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 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 Cohen–Macaulay rings have the unmixedness property.

For Noetherian local rings, there is the following chain of inclusions.

Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings

Definition

For a commutative Noetherian local ring R, the depth of R (the maximum length of a regular sequence in the maximal ideal of R) is at most the Krull dimension of R. The ring R is called Cohen–Macaulay if its depth is equal to its dimension.

More generally, a commutative ring is called Cohen–Macaulay if is Noetherian and all of its localizations at prime ideals are Cohen–Macaulay. In geometric terms, a scheme is called Cohen–Macaulay if it is locally Noetherian and its local ring at every point is Cohen–Macaulay.

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 Z, or a polynomial ring K[x₁,...,x_n] over a field K, or a power series ring K[[x₁,...,x_n]]. In geometric terms, every regular scheme, for example a smooth variety over a field, is Cohen–Macaulay.
• Any 0-dimensional ring (or equivalently, any Artinian 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 R^G when R is a Cohen–Macaulay algebra over a field of characteristic zero and G is a finite group (or more generally, a linear algebraic group whose identity component is reductive). 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 (or height) of I is equal to the "expected" codimension (p−r+1)(q−r+1), R is called a determinantal ring. In that case, R is Cohen−Macaulay.^[1]

Some more examples:

1. The ring K[x]/(x²) has dimension 0 and hence is Cohen–Macaulay, but it is not reduced and therefore not regular.
2. The subring K[t², t³] of the polynomial ring K[t], 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.
3. The subring K[t³, t⁴, t⁵] of the polynomial ring K[t], 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.^[2] 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 Cohen–Macaulay,^[3] One successful analog of rational singularities in positive characteristic is the notion of F-rational singularities; again, such singularities are Cohen–Macaulay.^[4]

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 homogeneous coordinate ring

$R=\oplus_{j\geq 0}H^0(X,L^j)$

is Cohen–Macaulay if and only if the cohomology group Hⁱ(X, L^j) is zero for all 1 ≤ i ≤ n−1 and all integers j.^[5] 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 (because H¹(X, O) is not zero).

Miracle Flatness

There is a remarkable characterization of Cohen–Macaulay rings, sometimes called Miracle Flatness. 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.^[6] 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.^[7]

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 Cohen–Macaulay if and only all fibers of f have the same degree.^[8] 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,

$R=K\oplus R_1 \oplus R_2 \oplus \cdots.$

There is always a graded polynomial subring A ⊂ R (with generators in various degrees) 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 local ring is Cohen–Macaulay if and only if its completion is Cohen–Macaulay.^[9]
• If a ring R is Cohen–Macaulay, then the polynomial ring R[x] and the power series ring R[[x]] are Cohen–Macaulay.^[10]
• 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/(u) is Cohen–Macaulay.^[11]
• The quotient of a Cohen–Macaulay ring by any ideal is universally catenary.^[12]
• Let (R, m, k) be a Noetherian local ring of embedding codimension c, meaning that c = dim_k(m/m²) − 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 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+1) × r matrix for some r.

The unmixedness theorem

An ideal I of a Noetherian ring A is called unmixed if the codimension (or height) of I is equal to the codimension of every associated prime P of A/I. (This is stronger than saying that A/I is equidimensional.) The unmixedness theorem is said to hold for the ring A if every ideal I generated by a number of elements equal to its codimension is unmixed. A Noetherian ring is Cohen–Macaulay if and only if the unmixedness theorem holds for it.^[13]

Counterexamples

1. If K is a field, then the ring R = K[x,y]/(x²,xy) (the coordinate ring of a line with an embedded point) is not Cohen–Macaulay. This follows, for example, by Miracle Flatness: R is finite over the polynomial ring A = K[y], with degree 1 over points of the affine line Spec A with y ≠ 0, but with degree 2 over the point y = 0 (because the K-vector space K[x]/(x²) has dimension 2).
2. If K is a field, then the ring K[x,y,z]/(xy,xz) (the coordinate ring of the union of a line and a plane) is reduced, but not equidimensional, and hence not Cohen–Macaulay. Taking the quotient by the non-zero-divisor x−z gives the previous example.
3. If K is a field, then the ring R = K[w,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 Cohen–Macaulay. To prove that, one can use Hartshorne's connectedness theorem: if R is a Cohen–Macaulay local ring of dimension at least 2, then Spec R minus its closed point is connected.^[14]

Grothendieck duality

One meaning of the Cohen–Macaulay condition is 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. In particular, every regular scheme is Gorenstein. Thus the statements of duality theorems such as Serre duality or Grothendieck local duality for Gorenstein or Cohen–Macaulay schemes retain some of the simplicity of what happens for regular schemes or smooth varieties.

Notes

1. Eisenbud (1995), Theorem 18.18.
2. Fulton (1993), p. 89.
3. Kollár & Mori (1998), Theorems 5.20 and 5.22.
4. Schwede & Tucker (2012), Appendix C.1.
5. Kollár (2013), (3.4).
6. Bruns & Herzog, Theorem A.22.
7. Eisenbud (1995), Corollary 18.17.
8. Eisenbud (1995), Exercise 18.17.
9. Matsumura (1989), Theorem 17.5.
10. Matsumura (1989), Theorem 23.5.
11. Matsumura (1989), Theorem 17.3.(ii).
12. Matsumura (1989), Theorem 17.9.
13. Matsumura (1989), Theorem 17.6.
14. Eisenbud (1995), Theorem 18.12.

References

• Bruns, Winfried; Herzog, Jürgen (1993), Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956
• Cohen, I. S. (1946), "On the structure and ideal theory of complete local rings", Transactions of the American Mathematical Society 59: 54–106, doi:10.2307/1990313, ISSN 0002-9947, MR 0016094 Cohen's paper was written when "local ring" meant what is now called a "Noetherian local ring".
• V.I. Danilov (2001), "Cohen–Macaulay ring", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Eisenbud, David (1995), Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960
• Fulton, William (1993), Introduction to Toric Varieties, Princeton University Press, ISBN 978-0-691-00049-7, MR 1234037
• Kollár, János; Mori, Shigefumi (1998), Birational Geometry of Algebraic Varieties, Cambridge University Press, ISBN 0-521-63277-3, MR 1658959
• Kollár, János (2013), Singularities of the Minimal Model Program, Cambridge University Press, ISBN 978-1-107-03534-8, MR 3057950
• Macaulay, F.S. (1994) [1916], The Algebraic Theory of Modular Systems, Cambridge University Press, ISBN 1-4297-0441-1, MR 1281612
• Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 0879273
• Schwede, Karl; Tucker, Kevin (2012), "A survey of test ideals", Progress in Commutative Algebra 2, Berlin: Walter de Gruyter, pp. 39–99, arXiv:1104.2000, MR 2932591