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In [[mathematics]], a '''Cohen–Macaulay ring''' is a [[commutative ring]] with some of the [[algebraic geometry|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 {{harvs|txt|authorlink=Francis Sowerby Macaulay|first=Francis Sowerby|last=Macaulay|year=1916}}, who proved the [[unmixedness theorem]] for polynomial rings, and for {{harvs|txt|authorlink=Irvin S. Cohen|last=Cohen|first=Irvin|year=1946}}, who proved the unmixedness theorem for [[formal power series ring]]s. All Cohen–Macaulay rings have the unmixedness property.

For Noetherian local rings, there is the following chain of inclusions.
{{Commutative local ring classes}}

==Definition==
For a [[commutative]] [[noetherian ring|Noetherian]] [[local ring]] ''R'', a finite (i.e. [[Finitely generated module|finitely generated]]) ''R''-module <math>M\neq 0</math> is a ''Cohen-Macaulay module'' if <math>\mathrm{depth}(M) = \mathrm{dim}(M)</math> (in general we have: <math>\mathrm{depth}(M) \leq \mathrm{dim}(M)</math>, see [[Auslander–Buchsbaum formula]] for the relation between [[Depth (ring theory)|depth]] and [[Krull dimension|dim]] of a certain kind of modules). On the other hand, <math>R</math> is a module on itself, so we call <math>R</math> a ''Cohen-Macaulay ring'' if it is a Cohen-Macaulay module as an <math>R</math>-module. A ''maximal'' Cohen-Macaulay module is a Cohen-Macaulay module ''M'' such that <math>\mathrm{dim}(M)=\mathrm{dim}(R)</math>.

The above definition was for a Noetherian local rings. But we can expand the definition for a more general Noetherian ring: If <math>R</math> is a commutative Noetherian ring, then an ''R''-module ''M'' is called '''Cohen–Macaulay module''' if <math>M_\mathrm{m}</math> is a Cohen-Macaulay module for all [[Maximal ideal|maximal ideals]] <math>\mathrm{m}\in \mathrm{Supp}(M)</math>. (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 <math>M_\mathrm{m}</math> to be such an <math>R_\mathrm{m}</math>-module for each maximal ideal <math>\mathrm{m}</math> of ''R''. As in the local case, ''R'' is a ''Cohen-Macaulay ring'' if it is a Cohen-Macaulay module (as an <math>R</math>-module on itself).<ref>Bruns & Herzog, from def. 2.1.1</ref>

==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 <math>\Z</math>, or a [[polynomial ring]] <math>K[x_1,\ldots ,x_n]</math> over a [[field (mathematics)|field]] ''K'', or a [[power series ring]] <math>K[[x_1,\ldots ,x_n]]</math> .  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 [[integral domain|domain]].
* Any 2-dimensional [[normal ring]].
* Any [[Gorenstein ring]].  In particular, any [[complete intersection ring]].
* The [[ring of invariants]] <math>R^G</math> when ''R'' is a Cohen–Macaulay algebra over a field of [[characteristic (algebra)|characteristic]] zero and ''G'' is a finite group (or more generally, a [[linear algebraic group]] whose identity component is [[reductive group|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'' [[minor (linear algebra)|minors]] of some ''p'' × ''q'' [[matrix (mathematics)|matrix]] of elements of ''S''. If the codimension (or [[height of an ideal|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.<ref>Eisenbud (1995), Theorem 18.18.</ref> Similarly, coordinate rings of [[determinantal variety|determinantal varieties]] are Cohen-Macaulay.

Some more examples:
# The ring ''K''[''x'']/(''x''²) has dimension 0 and hence is Cohen–Macaulay, but it is not reduced and therefore not regular.
# The subring ''K''[''t''<sup>2</sup>, ''t''<sup>3</sup>] of the polynomial ring ''K''[''t''], or its localization or [[completion (ring theory)|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 [[cusp (singularity)|cuspidal]] cubic curve ''y''<sup>2</sup> = ''x''<sup>3</sup> over ''K''.
# The subring ''K''[''t''<sup>3</sup>, ''t''<sup>4</sup>, ''t''<sup>5</sup>] 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 singularity|Rational singularities]] over a field of characteristic zero are Cohen–Macaulay. [[toric variety|Toric varieties]] over any field are Cohen–Macaulay.<ref>Fulton (1993), p. 89.</ref> The [[minimal model program]] makes prominent use of varieties with [[canonical singularity#Pairs|klt]] (Kawamata log terminal) singularities; in characteristic zero, these are rational singularities and hence are Cohen–Macaulay,<ref>Kollár & Mori (1998), Theorems 5.20 and 5.22.</ref> One successful analog of rational singularities in positive characteristic is the notion of '''F-rational singularities'''; again, such singularities are Cohen–Macaulay.<ref>Schwede & Tucker (2012), Appendix C.1.</ref>

