In
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a morphism between
algebraic varieties is a function between the varieties that is given locally by
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s. It is also called a regular map. A morphism from an algebraic variety to the
affine line is also called a regular function.
A regular map whose inverse is also regular is called biregular, and the biregular maps are the
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on
projective varieties – the concepts of
rational and
birational maps are widely used as well; they are
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
s that are defined locally by
rational fractions instead of polynomials.
An algebraic variety has naturally the structure of a
locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces.
Definition
If ''X'' and ''Y'' are closed subvarieties of
and
(so they are
affine varieties), then a regular map
is the restriction of a
polynomial map . Explicitly, it has the form:
:
where the
s are in the
coordinate ring
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space.
More formally, an affine algebraic set is the set of the common zeros over an algeb ...
of ''X'':
:
where ''I'' is the
ideal defining ''X'' (note: two polynomials ''f'' and ''g'' define the same function on ''X'' if and only if ''f'' − ''g'' is in ''I''). The image ''f''(''X'') lies in ''Y'', and hence satisfies the defining equations of ''Y''. That is, a regular map
is the same as the restriction of a polynomial map whose components satisfy the defining equations of
.
More generally, a map ''f'' : ''X''→''Y'' between two
varieties is regular at a point ''x'' if there is a neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ''f''(''x'') such that ''f''(''U'') ⊂ ''V'' and the restricted function ''f'' : ''U''→''V'' is regular as a function on some affine charts of ''U'' and ''V''. Then ''f'' is called regular, if it is regular at all points of ''X''.
*Note: It is not immediately obvious that the two definitions coincide: if ''X'' and ''Y'' are affine varieties, then a map ''f'' : ''X''→''Y'' is regular in the first sense if and only if it is so in the second sense. Also, it is not immediately clear whether regularity depends on a choice of affine charts (it does not.) This kind of a consistency issue, however, disappears if one adopts the formal definition. Formally, an (abstract) algebraic variety is defined to be a particular kind of a locally
ringed space. When this definition is used, a morphism of varieties is just a morphism of locally ringed spaces.
The composition of regular maps is again regular; thus, algebraic varieties form the
category of algebraic varieties where the morphisms are the regular maps.
Regular maps between affine varieties correspond contravariantly in one-to-one to
algebra homomorphisms between the coordinate rings: if ''f'' : ''X''→''Y'' is a morphism of affine varieties, then it defines the algebra homomorphism
:
where