Morphism of finite type

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-module unless A = 0 or n = 0. Another example of a finite-type morphism which is not finite is [math]\displaystyle{ \mathbb{C}[t] \to \mathbb{C}[t][x,y]/(y^2 - x^3 - t) }[/math].

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.

For example, for any natural number n and field k, affine n-space and projective n-space over k are of finite type over k (that is, over Spec k), while they are not finite over k unless n = 0. More generally, any quasi-projective scheme over k is of finite type over k.

The Noether normalization lemma says, in geometric terms, that every affine scheme X of finite type over a field k has a finite surjective morphism to affine space Aⁿ over k, where n is the dimension of X. Likewise, every projective scheme X over a field has a finite surjective morphism to projective space Pⁿ, where n is the dimension of X.

See also

• Finitely generated algebra

References

Bosch, Siegfried (2013). Algebraic Geometry and Commutative Algebra. London: Springer. pp. 360–365. ISBN 9781447148289.