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''
:<math>R=\bigoplus_{j\geq 0}H^0(X,L^j)</math>
is Cohen–Macaulay if and only if the [[coherent sheaf cohomology|cohomology]] group ''H''<sup>''i''</sup>(''X'', ''L''<sup>''j''</sup>) is zero for all 1 ≤ ''i'' ≤ ''n''−1 and all integers ''j''.<ref>Kollár (2013), (3.4).</ref> 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 ''H''<sup>1</sup>(''X'', ''O'') is not zero). See also [[Generalized Cohen–Macaulay ring]].

==Cohen–Macaulay schemes==
We say that a locally Noetherian [[scheme (mathematics)|scheme]] <math>X</math> is Cohen–Macaulay if at each point <math>x\in X</math> the local ring <math>\mathcal{O}_{X,x}</math> is Cohen–Macaulay.

===Cohen–Macaulay curves===
Cohen–Macaulay curves are a special case of Cohen–Macaulay schemes, but are useful for compactifying moduli spaces of curves<ref>{{Cite web|url=http://www.diva-portal.org/smash/get/diva2:13507/FULLTEXT01.pdf|title=Compactifying Locally Cohen–Macaulay Projective Curves|last=Honsen|first=Morten|url-status=live|archive-url=https://web.archive.org/web/20200305183519/http://www.diva-portal.org/smash/get/diva2:13507/FULLTEXT01.pdf|archive-date=5 Mar 2020}}</ref> where the boundary of the smooth locus <math>\mathcal{M}_g</math> is of Cohen–Macaulay curves. There is a useful criterion for deciding whether or not curves are Cohen–Macaulay. Schemes of dimension <math>\leq 1</math> are Cohen–Macaulay if and only if they have no embedded primes.<ref>{{Cite web|url=https://stacks.math.columbia.edu/tag/0BXG|title=Lemma 31.4.4 (0BXG)—The Stacks project|website=stacks.math.columbia.edu|access-date=2020-03-05}}</ref> The singularities present in Cohen–Macaulay curves can be classified completely by looking at the plane curve case.<ref>{{Cite journal|last=Wiegand|first=Roger|date=December 1991|title=Curve singularities of finite Cohen–Macaulay type|url=https://projecteuclid.org/euclid.afm/1485898045|journal=[[Arkiv för Matematik]]|language=EN|volume=29|issue=1–2|pages=339–357|doi=10.1007/BF02384346|bibcode=1991ArM....29..339W|issn=0004-2080|doi-access=free}}</ref>

==== Non-examples ====
Using the criterion, there are easy examples of non-Cohen–Macaulay curves from constructing curves with embedded points. For example, the scheme
:<math>X = \text{Spec}\left( \frac{\Complex[x,y]}{(x^2,xy)} \right)</math>
has the decomposition into prime ideals <math>(x)\cdot(x,y)</math>. Geometrically it is the <math>y</math>-axis with an embedded point at the origin, which can be thought of as a ''fat point''. Given a smooth projective plane curve <math>C \subset \mathbb{P}^2</math>, a curve with an embedded point can be constructed using the same technique: find the ideal <math>I_x</math> of a point in <math>x \in C</math> and multiply it with the ideal <math>I_C</math> of <math>C</math>. Then
:<math>X = \text{Proj}\left( \frac{\Complex[x,y,z]}{I_C \cdot I_x} \right)</math>
is a curve with an embedded point at <math>x</math>.

===Intersection theory===
Cohen–Macaulay schemes have a special relation with [[intersection theory]]. Precisely, let ''X'' be a smooth variety<ref>smoothness here is somehow extraneous and is used in part to make sense of a proper component.</ref> and ''V'', ''W'' closed subschemes of pure dimension. Let ''Z'' be a [[proper component]] of the scheme-theoretic intersection <math>V \times_X W</math>, that is, an irreducible component of expected dimension. If the local ring ''A'' of <math>V \times_X W</math> 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'':<ref>{{harvnb|Fulton|1998|loc=Proposition 8.2. (b)}}</ref>

:<math>i(Z, V \cdot W, X) = \operatorname{length}(A)</math>.

