Morphism of algebraic varieties

In algebraic geometry, 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 is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms 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 functions 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.

If X and Y are closed subvarieties of ${\displaystyle \mathbb {A} ^{n}}$ and ${\displaystyle \mathbb {A} ^{m}}$ (so they are affine varieties), then a regular map ${\displaystyle f\colon X\to Y}$ is the restriction of a polynomial map ${\displaystyle \mathbb {A} ^{n}\to \mathbb {A} ^{m}}$. Explicitly, it has the form:

where the ${\displaystyle f_{i}}$s are in the coordinate ring 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 ${\displaystyle f:X\to Y}$ is the same as the restriction of a polynomial map whose components satisfy the defining equations of ${\displaystyle Y}$.

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.

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