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.
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. All CohenâMacaulay 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 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]
Noetherian rings of the following types are CohenâMacaulay.
Some more examples:
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]
Let X be a projective variety of dimension n â„ 1 over a field, and let L be an ample line bundle on X. Then the section ring of L
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 variety X 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.
We say that a locally Noetherian scheme is CohenâMacaulay if at each point the local ring is CohenâMacaulay.
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]
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 .
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.
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.
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 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 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.
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 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.
See also: quasi-unmixed ring (a ring in which the unmixed theorem holds for integral closure of an ideal).
The Segre product of two Cohen-Macaulay rings need not be Cohen-Macaulay. [24]
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.
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.
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. All CohenâMacaulay 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 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]
Noetherian rings of the following types are CohenâMacaulay.
Some more examples:
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]
Let X be a projective variety of dimension n â„ 1 over a field, and let L be an ample line bundle on X. Then the section ring of L
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 variety X 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.
We say that a locally Noetherian scheme is CohenâMacaulay if at each point the local ring is CohenâMacaulay.
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]
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 .
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.
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.
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 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 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.
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 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.
See also: quasi-unmixed ring (a ring in which the unmixed theorem holds for integral closure of an ideal).
The Segre product of two Cohen-Macaulay rings need not be Cohen-Macaulay. [24]
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.