Theorem of the cube
Encyclopedia
In mathematics
, the theorem of the cube is a condition for a line bundle
of 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 . A treatment by means of sheaf cohomology
, and description in terms of the Picard functor, was given
by .
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 OX- module. If L is taken as a holomorphic line bundle, in the complex manifold
case, this is the same here as a trivial bundle, but in a holomorphic sense, not just topologically.
The theorem of the square 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the theorem of the cube is a condition for a line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
of 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
Italian school of algebraic geometry
In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...
. 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
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
; a discussion of the history has been given by . A treatment by means of sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
, and description in terms of the Picard functor, was given
by .
The theorem states that for any complete varieties U, V and W, and given points u, v and w on them, any invertible sheaf
Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
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
Ringed space
In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
X, an invertible sheaf L is trivial if isomorphic to OX, as OX- module. If L is taken as a holomorphic line bundle, in the complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
case, this is the same here as a trivial bundle, but in a holomorphic sense, not just topologically.
The theorem of the square is a corollary applying to an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
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.