Milnor K-theory (original) (raw)
In mathematics, Milnor K-theory is an algebraic invariant (denoted for a field ) defined by John Milnor as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic K-theory and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for and . Fortunately, it can be shown Milnor K-theory is a part of algebraic K-theory, which in general is the easiest part to compute.
| Property | Value |
|---|---|
| dbo:abstract | In mathematics, Milnor K-theory is an algebraic invariant (denoted for a field ) defined by John Milnor as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic K-theory and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for and . Fortunately, it can be shown Milnor K-theory is a part of algebraic K-theory, which in general is the easiest part to compute. (en) La K-théorie de Milnor, théorie mathématique introduite par John Milnor, fait partie des premières tentatives pour définir les groupes de K-théorie algébrique d'ordre supérieur. (fr) 대수적 K이론에서 밀너 환(Milnor環, 영어: Milnor ring)은 각 등급 성분이 대수적 K군으로 가는 자연스러운 군 준동형을 갖는 등급환이다. 그 0~2등급 성분은 대수적 K군과 동형이지만, 이는 고차 등급 성분에서 성립하지 않는다. (ko) ミルナーのK-理論(Milnor K-theory)は、高次代数的K-理論を定義する初期の試みであり、 Milnor により導入された。 (ja) Inom matematiken är Milnors K-teori ett tidigt försök att definiera högre algebraisk K-teori, introducerat av. (sv) |
| dbo:wikiPageExternalLink | http://math.rutgers.edu/~weibel/motiviclectures.html https://mathoverflow.net/questions/92290/about-tates-computation-of-k-2-rm-m-mathbb-q%3Frq=1 http://annals.math.princeton.edu/2011/174-1/p11 |
| dbo:wikiPageID | 2914128 (xsd:integer) |
| dbo:wikiPageLength | 21058 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1119723330 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Cambridge_University_Press dbr:Multiplicative_group dbr:Motivic_homotopy_theory dbr:Algebraic_K-theory dbr:Annals_of_Mathematics dbr:John_Tate_(mathematician) dbr:Vector_space dbr:Vladimir_Voevodsky dbr:Inventiones_Mathematicae dbr:Witt_group dbr:Mathematics dbr:Andrei_Suslin dbr:Long_exact_sequence dbr:Short_exact_sequence dbr:Commutative_ring dbr:Étale_cohomology dbr:Steinberg_group_(K-theory) dbc:K-theory dbr:Two-sided_ideal dbr:Galois_cohomology dbr:Alexander_Merkurjev dbr:American_Mathematical_Society dbr:Cyclic_group dbr:Daniel_Quillen dbr:Field_(mathematics) dbr:Finite_field dbr:Finitely_generated_module dbr:Dimension_(vector_space) dbr:Direct_sum dbr:Isomorphism dbr:Quadratic_form dbr:Mathematical_proof dbr:Characteristic_(algebra) dbr:Bloch's_higher_Chow_group dbr:Hideya_Matsumoto dbr:Higher_local_field dbr:Witt_ring_(forms) dbr:Azumaya_algebra dbr:Class_field_theory dbr:Grothendieck_group dbr:If_and_only_if dbr:Order_(group_theory) dbr:Real_closed_field dbr:Real_number dbr:Matsumoto's_theorem_(K-theory) dbr:Markus_Rost dbr:Tensor_algebra dbr:Supercommutative_algebra dbr:Steenrod_algebra dbr:Pfister_form dbr:Milnor_conjecture dbr:Motivic_cohomology dbr:Norm_residue_isomorphism_theorem dbr:Uncountable_set dbr:Generators_and_relations dbr:Arxiv:math/0311099 |
| dbp:authorLink | John Milnor (en) |
| dbp:first | John (en) |
| dbp:last | Milnor (en) |
| dbp:wikiPageUsesTemplate | dbt:Citation dbt:Cite_book dbt:Reflist dbt:Technical dbt:Harvs |
| dbp:year | 1970 (xsd:integer) |
| dcterms:subject | dbc:K-theory |
| gold:hypernym | dbr:Invariant |
| rdfs:comment | In mathematics, Milnor K-theory is an algebraic invariant (denoted for a field ) defined by John Milnor as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic K-theory and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for and . Fortunately, it can be shown Milnor K-theory is a part of algebraic K-theory, which in general is the easiest part to compute. (en) La K-théorie de Milnor, théorie mathématique introduite par John Milnor, fait partie des premières tentatives pour définir les groupes de K-théorie algébrique d'ordre supérieur. (fr) 대수적 K이론에서 밀너 환(Milnor環, 영어: Milnor ring)은 각 등급 성분이 대수적 K군으로 가는 자연스러운 군 준동형을 갖는 등급환이다. 그 0~2등급 성분은 대수적 K군과 동형이지만, 이는 고차 등급 성분에서 성립하지 않는다. (ko) ミルナーのK-理論(Milnor K-theory)は、高次代数的K-理論を定義する初期の試みであり、 Milnor により導入された。 (ja) Inom matematiken är Milnors K-teori ett tidigt försök att definiera högre algebraisk K-teori, introducerat av. (sv) |
| rdfs:label | K-théorie de Milnor (fr) 밀너 환 (ko) Milnor K-theory (en) ミルナーのK理論 (ja) Milnors K-teori (sv) |
| owl:sameAs | freebase:Milnor K-theory freebase:Milnor K-theory wikidata:Milnor K-theory dbpedia-fr:Milnor K-theory dbpedia-ja:Milnor K-theory dbpedia-ko:Milnor K-theory dbpedia-sv:Milnor K-theory https://global.dbpedia.org/id/4reKK |
| prov:wasDerivedFrom | wikipedia-en:Milnor_K-theory?oldid=1119723330&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Milnor_K-theory |
| is dbo:knownFor of | dbr:John_Milnor |
| is dbo:wikiPageRedirects of | dbr:Milnor_ring dbr:Milnor_K-group |
| is dbo:wikiPageWikiLink of | dbr:Norm_variety dbr:Stiefel–Whitney_class dbr:Alexander_Merkurjev dbr:Field_(mathematics) dbr:Hilbert's_Theorem_90 dbr:Hilbert_symbol dbr:John_Milnor dbr:Bloch_group dbr:Azumaya_algebra dbr:Milnor_ring dbr:Steinberg_symbol dbr:List_of_things_named_after_John_Milnor dbr:Pfister_form dbr:Milnor_conjecture dbr:Motivic_cohomology dbr:Norm_residue_isomorphism_theorem dbr:Milnor_K-group |
| is dbp:knownFor of | dbr:John_Milnor |
| is foaf:primaryTopic of | wikipedia-en:Milnor_K-theory |