In algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a morphism of schemes is a finite morphism, if has an open cover by affine schemes

such that for each ,


is an open affine subscheme , and the restriction of f to , which induces a map of rings


makes a finitely generated module over .

Morphisms of finite type

There is another finiteness condition on morphisms of schemes, morphisms of finite type, which is much weaker than being finite.

Morally, a morphism of finite type corresponds to a set of polynomial equations with finitely many variables. For example, the algebraic equation
corresponds to the map of (affine) schemes or equivalently to the inclusion of rings . This is an example of a morphism of finite type.

The technical definition is as follows: let be an open cover of by affine schemes, and for each let be an open cover of by affine schemes. The restriction of f to induces a morphism of rings .
The morphism f is called locally of finite type, if is a finitely generated algebra over (via the above map of rings). If in addition the open cover can be chosen to be finite, then f is called of finite type.

For example, if is a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, the scheme has a natural morphism to induced by the inclusion of rings

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

  This is a morphism of finite type, but if then it is not a finite morphism.

On the other hand, if we take the affine scheme , it has a natural morphism to given by the ring homomorphism Then this morphism is a finite morphism.

Properties of finite morphisms

In the following, f : X → Y denotes a finite morphism.

• The composition of two finite maps is finite.
• Any base change
  Grothendieck's relative point of view
  Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object...
  
   of a finite morphism is finite, i.e. if is another (arbitrary) morphism, then the canonical morphism is finite. This corresponds to the following algebraic statement: if A is a finitely generated B-module, then the tensor product
  Tensor product of algebras
  In mathematics, the tensor product of two R-algebras is also an R-algebra. This gives us a tensor product of algebras. The special case R = Z gives us a tensor product of rings, since rings may be regarded as Z-algebras....
  
    is a finitely generated C-module, where is any map. The generators are , where are the generators of A as a B-module.
• Closed immersion are finite, as they are locally given by , where I is the ideal
  Ideal (ring theory)
  In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
  
   corresponding to the closed subscheme.
• Finite morphisms are closed, hence (because of their stability under base change) proper. Indeed, replacing Y by the closure of f(X), one can assume that f is dominant. Further, one can assume that Y=Spec B is affine, hence so is X=Spec A. Then the morphism corresponds to an integral extension of rings B ⊂ A. Then the statement is a reformulation of the going up
  Going up and going down
  In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions....
  
   theorem of Cohen-Seidenberg.
• Finite morphisms have finite fibres (i.e. they are quasi-finite
  Quasi-finite morphism
  In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is finite type and satisfies any of the following equivalent conditions:...
  
  ). This follows from the fact that any finite k-algebra, for any field k is an Artinian ring
  Artinian ring
  In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
  
  . Slightly more generally, for a finite surjective morphism f, one has dim X=dim Y.
• Conversely, proper, quasi-finite maps are finite. This is a consequence of the Stein factorization.
• Finite morphisms are both projective and affine.