Finite morphism

In algebraic geometry, a morphism "f": "X" → "Y" of schemes is a finite morphism if "Y" has an open cover by affine schemes
such that for each "i",
is an open affine subscheme Spec "A""i", and the restriction of "f" to "U""i", which induces a ring homomorphism
makes "A""i" a finitely generated module over "B""i". One also says that "X" is finite over "Y".
In fact, "f" is finite if and only if for "every" open affine open subscheme "V" = Spec "B" in "Y", the inverse image of "V" in "X" is affine, of the form Spec "A", with "A" a finitely generated "B"-module.
For example, for any field "k", formula_4 is a finite morphism since formula_5 as formula_6-modules. Geometrically, this is obviously finite since this is a ramified n-sheeted cover of the affine line which degenerates at the origin. By contrast, the inclusion of "A"1 − 0 into "A"1 is not finite. (Indeed, the Laurent polynomial ring "k"["y", "y"−1] is not finitely generated as a module over "k"["y"].) This restricts our geometric intuition to surjective families with finite fibers.
Morphisms 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"["x"1, ..., "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 formula_7.
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 A"n" 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 P"n", where "n" is the dimension of "X".