In commutative algebra, given a homomorphism
of commutative rings,
-algebra of finite type if
is a finitely generated as an
is finitely generated as an
For example, for any commutative ring
and natural number
, the polynomial ring
Another example of a finite-type homomorphism that is not finite is
The analogous notion in terms of schemes is: a morphism
of schemes is of finite type if
has a covering by affine open subschemes
has a finite covering by affine open subschemes
{\displaystyle U_{ij}=\operatorname {Spec} (B_{ij})}
-algebra of finite type.
is of finite type over
For example, for any natural number
are of finite type over
More generally, any quasi-projective scheme over
is of finite type over
The Noether normalization lemma says, in geometric terms, that every affine scheme
of finite type over a field
has a finite surjective morphism to affine space
Likewise, every projective scheme
over a field has a finite surjective morphism to projective space
Bosch, Siegfried (2013).
Algebraic Geometry and Commutative Algebra.
London: Springer.
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