For an irreducible algebraic variety with function field , this sheaf of abelian groups assigns to every nonempty open set, with identity restriction maps, and assigns the trivial group to the empty set. Irreducibility ensures that any two nonempty open subsets meet, so these sections satisfy the sheaf gluing axiom. It is a flasque sheaf and has zero higher sheaf cohomology.
For an irreducible variety over a field ,Consequently birational irreducible varieties have the same dimension.
If is a finitely generated field extension, write for a finitely generated integral -algebra . The projective closure of the affine variety is an irreducible projective variety with function field .
Articles by others on the same topic
The function field of an algebraic variety is a concept that arises in algebraic geometry. It can be thought of as the "field of rational functions" defined on the variety. Here’s a more detailed explanation: 1. **Algebraic Variety**: An algebraic variety is a geometric object that is defined as the solution set to a system of polynomial equations over a given field (typically the field of complex numbers or the rationals).