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.

===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
:<math> \frac{\Complex[x,y]}{(y - x^2)} \otimes_{\Complex[x,y]}\frac{\Complex[x,y]}{(y)} \cong \frac{\Complex[x]}{(x^2)} </math>
which is Cohen–Macaulay of length two, hence the intersection multiplicity is two, as expected.<!--If <math>Y_1,Y_2</math> are two Cohen–Macaulay subschemes in <math>X</math>, and their intersection has expected [[Krull dimension|dimension]], then the scheme structure of <math>Y_1\times_XY_2</math> contains the [[intersection number|intersection multiplicity]]. In general, the multiplicity is contained in the data of the "derived intersection", as explained by the [[intersection number#Serre's_Tor_formula|Serre's Tor formula]]. For example, if we take the intersection of a [[parabola]] with a line tangent to it, we find that
:<math> \frac{\mathbb{C}[x,y]}{(y - x^2)} \otimes_{\mathbb{C}[x,y]}\frac{\mathbb{C}[x,y]}{(y)} \cong \frac{\mathbb{C}[x]}{(x^2)} </math>
hence the intersection multiplicity is two. If we perturb our equation <math>(y) \to (y -a)</math> such that the varieties still intersect in <math>\mathbb{A}^2</math>, then the intersection will be two points, each with multiplicity one, so total multiplicity two.
 Cite 3264 and all that... -->

==Miracle flatness or Hironaka's criterion==
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 module|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.<ref>Bruns & Herzog, Theorem A.22.</ref> Then ''R'' is Cohen–Macaulay if and only if it is [[flat module|flat]] as an ''A''-module; it is also equivalent to say that ''R'' is [[free module|free]] as an ''A''-module.<ref>Eisenbud (1995), Corollary 18.17.</ref>

A geometric reformulation is as follows. Let ''X'' be a [[connected space|connected]] [[affine scheme]] of [[morphism of finite type|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 Cohen–Macaulay if and only if all fibers of ''f'' have the same degree.<ref>Eisenbud (1995), Exercise 18.17.</ref> 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'',
:<math>R=K\oplus R_1 \oplus R_2 \oplus \cdots.</math>
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 Noetherian local ring is Cohen–Macaulay if and only if its completion is Cohen–Macaulay.<ref>Matsumura (1989), Theorem 17.5.</ref>
*If ''R'' is a Cohen–Macaulay ring, then the polynomial ring ''R''[''x''] and the power series ring ''R''<nowiki>[[</nowiki>''x''<nowiki>]]</nowiki> are Cohen–Macaulay.<ref>Matsumura (1989), Theorem 17.7.</ref><ref>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.</ref>
*For a [[regular sequence|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.<ref>Matsumura (1989), Theorem 17.3.(ii).</ref>
*The quotient of a Cohen–Macaulay ring by any [[ideal (ring theory)|ideal]] is [[universally catenary ring|universally catenary]].<ref>Matsumura (1989), Theorem 17.9.</ref>
*If ''R'' is a quotient of a Cohen–Macaulay ring, then the locus { ''p'' ∈ Spec ''R'' | ''R''<sub>''p''</sub> is Cohen–Macaulay } is an open subset of Spec ''R''.<ref>Matsumura (1989), Exercise 24.2.</ref>
*Let (''R'', ''m'', ''k'') be a Noetherian local ring of embedding codimension ''c'', meaning that ''c'' = dim<sub>''k''</sub>(''m''/''m''<sup>2</sup>) − 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 [[complete intersection ring|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''.
*For a Noetherian local ring (''R'', ''m''), the following are equivalent:<ref>Matsumura (1989), Theorem 17.11.</ref>
*#''R'' is Cohen–Macaulay.
*#For every [[parameter ideal]] ''Q'' (an ideal generated by a [[system of parameters]]),
*#:<math>\operatorname{length}(R/Q) = e(Q)</math> := the [[Hilbert–Samuel multiplicity]] of ''Q''.
*#For some parameter ideal ''Q'', <math>\operatorname{length}(R/Q) = e(Q)</math>.
:(See [[Generalized Cohen–Macaulay ring]] as well as [[Buchsbaum ring]] for rings that generalize this characterization.)

==The unmixedness theorem{{anchor|unmixed}}==
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 [[equidimensionality|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 Cohen–Macaulay if and only if the unmixedness theorem holds for it.<ref>Matsumura (1989), Theorem 17.6.</ref>

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.

See also: [[quasi-unmixed ring]] (a ring in which the unmixed theorem holds for [[integral closure of an ideal]]).

