Finite morphism

In algebraic geometry, a finite morphism between two affine varieties is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over . This definition can be extended to the quasi-projective varieties, such that a regular map between quasiprojective varieties is finite if any point like has an affine neighbourhood V such that is affine and is a finite map (in view of the previous definition, because it is between affine varieties). A morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes such that for each i, is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism makes Ai a finitely generated module over Bi. One also says that X is finite over Y. In fact, f is finite if and only if for every open affine subscheme V = Spec B in Y, the inverse image of V in X is affine, of the form Spec A, with A a finitely generated B-module. For example, for any field k, is a finite morphism since as -modules. Geometrically, this is obviously finite since this is a ramified n-sheeted cover of the affine line which degenerates at the origin. By contrast, the inclusion of A1 − 0 into A1 is not finite. (Indeed, the Laurent polynomial ring k[y, y−1] is not finitely generated as a module over k[y].) This restricts our geometric intuition to surjective families with finite fibers. The composition of two finite morphisms is finite. Any base change of a finite morphism f: X → Y is finite. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y Z → Z is finite. This corresponds to the following algebraic statement: if A and C are (commutative) B-algebras, and A is finitely generated as a B-module, then the tensor product A ⊗B C is finitely generated as a C-module. Indeed, the generators can be taken to be the elements ai ⊗ 1, where ai are the given generators of A as a B-module. Closed immersions are finite, as they are locally given by A → A/I, where I is the ideal corresponding to the closed subscheme.

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