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 they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well.
Definition.
If "X" and "Y" are closed subvarieties of A"n" and A"m" (so they are affine varieties), then a regular map ƒ: "X"→"Y" is the restriction of a polynomial map A"n"→A"m". Explicitly, it has the form
where the formula_2s 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 formula_4 is the same as the restriction of a polynomial map whose components satisfy the defining equations of formula_5.
More generally, a map ƒ:"X"→"Y" between two varieties is regular at a point "x" if there is a neighbourhood "U" of "x" and a neighbourhood "V" of ƒ("x") such that ƒ("U") ⊂ "V" and the restricted function ƒ:"U"→"V" is regular as a function on some affine charts of "U" and "V". Then ƒ 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 ƒ:"X"→"Y" is a morphism of affine varieties, then it defines the algebra homomorphism
where formula_7 are the coordinate rings of "X" and "Y"; it is well-defined since formula_8 is a polynomial in elements of formula_9. Conversely, if formula_10 is an algebra homomorphism, then it induces the morphism
given by: writing formula_12
where formula_14 are the images of formula_15's. Note formula_16 as well as formula_17 In particular, "f" is an isomorphism of affine varieties if and only if "f"# is an isomorphism of the coordinate rings.
For example, if "X" is a closed subvariety of an affine variety "Y" and ƒ is the inclusion, then ƒ# is the restriction of regular functions on "Y" to "X". See #Examples below for more examples.
Regular functions.
In the particular case that Y equals A1 the regular map ƒ:"X"→A1 is called a regular function, and are algebraic analogs of smooth functions studied in differential geometry. The ring of regular functions (that is the coordinate ring or more abstractly the ring of global sections of the structure sheaf) is a fundamental object in affine algebraic 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).
A scalar function ƒ:"X"→A1 is regular at a point "x" if, in some open affine neighborhood of "x", it is a rational function that is regular at "x"; i.e., there are regular functions "g", "h" near "x" such that "f" = "g"/"h" and "h" does not vanish at "x". Caution: the condition is for some pair ("g", "h") not for all pairs ("g", "h"); see Examples.
If "X" is a quasi-projective variety; i.e., an open subvariety of a projective variety, then the function field "k"("X") is the same as that of the closure formula_18 of "X" and thus a rational function on "X" is of the form "g"/"h" for some homogeneous elements "g", "h" of the same degree in the homogeneous coordinate ring formula_19 of formula_18 (cf. Projective variety#Variety structure.) 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 formula_19 such that "f" = "g"/"h" and "h" does not vanish at "x". This characterization is sometimes taken as the definition of a regular function.
Comparison with a morphism of schemes.
If "X" = Spec "A" and "Y" = Spec "B" are affine schemes, then each ring homomorphism determines a morphism
by taking the pre-images of prime ideals. All morphisms between affine schemes are of this type and gluing such morphisms gives a morphism of schemes in general.
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. (Proof: If is a morphism, then writing formula_23, we need to show
where formula_25 are the maximal ideals corresponding to the points "x" and "f"("x"); i.e., formula_26. This is immediate.)
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 .
Properties.
A morphism between varieties is continuous with respect to Zariski topologies on the source and the target.
The image of a morphism of varieties need not be open nor closed (for example, the image of formula_46 is neither open nor closed). 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. constructible set.)
A morphism ƒ:"X"→"Y" of algebraic varieties is said to be a "dominant" if it has dense image. For such an "f", if "V" is a nonempty open affine subset of "Y", then there is a nonempty open affine subset "U" of "X" such that ƒ("U") ⊂ "V" and then formula_47 is injective. Thus, the dominant map ƒ induces an injection on the level of function fields:
where the 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".) Conversely, every inclusion of fields formula_49 is induced by a dominant rational map from "X" to "Y". 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".
If "X" is a smooth complete curve (for example, P1) and if "f" is a rational map from "X" to a projective space P"m", then "f" is a regular map "X" → P"m". In particular, when "X" is a smooth complete curve, any rational function on "X" may be viewed as a morphism "X" → P1 and, conversely, such a morphism as a rational function on "X".
On a normal variety (in particular, a smooth variety), a rational function is regular if and only if it has no poles of codimension one. This is an algebraic analog of Hartogs' extension theorem. There is also a relative version of this fact; see .
A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism (a counterexample is given by a Frobenius morphism formula_50.) On the other hand, if "f" is bijective birational and the target space of "f" is a normal variety, then "f" is biregular. (cf. Zariski's main theorem.)
A regular map between complex algebraic varieties is a holomorphic map. (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.) In particular, a regular map into the complex numbers is just a usual holomorphic function (complex-analytic function).
Morphisms to a projective space.
Let
be a morphism from a projective variety to a projective space. Let "x" be a point of "X". Then some "i"-th homogeneous coordinate of "f"("x") is nonzero; say, "i" = 0 for simplicity. Then, by continuity, there is an open affine neighborhood "U" of "x" such that
is a morphism, where "y""i" are the homogeneous coordinates. Note the target space is the affine space A"m" through the identification formula_53. Thus, by definition, the restriction "f" |"U" is given by
where "g""i"'s are regular functions on "U". Since "X" is projective, each "g""i" is a fraction of homogeneous elements of the same degree in the homogeneous coordinate ring "k"["X"] of "X". We can arrange the fractions so that they all have the same homogeneous denominator say "f"0. Then we can write "g""i" = "f""i"/"f"0 for some homogeneous elements "f""i"'s in "k"["X"]. Hence, going back to the homogeneous coordinates,
for all "x" in "U" and by continuity for all "x" in "X" as long as the "f""i"'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 "f""i"'s that do not vanish at "x" simultaneously (see Note at the end of the section.)
In fact, the above description is valid for any quasi-projective variety "X", an open subvariety of a projective variety formula_18; the difference being that "f""i"'s are in the homogeneous coordinate ring of formula_18.
Note: The above does not say a morphism from a projective variety to a projective space is given by a single set of polynomials (unlike the affine case). For example, let "X" be the conic formula_58 in P2. Then two maps formula_59 and formula_60 agree on the open subset formula_61 of "X" (since formula_62) and so defines a morphism formula_63.
Fibers of a morphism.
The important fact is:
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. 14 of "Commutative algebra with a view toward algebraic geometry." 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).
Degree of a finite morphism.
Let "f": "X" → "Y" be a finite surjective morphism between algebraic varieties over a field "k". Then, by definition, the degree of "f" is the degree of the finite field extension of the function field "k"("X") over "f"*"k"("Y"). By generic freeness, there is some nonempty open subset "U" in "Y" such that the restriction of the structure sheaf "O""X" to is free as "O""Y""U"-module. The degree of "f" is then also the rank of this free 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, 
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 formula_65, one gets:
In particular, if "F" is a tensor power formula_67 of a line bundle, then formula_68 and since the support of formula_69 has positive codimension if "q" is positive, comparing the leading terms, one has:
If "f" is étale and "k" is algebraically closed, then each geometric fiber "f"−1("y") consists exactly of deg("f") points.