In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a morphism between
algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the
affine line
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties rela ...
is also called a regular function.
A regular map whose inverse is also regular is called biregular, and they are
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
s in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on
projective varieties
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
– the weaker condition of a
rational map and
birational maps are frequently used as well.
Definition
If ''X'' and ''Y'' are closed
subvarieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
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, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal ...
of ''X'':
:
where ''I'' is the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
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
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
. 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 homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF( ...
s between the coordinate rings: if ''f'':''X''→''Y'' is a morphism of affine varieties, then it defines the algebra homomorphism
:
where