Koszul complex (original) (raw)
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.
| Property | Value |
|---|---|
| dbo:abstract | 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) |
| dbo:wikiPageExternalLink | http://www.math.lsa.umich.edu/~hochster/711F07/L10.03.pdf |
| dbo:wikiPageID | 519992 (xsd:integer) |
| dbo:wikiPageLength | 23886 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1073177248 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Cambridge_University_Press dbr:Module_(mathematics) dbr:David_Eisenbud dbr:Homomorphism dbr:Volume_form dbr:Depth_(ring_theory) dbr:Jacobson_radical dbr:Lie_algebra_cohomology dbr:Mathematics dbr:Melvin_Hochster dbr:Free_resolution dbr:Ergebnisse_der_Mathematik_und_ihrer_Grenzgebiete dbr:Annihilator_(ring_theory) dbr:Lie_algebra dbr:Krull_dimension dbr:Banach_space dbr:Local_ring dbr:Exterior_algebra dbr:Exterior_power dbr:Graduate_Texts_in_Mathematics dbr:Hilbert's_syzygy_theorem dbr:Mapping_cone_(homological_algebra) dbr:Jean-Louis_Koszul dbc:Homological_algebra dbr:Syzygy_(mathematics) dbr:Homological_algebra dbr:Tor_functor dbr:Module_homomorphism dbr:Regular_sequence dbr:Chain_complex dbr:Nakayama's_lemma dbr:Koszul–Tate_complex dbr:Springer-Verlag dbr:Regular_sequence_(algebra) dbr:Cochain_complex dbr:Cohomology_theory dbr:Bounded_linear_operator dbr:Cycle_(homology_theory) |
| dbp:mathStatement | 3.15576E7 (dbd: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) |
| dbp:name | Corollary (en) Proposition (en) |
| dbp:note | depth-sensitivity (en) |
| dbp:wikiPageUsesTemplate | dbt:' dbt:Citation dbt:Fact dbt:Harv dbt:Reflist dbt:Short_description dbt:Technical dbt:Context dbt:Isbn dbt:Math_theorem |
| dct:subject | dbc:Homological_algebra |
| 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) |
| rdfs:label | Koszul-Komplex (de) Koszul complex (en) 코쥘 복합체 (ko) Комплекс Кошуля (ru) |
| owl:sameAs | freebase:Koszul complex wikidata:Koszul complex dbpedia-de:Koszul complex dbpedia-ko:Koszul complex dbpedia-ru:Koszul complex https://global.dbpedia.org/id/4piLy |
| prov:wasDerivedFrom | wikipedia-en:Koszul_complex?oldid=1073177248&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Koszul_complex |
| is dbo:knownFor of | dbr:Jean-Louis_Koszul |
| is dbo:wikiPageWikiLink of | dbr:List_of_algebraic_geometry_topics dbr:List_of_commutative_algebra_topics dbr:Derived_scheme dbr:List_of_homological_algebra_topics dbr:Standard_complex dbr:Glossary_of_arithmetic_and_diophantine_geometry dbr:Glossary_of_commutative_algebra dbr:Glossary_of_module_theory dbr:Linear_relation dbr:Commutative_ring dbr:Complete_intersection_ring dbr:Koszul_algebra dbr:Koszul_cohomology dbr:Koszul–Tate_resolution dbr:January_1921 dbr:Local_cohomology dbr:Exterior_algebra dbr:Hilbert's_syzygy_theorem dbr:Projective_module dbr:Resolution_(algebra) dbr:Jean-Louis_Koszul dbr:BRST_quantization dbr:Homological_conjectures_in_commutative_algebra dbr:Tor_functor dbr:Regular_embedding dbr:Regular_sequence dbr:Differential_graded_algebra dbr:Donald_C._Spencer dbr:Chain_complex dbr:Ext_functor |
| is dbp:knownFor of | dbr:Jean-Louis_Koszul |
| is foaf:primaryTopic of | wikipedia-en:Koszul_complex |