About: Cohen–Macaulay ring

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

Attributes	Values
rdf:type	Thing anatomical structure
rdfs:label	Cohen–Macaulay ring (en) Cohen-Macaulay-Ring (de) Anello di Cohen-Macaulay (it) コーエン・マコーレー環 (ja) 코언-매콜리 환 (ko) 科恩-麥考利環 (zh) Кільце Коена — Маколея (uk)
rdfs:comment	数学において、コーエン・マコーレー環 (英: Cohen–Macaulay ring, CM ring) は局所のような非特異多様体の代数幾何的な性質のいくつかをもった可換環のタイプである。名称は純性定理を多項式環に対して証明したと、純性定理を形式的冪級数環に対して証明したによる。すべての Cohen–Macaulay 環は純性定理が成り立つ。可換ネーター局所環については次の包含関係が成り立つ。強鎖状環 ⊃ コーエン・マコーレー環 ⊃ ゴレンシュタイン環 ⊃ 完全交叉環 ⊃ 正則局所環 (ja) 가환대수학과 대수기하학에서 코언-매콜리 환(Cohen-Macaulay環, 영어: Cohen–Macaulay ring)은 국소적으로 어느 곳에서나 차원이 동일한 아핀 스킴의 개념을 형식화한 개념이다. (ko) In matematica, in particolare in algebra commutativa, un anello di Cohen-Macaulay è un anello commutativo unitario noetheriano tale che, per ogni ideale massimale , la profondità e la dimensione di Krull della localizzazione sono uguali. La classe degli anelli di Cohen-Macaulay contiene al suo interno tutti gli anelli regolari e gli anelli di Gorenstein. Prendono nome da e , che dimostrarono il rispettivamente per gli anelli di polinomi (Macaulay, 1916) e gli anelli di serie formali (Cohen, 1946). (it) У комутативній алгебрі кільцями Коена — Маколея називається клас комутативних кілець, що є зокрема важливим у алгебричній геометрії, завдяки властивостям локальної рівнорозмірності. Названі на честь англійського математика Френсіса Маколея і американського математика Ірвінга Коена. (uk) 在交換代數中，Cohen-Macaulay環是對應到一類代數幾何性質（例如局部等維性）的交換環。此概念依數學家（Francis Sowerby Macaulay）與（Irvin S. Cohen）命名，麦考利（1916年）證明了多項式環的純粹性定理，科恩（1946年）則證明了冪級數環的情形；事實上所有Cohen-Macaulay環都具純粹性。 (zh) Im mathematischen Teilgebiet der kommutativen Algebra versteht man unter einem Cohen-Macaulay-Ring einen noetherschen Ring, der nicht mehr unbedingt regulär ist, dessen Tiefe aber gleich seiner Krulldimension ist. Eine Cohen-Macaulay-Singularität ist eine Singularität, deren lokaler Ring ein Cohen-Macaulay-Ring ist. Benannt wurden die Ringe nach Irvin Cohen und Francis Macaulay. (de) 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. For Noetherian local rings, there is the following chain of inclusions. (en)
differentFrom	Cohen algebra Cohen ring
dcterms:subject	Commutative algebra Algebraic geometry
Wikipage page ID	497407 (xsd:integer)
Wikipage revision ID	1099047862 (xsd:integer)
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