Finite morphism

In algebraic geometry, 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 A_i, and the restriction of f to U_i, which induces a ring homomorphism

makes A_i a finitely generated module over B_i. One also says that X is finite over Y.

In fact, f is finite if and only if for every open affine open 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, the morphism from the affine line A¹ over k to itself given by x ↦ x² is finite. (Indeed, the polynomial ring k[x] is finitely generated as a module over k[y] by y ↦ x², with generators 1 and x.) 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].)

For a homomorphism A → B of commutative rings, B is called an A-algebra of finite type if B is a finitely generated as an A-algebra. It is much stronger for B to be a finite A-algebra, which means that B is finitely generated as an A-module. For example, for any commutative ring A and natural number n, the polynomial ring A[x₁, ..., x_n] is an A-algebra of finite type, but it is not a finite A-algebra unless A = 0 or n = 0.

The analogous notion in terms of schemes is: a morphism f: X → Y of schemes is of finite type if Y has a covering by affine open subschemes V_i = Spec A_i such that f⁻¹(V_i) has a finite covering by affine open subschemes U_ij = Spec B_ij with B_ij an A_i-algebra of finite type. One also says that X is of finite type over Y.

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