In commutative algebra, given a homomorphism A → B {\displaystyle A\to B} of commutative rings, B {\displaystyle B} is called an A {\displaystyle A}-algebra of finite type if B {\displaystyle B} can be finitely generated as an A {\displaystyle A}-algebra. It is much stronger for B {\displaystyle B} to be a finite A {\displaystyle A}-algebra, which means that B {\displaystyle B} is finitely generated as an A {\displaystyle A}-module. For example, for any commutative ring A {\displaystyle A} and natural number n {\displaystyle n}, the polynomial ring A [ x 1 , … , x n ] {\displaystyle A[x_{1},\dots ,x_{n}]} is an A {\displaystyle A}-algebra of finite type, but it is not a finite A {\displaystyle A}-algebra unless A {\displaystyle A} = 0 or n {\displaystyle n} = 0. Another example of a finite-type homomorphism that is not finite is C [ t ] → C [ t ] [ x , y ] / ( y 2 − x 3 − t ) {\displaystyle \mathbb {C} [t]\to \mathbb {C} [t][x,y]/(y^{2}-x^{3}-t)}.

The analogous notion in terms of schemes is that a morphism f : X → Y {\displaystyle f:X\to Y} of schemes is of finite type if Y {\displaystyle Y} has a covering by affine open subschemes V i = Spec ⁡ ( A i ) {\displaystyle V_{i}=\operatorname {Spec} (A_{i})} such that f − 1 ( V i ) {\displaystyle f^{-1}(V_{i})} has a finite covering by affine open subschemes U i j = Spec ⁡ ( B i j ) {\displaystyle U_{ij}=\operatorname {Spec} (B_{ij})} of X {\displaystyle X} with B i j {\displaystyle B_{ij}} an A i {\displaystyle A_{i}}-algebra of finite type. One also says that X {\displaystyle X} is of finite type over Y {\displaystyle Y}.

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

The Noether normalization lemma says, in geometric terms, that every affine scheme X {\displaystyle X} of finite type over a field k {\displaystyle k} has a finite surjective morphism to affine space A n {\displaystyle \mathbf {A} ^{n}} over k {\displaystyle k}, where n {\displaystyle n} is the dimension of X {\displaystyle X}. Likewise, every projective scheme X {\displaystyle X} over a field has a finite surjective morphism to projective space P n {\displaystyle \mathbf {P} ^{n}}, where n {\displaystyle n} is the dimension of X {\displaystyle X}.

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