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- Theorem_of_the_cube abstract "In mathematics, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial. It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry. The specific result was proved under this name, in the early 1950s, in the course of his fundamental work on abstract algebraic geometry by André Weil; a discussion of the history has been given by Kleiman (2005). A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given by Mumford (2008).The theorem states that for any complete varieties U, V and W, and given points u, v and w on them, any invertible sheaf L which has a trivial restriction to each of U× V × {w}, U× {v} × W, and {u} × V × W, is itself trivial. (Mumford p. 55; the result there is slightly stronger, in that one of the varieties need not be complete and can be replaced by a connected scheme.) Note: On a ringed space X, an invertible sheaf L is trivial if isomorphic to OX, as an OX-module. If the base X is a complex manifold, then an invertible sheaf is (the sheaf of sections of) a holomorphic line bundle, and trivial means holomorphically equivalent to a trivial bundle, not just topologically equivalent. The theorem of the square (Mumford 2008, p.59) is a corollary applying to an abelian variety A, defining a group homomorphism from A to Pic(A), in terms of the change in L by translation on A.Weil's result has been restated in terms of biextensions, a concept now generally used in the duality theory of abelian varieties.".
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- Theorem_of_the_cube wikiPageRevisionID "598602565".
- Theorem_of_the_cube wikiPageWikiLink Abelian_variety.
- Theorem_of_the_cube wikiPageWikiLink American_Mathematical_Society.
- Theorem_of_the_cube wikiPageWikiLink André_Weil.
- Theorem_of_the_cube wikiPageWikiLink Biextension.
- Theorem_of_the_cube wikiPageWikiLink Category:Algebraic_varieties.
- Theorem_of_the_cube wikiPageWikiLink Category:Theorems_in_geometry.
- Theorem_of_the_cube wikiPageWikiLink Complete_variety.
- Theorem_of_the_cube wikiPageWikiLink Complex_manifold.
- Theorem_of_the_cube wikiPageWikiLink Dual_abelian_variety.
- Theorem_of_the_cube wikiPageWikiLink Duality_theory_of_abelian_varieties.
- Theorem_of_the_cube wikiPageWikiLink Fiber_bundle.
- Theorem_of_the_cube wikiPageWikiLink Holomorphic_line_bundle.
- Theorem_of_the_cube wikiPageWikiLink Holomorphic_vector_bundle.
- Theorem_of_the_cube wikiPageWikiLink Invertible_sheaf.
- Theorem_of_the_cube wikiPageWikiLink Italian_school_of_algebraic_geometry.
- Theorem_of_the_cube wikiPageWikiLink Line_bundle.
- Theorem_of_the_cube wikiPageWikiLink Linear_equivalence.
- Theorem_of_the_cube wikiPageWikiLink Linear_system_of_divisors.
- Theorem_of_the_cube wikiPageWikiLink Mathematics.
- Theorem_of_the_cube wikiPageWikiLink Picard_functor.
- Theorem_of_the_cube wikiPageWikiLink Picard_group.
- Theorem_of_the_cube wikiPageWikiLink Ringed_space.
- Theorem_of_the_cube wikiPageWikiLink Sheaf_cohomology.
- Theorem_of_the_cube wikiPageWikiLink Trivial_bundle.
- Theorem_of_the_cube wikiPageWikiLinkText "Theorem of the cube".
- Theorem_of_the_cube hasPhotoCollection Theorem_of_the_cube.
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- Theorem_of_the_cube subject Category:Algebraic_varieties.
- Theorem_of_the_cube subject Category:Theorems_in_geometry.
- Theorem_of_the_cube hypernym Condition.
- Theorem_of_the_cube type Disease.
- Theorem_of_the_cube type Theorem.
- Theorem_of_the_cube type Variety.
- Theorem_of_the_cube comment "In mathematics, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial. It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry. The specific result was proved under this name, in the early 1950s, in the course of his fundamental work on abstract algebraic geometry by André Weil; a discussion of the history has been given by Kleiman (2005).".
- Theorem_of_the_cube label "Theorem of the cube".
- Theorem_of_the_cube sameAs m.0h1qc5.
- Theorem_of_the_cube sameAs Q7782346.
- Theorem_of_the_cube sameAs Q7782346.
- Theorem_of_the_cube wasDerivedFrom Theorem_of_the_cube?oldid=598602565.
- Theorem_of_the_cube isPrimaryTopicOf Theorem_of_the_cube.