Theorem of the cube


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 final version of the theorem of the cube was first published by, who credited it to André Weil. A discussion of the history has been given by. A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given
by.

Statement

The theorem states that for any complete varieties U, V and W over an algebraically closed field, and given points u, v and w on them, any invertible sheaf L which has a trivial restriction to each of U× V ×, U× × W, and × V × W, is itself trivial.

Special cases

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 a holomorphic line bundle, and trivial means holomorphically equivalent to a trivial bundle, not just topologically equivalent.

Restatement using biextensions

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 square

The theorem of the square is a corollary applying to an abelian variety A. One version of it states that the function φL taking xA to TLL−1 is a group homomorphism from A to Pic.