A morphism from an algebraic variety to the affine line is also called a regular function.
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.
is the same as the restriction of a polynomial map whose components satisfy the defining equations of
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
For example, if X is a closed subvariety of an affine variety Y and f is the inclusion, then f# is the restriction of regular functions on Y to X.
In the particular case that Y equals A1 the regular maps f : X→A1 are called regular functions, and are algebraic analogs of smooth functions studied in differential geometry.
The only regular function on a projective variety is constant (this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis).
Then a rational function f on X is regular at a point x if and only if there are some homogeneous elements g, h of the same degree in
[2] If X = Spec A and Y = Spec B are affine schemes, then each ring homomorphism ϕ : B → A determines a morphism by taking the pre-images of prime ideals.
Now, if X, Y are affine varieties; i.e., A, B are integral domains that are finitely generated algebras over an algebraically closed field k, then, working with only the closed points, the above coincides with the definition given at #Definition.
This fact means that the category of affine varieties can be identified with a full subcategory of affine schemes over k. Since morphisms of varieties are obtained by gluing morphisms of affine varieties in the same way morphisms of schemes are obtained by gluing morphisms of affine schemes, it follows that the category of varieties is a full subcategory of the category of schemes over k. For more details, see [1].
A morphism between varieties is continuous with respect to Zariski topologies on the source and the target.
However, one can still say: if f is a morphism between varieties, then the image of f contains an open dense subset of its closure (cf.
A morphism f:X→Y of algebraic varieties is said to be dominant if it has dense image.
Thus, the dominant map f induces an injection on the level of function fields: where the direct limit runs over all nonempty open affine subsets of Y.
(More abstractly, this is the induced map from the residue field of the generic point of Y to that of X.)
[3] Hence, the above construction determines a contravariant-equivalence between the category of algebraic varieties over a field k and dominant rational maps between them and the category of finitely generated field extension of k.[4] If X is a smooth complete curve (for example, P1) and if f is a rational map from X to a projective space Pm, then f is a regular map X → Pm.
On the other hand, if f is bijective birational and the target space of f is a normal variety, then f is biregular.
(There is actually a slight technical difference: a regular map is a meromorphic map whose singular points are removable, but the distinction is usually ignored in practice.)
Then, by continuity, there is an open affine neighborhood U of x such that is a morphism, where yi are the homogeneous coordinates.
Hence, going back to the homogeneous coordinates, for all x in U and by continuity for all x in X as long as the fi's do not vanish at x simultaneously.
If they vanish simultaneously at a point x of X, then, by the above procedure, one can pick a different set of fi's that do not vanish at x simultaneously (see Note at the end of the section.)
In Mumford's red book, the theorem is proved by means of Noether's normalization lemma.
For an algebraic approach where the generic freeness plays a main role and the notion of "universally catenary ring" is a key in the proof, see Eisenbud, Ch.
In fact, the proof there shows that if f is flat, then the dimension equality in 2. of the theorem holds in general (not just generically).
By generic freeness, there is some nonempty open subset U in Y such that the restriction of the structure sheaf OX to f−1(U) is free as OY|U-module.
If f is étale and if X, Y are complete, then for any coherent sheaf F on Y, writing χ for the Euler characteristic, (The Riemann–Hurwitz formula for a ramified covering shows the "étale" here cannot be omitted.)
In general, if f is a finite surjective morphism, if X, Y are complete and F a coherent sheaf on Y, then from the Leray spectral sequence
is the degree of f.) If f is étale and k is algebraically closed, then each geometric fiber f−1(y) consists exactly of deg(f) points.