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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.".
- Theorem_of_the_cube wikiPageID "7044103".
- Theorem_of_the_cube wikiPageRevisionID "598602565".
- Theorem_of_the_cube hasPhotoCollection Theorem_of_the_cube.
- Theorem_of_the_cube subject Category:Algebraic_varieties.
- Theorem_of_the_cube subject Category:Theorems_in_geometry.
- Theorem_of_the_cube type Abstraction100002137.
- Theorem_of_the_cube type AlgebraicVarieties.
- Theorem_of_the_cube type Assortment108398773.
- Theorem_of_the_cube type Collection107951464.
- Theorem_of_the_cube type Communication100033020.
- Theorem_of_the_cube type Group100031264.
- Theorem_of_the_cube type Message106598915.
- Theorem_of_the_cube type Proposition106750804.
- Theorem_of_the_cube type Statement106722453.
- Theorem_of_the_cube type Theorem106752293.
- Theorem_of_the_cube type TheoremsInGeometry.
- 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 sameAs Theorem_of_the_cube.
- Theorem_of_the_cube wasDerivedFrom Theorem_of_the_cube?oldid=598602565.
- Theorem_of_the_cube isPrimaryTopicOf Theorem_of_the_cube.