Morphism of finite type

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

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

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

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