In commutative algebra, given 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 finiteA-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, ..., xn] 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 homomorphism that is not finite is .
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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 Vi = Spec Ai such that f−1(Vi) has a finite covering by affine open subschemes Uij = Spec Bij with Bij an Ai-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.
and 21 Related for: Morphism of finite type information
y]/(y^{2}-x^{3}-t)} . The analogous notion in terms of schemes is: a morphism f: X → Y of schemes is offinitetype if Y has a covering by affine open subschemes...
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closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite. A morphism f: X → Y of schemes is called universally closed...
finitely generated algebra (also called an algebra offinitetype) is a commutative associative algebra A over a field K where there exists a finite set...
in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on...
_{S}^{n}\to S} where g is étale. A morphismoffinitetype is étale if and only if it is smooth and quasi-finite. A smooth morphism is stable under base change...
{\displaystyle 2} . A morphism of finitetype f : X → Y {\displaystyle f:X\to Y} is called a (local) complete intersection morphism if each point x in X has an...
being an algebra offinitetype. This concept is closely related to that offinitemorphism in algebraic geometry; in the simplest case of affine varieties...
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classes each determined by a morphism f: X → Ω, the characteristic morphismof that class, which we take to be the subobject of X characterized or named by...
{O}}_{X}} is a morphism from the structure sheaf of Y {\displaystyle Y} to the direct image of the structure sheaf of X. In other words, a morphism from ( X...
of technical conditions: f needs to be a separated morphismoffinitetype, the schemes involved need to be Noetherian). A far reaching extension of flat...
the topology associated to finite families { p i : U i → X } {\displaystyle \{p_{i}:U_{i}\to X\}} ofmorphismsoffinitetype such that ⨿ U i → X {\displaystyle...
has naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphismof the underlying locally ringed...
roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories...
x in X. Morphismsof G-sets are also called equivariant maps or G-maps. The composition of two morphisms is again a morphism. If a morphism f is bijective...
some category, an automorphism is a morphismof the object to itself that has an inverse morphism; that is, a morphism f : X → X {\displaystyle f:X\to X}...
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