Matches in DBpedia 2014 for { <http://dbpedia.org/resource/Cartan's_theorems_A_and_B> ?p ?o. }
Showing items 1 to 19 of
19
with 100 items per page.
- Cartan's_theorems_A_and_B abstract "In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.Theorem A. F is spanned by its global sections. Theorem B is stated in cohomological terms (a formulation that Cartan (1953, p.51) attributes to J.-P. Serre):Theorem B. H p(X, F) = 0 for all p > 0.Analogous properties were established by Serre (1955) for coherent sheaves in algebraic geometry, when X is an affine scheme. The analogue of Theorem B in this context is as follows (Hartshorne 1977, Theorem III.3.7):Theorem B (Scheme theoretic analogue). Let X be an affine scheme, F a quasi-coherent sheaf of OX-modules for the Zariski topology on X. Then H p(X, F) = 0 for all p > 0.These theorems have many important applications. Naively, they imply that a holomorphic function on a closed complex submanifold, Z, of a Stein manifold X can be extended to a holomorphic function on all of X. At a deeper level, these theorems were used by Jean-Pierre Serre to prove the GAGA theorem.Theorem B is sharp in the sense that if H 1(X, F) = 0 for all coherent sheaves F on a complex manifold X (resp. quasi-coherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see Serre (1952) (resp. Serre (1957) and Hartshorne|1977|loc=Theorem III.3.7). See also Cousin problems".
- Cartan's_theorems_A_and_B wikiPageExternalLink item?id=AIF_1956__6__1_0.
- Cartan's_theorems_A_and_B wikiPageID "621677".
- Cartan's_theorems_A_and_B wikiPageRevisionID "596293381".
- Cartan's_theorems_A_and_B first "E.M.".
- Cartan's_theorems_A_and_B hasPhotoCollection Cartan's_theorems_A_and_B.
- Cartan's_theorems_A_and_B id "c/c020570".
- Cartan's_theorems_A_and_B last "Chirka".
- Cartan's_theorems_A_and_B title "Cartan theorem".
- Cartan's_theorems_A_and_B subject Category:Several_complex_variables.
- Cartan's_theorems_A_and_B subject Category:Theorems_in_algebraic_geometry.
- Cartan's_theorems_A_and_B subject Category:Topological_methods_of_algebraic_geometry.
- Cartan's_theorems_A_and_B comment "In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.Theorem A. F is spanned by its global sections. Theorem B is stated in cohomological terms (a formulation that Cartan (1953, p.51) attributes to J.-P. Serre):Theorem B.".
- Cartan's_theorems_A_and_B label "Cartan's theorems A and B".
- Cartan's_theorems_A_and_B sameAs m.02xl42.
- Cartan's_theorems_A_and_B sameAs Q5047043.
- Cartan's_theorems_A_and_B sameAs Q5047043.
- Cartan's_theorems_A_and_B wasDerivedFrom Cartan's_theorems_A_and_B?oldid=596293381.
- Cartan's_theorems_A_and_B isPrimaryTopicOf Cartan's_theorems_A_and_B.