About: Popescu's theorem

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In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu,states: Let A be a Noetherian ring and B a Noetherian algebra over it. Then, the structure map A → B is a regular homomorphism if and only if B is a direct limit of smooth A-algebras. For example, if A is a local G-ring (e.g., a local excellent ring) and B its completion, then the map A → B is regular by definition and the theorem applies. Another proof of Popescu's theorem was given by Tetsushi Ogoma, while an exposition of the result was provided by Richard Swan.

Attributes	Values
rdfs:label	Popescu's theorem (en) Popescus sats (sv)
rdfs:comment	Inom matematiken är Popescus sats, introducerad av , ett resultat som säger att om A är en noethersk ring och B en noethersk algebra över den, då är strukturavbildningen A →B en om och bara om B är ett av . Om exempelvis A är en lokal och B dess kompletition, då är avbildningen A →B per definition regelbundenoch satsen kan användas. (sv) In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu,states: Let A be a Noetherian ring and B a Noetherian algebra over it. Then, the structure map A → B is a regular homomorphism if and only if B is a direct limit of smooth A-algebras. For example, if A is a local G-ring (e.g., a local excellent ring) and B its completion, then the map A → B is regular by definition and the theorem applies. Another proof of Popescu's theorem was given by Tetsushi Ogoma, while an exposition of the result was provided by Richard Swan. (en)
dcterms:subject	Theorems in algebraic geometry
Wikipage page ID	44549263 (xsd:integer)
Wikipage revision ID	1097202896 (xsd:integer)
Link from a Wikipage to another Wikipage	Approximation property (ring theory) Noetherian ring Commutative algebra Completion of a ring G-ring Local ring Algebra over a ring Algebraic geometry Direct limit Mathematical proof Theorems in algebraic geometry Artin approximation theorem If and only if Excellent ring Smooth algebra Richard Swan Regular homomorphism
Link from a Wikipage to an external page	https://mathoverflow.net/q/155235
sameAs	Popescu's theorem Popescu's theorem Popescu's theorem
dbp:wikiPageUsesTemplate	dbt:Reflist dbt:Algebraic-geometry-stub
has abstract	In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu,states: Let A be a Noetherian ring and B a Noetherian algebra over it. Then, the structure map A → B is a regular homomorphism if and only if B is a direct limit of smooth A-algebras. For example, if A is a local G-ring (e.g., a local excellent ring) and B its completion, then the map A → B is regular by definition and the theorem applies. Another proof of Popescu's theorem was given by Tetsushi Ogoma, while an exposition of the result was provided by Richard Swan. The usual proof of the Artin approximation theorem relies crucially on Popescu's theorem. Popescu's result was proved by an alternate method, and somewhat strengthened, by Mark Spivakovsky. (en) Inom matematiken är Popescus sats, introducerad av , ett resultat som säger att om A är en noethersk ring och B en noethersk algebra över den, då är strukturavbildningen A →B en om och bara om B är ett av . Om exempelvis A är en lokal och B dess kompletition, då är avbildningen A →B per definition regelbundenoch satsen kan användas. (sv)
prov:wasDerivedFrom	wikipedia-en:Popescu's_theorem?oldid=1097202896&ns=0
page length (characters) of wiki page	3529 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Popescu's_theorem
is Link from a Wikipage to another Wikipage of	Approximation property (ring theory) Commutative ring G-ring Artin approximation theorem Cartier isomorphism Popescu’s theorem Smooth algebra
is Wikipage redirect of	Popescu’s theorem
is foaf:primaryTopic of	wikipedia-en:Popescu's_theorem

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