In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an M-regular sequence, and hence it can be used to prove basic facts about the depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of Krull dimension. Moreover, in certain circumstances, the complex is the complex of syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth.
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| rdfs:label
| - Koszul-Komplex (de)
- Koszul complex (en)
- 코쥘 복합체 (ko)
- Комплекс Кошуля (ru)
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| rdfs:comment
| - In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an M-regular sequence, and hence it can be used to prove basic facts about the depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of Krull dimension. Moreover, in certain circumstances, the complex is the complex of syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth. (en)
- 가환대수학에서 코쥘 복합체(Koszul複合體, 영어: Koszul complex)는 가환환의 가군 및 가군의 특별한 원소로부터 정의되는 미분 등급 대수이다. 이를 통하여 가군의 코쥘 코호몰로지(영어: Koszul cohomology)를 정의할 수 있다. (ko)
- Комплекс Кошуля был впервые введён в математике , чтобы определить теорию когомологий алгебр Ли. Впоследствии он оказался полезной общей конструкцией гомологической алгебры. Его гомологии могут быть использованы для того, чтобы определить, является ли последовательность элементов кольца , и, как следствие, он может быть использован ля того, чтобы доказать базовые свойства или . (ru)
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| - Corollary (en)
- Proposition (en)
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| - In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an M-regular sequence, and hence it can be used to prove basic facts about the depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of Krull dimension. Moreover, in certain circumstances, the complex is the complex of syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth. (en)
- 가환대수학에서 코쥘 복합체(Koszul複合體, 영어: Koszul complex)는 가환환의 가군 및 가군의 특별한 원소로부터 정의되는 미분 등급 대수이다. 이를 통하여 가군의 코쥘 코호몰로지(영어: Koszul cohomology)를 정의할 수 있다. (ko)
- Комплекс Кошуля был впервые введён в математике , чтобы определить теорию когомологий алгебр Ли. Впоследствии он оказался полезной общей конструкцией гомологической алгебры. Его гомологии могут быть использованы для того, чтобы определить, является ли последовательность элементов кольца , и, как следствие, он может быть использован ля того, чтобы доказать базовые свойства или . (ru)
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| - 3.15576E7 (second)
- Let R be a Noetherian ring, x1, ..., xn elements of R and I = the ideal generated by them. For a finitely generated module M over R, if, for some integer m,
: for all i > m,
while
:
then every maximal M-regular sequence in I has length n - m . As a consequence,
:. (en)
- Let R, M be as above and a sequence of elements of R. Suppose there are a ring S, an S-regular sequence in S and a ring homomorphism S → R that maps to . Then
:
where Tor denotes the Tor functor and M is an S-module through S → R. (en)
- Let R, M be as above and a sequence of elements of R. Then both the ideal and the annihilator of M annihilate
:
for all i. (en)
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