==Counterexamples==
# If ''K'' is a field, then the ring ''R'' = ''K''[''x'',''y'']/(''x''<sup>2</sup>,''xy'') (the coordinate ring of a line with an embedded point) is not Cohen–Macaulay. This follows, for example, by [[Miracle flatness theorem|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'']/(''x''<sup>2</sup>) has dimension 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.
# 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 [[Robin Hartshorne|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.<ref>Eisenbud (1995), Theorem 18.12.</ref>

The [[Segre product]] of two [[Cohen-Macaulay ring]]s need not be Cohen-Macaulay.<ref>{{citation
 | last = Chow | first = Wei Liang | author-link = Wei-Liang Chow
 | doi = 10.2307/2373158
 | journal = [[American Journal of Mathematics]]
 | jstor = 2373158
 | mr = 171804
 | pages = 799–822
 | title = On unmixedness theorem
 | volume = 86
 | year = 1964}}</ref>

==Grothendieck duality==
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 [[coherent sheaf|sheaves]] on ''X'', is represented by a single sheaf. The stronger property of being [[Gorenstein ring|Gorenstein]] means that this sheaf is a [[line bundle]]. In particular, every [[regular local ring|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==
{{reflist|2}}

==References==
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*{{Citation | author1-last=Cohen | author1-first=I. S. | author1-link=Irvin Cohen | title=On the structure and ideal theory of complete local rings | jstor= 1990313 |mr=0016094 | year=1946 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=59 | issue=1 | pages=54–106 | doi=10.2307/1990313| doi-access=free }} Cohen's paper was written when "local ring" meant what is now called a "Noetherian local ring". 
*{{springer|id=c/c022970|author=V.I. Danilov|title=Cohen–Macaulay ring}}
* {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative Algebra with a View toward Algebraic Geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-94268-1 | mr=1322960 | year=1995 | volume=150 | doi=10.1007/978-1-4612-5350-1}}
* {{Citation | author1-last=Fulton | author1-first=William | author1-link=William Fulton (mathematician) | title=Introduction to Toric Varieties | publisher=[[Princeton University Press]] | isbn=978-0-691-00049-7 | year=1993 | mr=1234037 | doi=10.1515/9781400882526}}
* {{Citation | author1-last=Fulton | author1-first=William | author1-link=William Fulton (mathematician) | title=Intersection theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]]. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd }} 
*{{Citation | author1-first=János | author1-last=Kollár | author1-link=János Kollár | author2-first=Shigefumi | author2-last=Mori | author2-link=Shigefumi Mori| title=Birational Geometry of Algebraic Varieties | publisher=[[Cambridge University Press]] | year=1998 | isbn=0-521-63277-3 | mr=1658959 | doi=10.1017/CBO9780511662560}}
*{{Citation | author1-first=János | author1-last=Kollár | author1-link=János Kollár | title=Singularities of the Minimal Model Program | publisher=[[Cambridge University Press]] | year=2013 | isbn=978-1-107-03534-8 | mr=3057950 | doi=10.1017/CBO9781139547895}}
*{{citation| author1-last=Macaulay| author1-first=F.S. | author1-link=Francis Sowerby Macaulay | title=The Algebraic Theory of Modular Systems|publisher=[[Cambridge University Press]] | year=1916 | mr=1281612 | doi=10.3792/chmm/1263317740|url=https://archive.org/details/algebraictheoryo00macauoft|isbn=1-4297-0441-1}}
*{{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative Ring Theory | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36764-6 | year=1989 | mr=0879273}}
*{{Citation | author1-first=Karl | author1-last=Schwede | author2-first=Kevin | author2-last=Tucker | chapter=A survey of test ideals | title=Progress in Commutative Algebra 2 | pages=39–99 | publisher=Walter de Gruyter | location=Berlin | year=2012 | mr=2932591 | arxiv=1104.2000| bibcode=2011arXiv1104.2000S }}

== External links ==
*[https://math.stackexchange.com/q/1432721 Examples of Cohen-Macaulay integral domains]
*[https://math.stackexchange.com/q/751670 Examples of Cohen-Macaulay rings]

== See also ==
* [[Ring theory]]
* [[Local ring]]s
* [[Gorenstein local ring]]s
* [[Wiles's proof of Fermat's Last Theorem]]

{{DEFAULTSORT:Cohen-Macaulay ring}}
[[Category:Algebraic geometry]]
[[Category:Commutative algebra]]