Correspondences

Summary

A correspondence from X to Y, denoted α: X ⊢ Y, is a subvariety, cycle, or equivalence class of cycles on X × Y. The graph of a morphism, or the closure of the graph of a rational map, are basic examples, but more general correspondences have played an important role in the development of algebraic geometry. On complete non-singular varieties correspondences have a product β ∘ α, and a correspondence α: X ⊢ Y determines homomorphisms α_* from A(X) to A(Y), and α* from A(Y) to A(X), these notions generalizing composition, push-forward, and pull-back for morphisms. The basic algebra of correspondences is deduced easily from the general theory of Chap. 8.