In mathematics, in algebraic geometry, a morphism of schemes is a finite morphism, if Y has an open cover by affine schemes Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...

V_i = SpecB_i

such that for each i,

f ^{− 1}(V_i) = U_i

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

makes A_i a finitely generated module over B_i. In mathematics, a module is a finitely-generated module if it has a finite generating set. ...

Morphisms of finite type

There is another (mostly technical) 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

y³ = x⁴ − z

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 {V_i = SpecB_i} be an open cover of Y by affine schemes, and for each i let {U_ij = SpecA_ij} be an open cover of f ^{− 1}(V_i) by affine schemes. The restriction of f to {U_ij} induces a morphism of rings . The morphism f is called locally of finite type, if A_ij is a finitely generated algebra over B_i (via the above map of rings). If in addition the open cover can be chose to be finite, then f is called of finite type. In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset...

For example, if k is a field, the scheme has a natural morphism to Speck induced by the inclusion of rings This is a morphism of finite type, but if n > 0 then it is not a finite morphism. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

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

• Finite morphisms have finite fibres (i.e. they are quasi-finite).
• Finite morphisms are proper, in particular closed.
• Proper, quasi-finite maps are finite. This is a deep theorem.
• Closed immersions are finite, as they are locally given by , where I is the ideal corresponding to the closed subset.
• Any base-change of a finite morphism is finite, i.e. if is finite and is any morphism, then the canonical morphism is finite. This corresponds to the following algebraic statement: if A is a finitely generated B-module, then is a finitely generated C-module, where is any map. The generators are , where a_i are the generators of A as a B-module.
• The composition of two finite maps is finite.

A morphism of of schemes) is called quasi-finite if for every point the fibre (where is the residue field of and is the canonical morphism) has only a finite number of points. ... This is a glossary of scheme theory. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

See also

Glossary of scheme theory This is a glossary of scheme theory. ...