Geometrically regular ring (original) (raw)
In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are defined in a similar way. In older terminology, points with regular local rings were called simple points, and points with geometrically regular local rings were called absolutely simple points. Over fields that are of characteristic 0, or algebraically closed, or more generally perfect, geometrically regular rings are the same as regular rings. Geometric regularity originated when Claude Chevalley and André Weil pointed out to Oscar Zariski that, over non-perfect fields, the for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular.
| Property | Value |
|---|---|
| dbo:abstract | In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are defined in a similar way. In older terminology, points with regular local rings were called simple points, and points with geometrically regular local rings were called absolutely simple points. Over fields that are of characteristic 0, or algebraically closed, or more generally perfect, geometrically regular rings are the same as regular rings. Geometric regularity originated when Claude Chevalley and André Weil pointed out to Oscar Zariski that, over non-perfect fields, the for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular. A Noetherian local ring containing a field k is geometrically regular over k if and only if it is formally smooth over k. (en) |
| dbo:wikiPageID | 40023277 (xsd:integer) |
| dbo:wikiPageLength | 2511 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1022204973 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Noetherian_ring dbr:Claude_Chevalley dbr:André_Weil dbr:Local_ring dbr:Algebraic_geometry dbr:Field_(algebra) dbr:Regular_ring dbc:Commutative_algebra dbc:Algebraic_geometry dbr:Regular_scheme dbr:Perfect_field dbr:Transactions_of_the_American_Mathematical_Society dbr:Formally_smooth dbr:Jacobian_criterion |
| dbp:authorlink | Oscar Zariski (en) |
| dbp:first | Oscar (en) |
| dbp:last | Zariski (en) |
| dbp:wikiPageUsesTemplate | dbt:Citation dbt:Harvtxt dbt:Harvs dbt:EGA |
| dbp:year | 1947 (xsd:integer) |
| dcterms:subject | dbc:Commutative_algebra dbc:Algebraic_geometry |
| gold:hypernym | dbr:Ring |
| rdf:type | dbo:AnatomicalStructure |
| rdfs:comment | In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are defined in a similar way. In older terminology, points with regular local rings were called simple points, and points with geometrically regular local rings were called absolutely simple points. Over fields that are of characteristic 0, or algebraically closed, or more generally perfect, geometrically regular rings are the same as regular rings. Geometric regularity originated when Claude Chevalley and André Weil pointed out to Oscar Zariski that, over non-perfect fields, the for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular. (en) |
| rdfs:label | Geometrically regular ring (en) |
| owl:sameAs | freebase:Geometrically regular ring wikidata:Geometrically regular ring https://global.dbpedia.org/id/fhhr |
| prov:wasDerivedFrom | wikipedia-en:Geometrically_regular_ring?oldid=1022204973&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Geometrically_regular_ring |
| is dbo:wikiPageRedirects of | dbr:Geometrically_regular dbr:Absolutely_regular dbr:Absolutely_regular_local_ring dbr:Absolutely_regular_ring dbr:Absolutely_simple dbr:Absolutely_simple_point dbr:Geometric_regularity dbr:Geometrically_regular_local_ring |
| is dbo:wikiPageWikiLink of | dbr:Glossary_of_commutative_algebra dbr:Regular_local_ring dbr:Regular_scheme dbr:Geometrically_regular dbr:Absolutely_regular dbr:Absolutely_regular_local_ring dbr:Absolutely_regular_ring dbr:Absolutely_simple dbr:Absolutely_simple_point dbr:Geometric_regularity dbr:Geometrically_regular_local_ring |
| is foaf:primaryTopic of | wikipedia-en:Geometrically_regular_ring |