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...

In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring ( A , m ) {\displaystyle (A,{\mathfrak {m}})} of Krull dimension d...

Hebrides, in Scotland. Cohen–Macaulay ring, a commutative ring, named after Irvin Cohen and Francis Sowerby Macaulay (1862-1937). Macaulay computer algebra...

intersection rings, and a fortiori, regular local rings are Cohen–Macaulay, but not conversely. Cohen–Macaulay combine desirable properties of regular rings (such...

rings. For Noetherian local rings, there is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃...

has a finite injective resolution then R {\displaystyle R} is a Cohen–Macaulay ring. The Intersection Theorem. If M ⊗ R N ≠ 0 {\displaystyle M\otimes...

equidimensional. An affine algebraic variety whose coordinate ring is a Cohen–Macaulay ring is equidimensional.[clarification needed] Wirthmüller, Klaus...

An abstract simplicial complex Δ is called Cohen–Macaulay over k if its face ring is a Cohen–Macaulay ring. In his 1974 thesis, Gerald Reisner gave a...

. Depth is used to define classes of rings and modules with good properties, for example, Cohen-Macaulay rings and modules, for which equality holds...

Cohen–Macaulay ring. A regular local ring is an example of a Cohen–Macaulay ring. It is a theorem of Serre that R is a regular local ring if and only if it has finite...

algebra. Cohen–Macaulay rings, Macaulay duality, the Macaulay resultant and the Macaulay and Macaulay2 computer algebra systems are named for Macaulay. Macaulay...

inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings Suppose that A is a Noetherian...

David Buchsbaum. Every Cohen–Macaulay local ring is a Buchsbaum ring. Every Buchsbaum ring is a generalized Cohen–Macaulay ring. Buchsbaum, D. (1966),...

local ring is called a Cohen–Macaulay ring if its dimension is equal to its depth. A regular local ring is an example of such a ring. A Noetherian integral...

of a local ring R is equal to its Krull dimension: R is then said to be Cohen-Macaulay. The three examples shown are all Cohen-Macaulay rings. Similarly...

catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings A local complete intersection ring is a Noetherian...

rings. For Noetherian local rings, there is the following chain of inclusions: Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃...

with a dualizing module must be Cohen–Macaulay. Conversely if a Cohen–Macaulay ring is a quotient of a Gorenstein ring then it has a dualizing module....

a result, Cohen–Macaulay rings are named after him and Francis Sowerby Macaulay. Cohen and Abraham Seidenberg published their Cohen–Seidenberg theorems...

Flat map Flat map (ring theory) Projective module Injective module Cohen-Macaulay ring Gorenstein ring Complete intersection ring Koszul complex Hilbert's...

in 1913 by Francis Sowerby Macaulay in connection to what nowadays is called a Cohen-Macaulay ring, but for which Macaulay did not have a name for yet...

related to earlier work by Francis Sowerby Macaulay on polynomial rings and is sometimes called Macaulay duality, and the general case was introduced...

Noetherian ring is called a Cohen–Macaulay ring if the localizations at all maximal ideals are Cohen–Macaulay. We note that a Cohen–Macaulay ring is universally...

Krull dimension Regular local ring Regular sequence Cohen–Macaulay ring Gorenstein ring Koszul complex Spectrum of a ring Zariski topology Kähler differential...

integral domain and a local Cohen–Macaulay ring are equidimensional. See also equidimensional scheme and quasi-unmixed ring. Extension and contraction...

in place of a single dualizing sheaf became apparent. In fact the Cohen–Macaulay ring condition, a weakening of non-singularity, corresponds to the existence...

has no embedded prime for any non-zerodivisor f. In particular, a Cohen-Macaulay ring satisfies (ii). Geometrically, we have the following: if X is a local...

ring Quasi-Frobenius ring Hereditary ring, Semihereditary ring Local ring, Semi-local ring Discrete valuation ring Regular local ring Cohen–Macaulay ring...

1 , S 2 {\displaystyle \Leftrightarrow R_{1},S_{2}} hold. A is a Cohen–Macaulay ring ⇔ S k {\displaystyle \Leftrightarrow S_{k}} hold for all k. Items...

polynomial ring and Stanley–Reisner ring of a simplicial complex. Cohen–Macaulay rings. Monomial ring, closely related to an affine semigroup ring and to...