is a 1-dimensional domain which is CohenâMacaulay but not Gorenstein.
Rational singularities over a field of characteristic zero are CohenâMacaulay.
Toric varieties over any field are CohenâMacaulay.[3] 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 CohenâMacaulay,[4] One successful analog of rational singularities in positive characteristic is the notion of F-rational singularities; again, such singularities are CohenâMacaulay.[5]
is CohenâMacaulay if and only if the
cohomology group Hi(X, Lj) is zero for all 1 †i †nâ1 and all integers j.[6] It follows, for example, that the affine cone Spec R over an
abelian varietyX is CohenâMacaulay when X has dimension 1, but not when X has dimension at least 2 (because H1(X, O) is not zero). See also
Generalized CohenâMacaulay ring.
CohenâMacaulay schemes
We say that a locally Noetherian
scheme is CohenâMacaulay if at each point the local ring is CohenâMacaulay.
CohenâMacaulay curves
CohenâMacaulay curves are a special case of CohenâMacaulay schemes, but are useful for compactifying moduli spaces of curves[7] where the boundary of the smooth locus is of CohenâMacaulay curves. There is a useful criterion for deciding whether or not curves are CohenâMacaulay. Schemes of dimension are CohenâMacaulay if and only if they have no embedded primes.[8] The singularities present in CohenâMacaulay curves can be classified completely by looking at the plane curve case.[9]
Non-examples
Using the criterion, there are easy examples of non-CohenâMacaulay 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
CohenâMacaulay schemes have a special relation with
intersection theory. Precisely, let X be a smooth variety[10] 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:[11]
.
In general, that multiplicity is given as a length essentially characterizes CohenâMacaulay 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 CohenâMacaulay of length two, hence the intersection multiplicity is two, as expected.
Miracle flatness or Hironaka's criterion
There is a remarkable characterization of CohenâMacaulay 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.[12] Then R is CohenâMacaulay 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.[13]
A geometric reformulation is as follows. Let X be a
connectedaffine 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 morphismf from X to affine space An over K. Then X is CohenâMacaulay if and only if all fibers of f have the same degree.[14] 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 CohenâMacaulay 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 CohenâMacaulay if and only if its completion is CohenâMacaulay.[15]
If R is a CohenâMacaulay ring, then the polynomial ring Rx] and the power series ring R[[x]] are CohenâMacaulay.[16][17]
For a
non-zero-divisoru in the maximal ideal of a Noetherian local ring R, R is CohenâMacaulay if and only if R/(u) is CohenâMacaulay.[18]
If R is a quotient of a CohenâMacaulay ring, then the locus { p â Spec R | Rp is CohenâMacaulay } is an open subset of Spec R.[20]
Let (R, m, k) be a Noetherian local ring of embedding codimension c, meaning that c = dimk(m/m2) â dim(R). In geometric terms, this holds for a local ring of a subscheme of codimension c in a regular scheme. For c=1, R is CohenâMacaulay if and only if it is a
hypersurface ring. There is also a structure theorem for CohenâMacaulay rings of codimension 2, the
HilbertâBurch theorem: they are all determinantal rings, defined by the r Ă r minors of an (r+1) Ă r matrix for some r.
For a Noetherian local ring (R, m), the following are equivalent:[21]
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 primeP 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 CohenâMacaulay if and only if the unmixedness theorem holds for it.[22]
The unmixed theorem applies in particular to the zero ideal (an ideal generated by zero elements) and thus it says a CohenâMacaulay ring is an
equidimensional ring; in fact, in the strong sense: there is no embedded component and each component has the same codimension.
If K is a field, then the ring R = Kx,y]/(x2,xy) (the coordinate ring of a line with an embedded point) is not CohenâMacaulay. This follows, for example, by
Miracle Flatness: R is finite over the polynomial ring A = Ky], with degree 1 over points of the affine line Spec A with y â 0, but with degree 2 over the point y = 0 (because the K-vector space Kx]/(x2) has dimension 2).
If K is a field, then the ring Kx,y,z]/(xy,xz) (the coordinate ring of the union of a line and a plane) is reduced, but not equidimensional, and hence not CohenâMacaulay. Taking the quotient by the non-zero-divisor xâz gives the previous example.
If K is a field, then the ring R = Kw,x,y,z]/(wy,wz,xy,xz) (the coordinate ring of the union of two planes meeting in a point) is reduced and equidimensional, but not CohenâMacaulay. To prove that, one can use
Hartshorne's connectedness theorem: if R is a CohenâMacaulay local ring of dimension at least 2, then Spec R minus its closed point is connected.[23]
One meaning of the CohenâMacaulay condition can be seen in
coherent duality theory. A variety or scheme X is CohenâMacaulay 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 CohenâMacaulay schemes retain some of the simplicity of what happens for regular schemes or smooth varieties.
^Matsumura (1989), Theorem 23.5.; NB: although the reference is somehow vague on whether a ring there is assumed to be local or not, the proof there does not need the ring to be local.