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 Francis Sowerby Macaulay (1916), who proved the unmixedness theorem for polynomial rings, and for Irvin Cohen (1946), who proved the unmixedness theorem for formal power series rings.
For Noetherian local rings, there is the following chain of inclusions.
, see Auslander–Buchsbaum formula for the relation between depth and dim of a certain kind of modules).
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
(This is a kind of circular definition unless we define zero modules as Cohen-Macaulay.
Now, to define maximal Cohen-Macaulay modules for these rings, we require that
Some more examples: Rational singularities over a field of characteristic zero are Cohen–Macaulay.
[5] 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(X, Lj) is zero for all 1 ≤ i ≤ n−1 and all integers j.
Cohen–Macaulay curves are a special case of Cohen–Macaulay schemes, but are useful for compactifying moduli spaces of curves[7] where the boundary of the smooth locus
For example, the scheme has the decomposition into prime ideals
, a curve with an embedded point can be constructed using the same technique: find the ideal
Cohen–Macaulay schemes have a special relation with intersection theory.
Precisely, let X be a smooth variety[10] and V, W closed subschemes of pure dimension.
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 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.
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.
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.
[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.
Again, it follows that this freeness is independent of the choice of the polynomial subring A.
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.
[22] The unmixed theorem applies in particular to the zero ideal (an ideal generated by zero elements) 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.
[24] One meaning of the Cohen–Macaulay condition can be seen in coherent duality theory.
A variety or scheme X is Cohen–Macaulay 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.