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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 geometrica ...
, a morphism between
algebraic varieties 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. ...
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 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 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 In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
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 \mathbb^n and \mathbb^m (so they are
affine varieties 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. ...
), then a regular map f\colon X\to Y is the restriction of a polynomial map \mathbb^n\to \mathbb^m. Explicitly, it has the form: :f = (f_1, \dots, f_m) where the f_is 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'': :k = k _1, \dots, x_nI, 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 f: X \to Y is the same as the restriction of a polynomial map whose components satisfy the defining equations of 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''. *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 :f^: k \to k \, g \mapsto g \circ f where k k /math> are the coordinate rings of ''X'' and ''Y''; it is well-defined since g \circ f = g(f_1, \dots, f_m) is a polynomial in elements of k /math>. Conversely, if \phi: k \to k /math> is an algebra homomorphism, then it induces the morphism :\phi^a: X \to Y given by: writing k = k _1, \dots, y_mJ, :\phi^a = (\phi(\overline), \dots, \phi(\overline)) where \overline_i are the images of y_i's. Note ^ = \phi as well as ^a = f. 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 ''f'' is the inclusion, then ''f''# 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 ''f'':''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 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. ...
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 ''f'':''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 \overline 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 k overline/math> of \overline (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 k overline/math> 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 scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
s, then each ring homomorphism determines a morphism :\phi^a: X \to Y, \, \mathfrak \mapsto \phi^(\mathfrak) 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 algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a ...
in general. Now, if ''X'', ''Y'' are affine varieties; i.e., ''A'', ''B'' are
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
s that are finitely generated algebras over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
''k'', then, working with only the closed points, the above coincides with the definition given at #Definition. (Proof: If is a morphism, then writing \phi = f^, we need to show : \mathfrak_ = \phi^(\mathfrak_x) where \mathfrak_x, \mathfrak_ are the maximal ideals corresponding to the points ''x'' and ''f''(''x''); i.e., \mathfrak_x = \. 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, se


Examples

*The regular functions on A''n'' are exactly the polynomials in ''n'' variables and the regular functions on P''n'' are exactly the constants. *Let ''X'' be the affine curve y = x^2. Then f: X \to \mathbf^1, \, (x, y) \mapsto x is a morphism; it is bijective with the inverse g(x) = (x, x^2). Since ''g'' is also a morphism, ''f'' is an isomorphism of varieties. *Let ''X'' be the affine curve y^2 = x^3 + x^2. Then f: \mathbf^1 \to X, \, t \mapsto (t^2 - 1, t^3 - t) is a morphism. It corresponds to the ring homomorphism f^: k \to k \, g \mapsto g(t^2 - 1, t^3 - t), which is seen to be injective (since ''f'' is surjective). *Continuing the preceding example, let ''U'' = A1 − . Since ''U'' is the complement of the hyperplane ''t'' = 1, ''U'' is affine. The restriction f: U \to X is bijective. But the corresponding ring homomorphism is the inclusion k = k ^2 - 1, t^3 - t\hookrightarrow k , (t - 1)^/math>, which is not an isomorphism and so the restriction ''f'' , ''U'' is not an isomorphism. *Let ''X'' be the affine curve ''x''2 + ''y''2 = 1 and let f(x, y) = . Then ''f'' is a rational function on ''X''. It is regular at (0, 1) despite the expression since, as a rational function on ''X'', ''f'' can also be written as f(x, y) = . *Let . Then ''X'' is an algebraic variety since it is an open subset of a variety. If ''f'' is a regular function on ''X'', then ''f'' is regular on D_(x) = \mathbf^2 - \ and so is in k _(x)= k mathbf^2x^] = k , x^, y/math>. Similarly, it is in k , y, y^/math>. Thus, we can write: f = = where ''g'', ''h'' are polynomials in ''k'' 'x'', ''y'' But this implies ''g'' is divisible by ''x''''n'' and so ''f'' is in fact a polynomial. Hence, the ring of regular functions on ''X'' is just ''k'' 'x'', ''y'' (This also shows that ''X'' cannot be affine since if it were, ''X'' is determined by its coordinate ring and thus ''X'' = A2.) *Suppose \mathbf^1 = \mathbf^1 \cup \ by identifying the points (''x'' : 1) with the points ''x'' on A1 and ∞ = (1 : 0). There is an automorphism σ of P1 given by σ(x : y) = (y : x); in particular, σ exchanges 0 and ∞. If ''f'' is a rational function on P1, then \sigma^(f) = f(1/z) and ''f'' is regular at ∞ if and only if ''f''(1/''z'') is regular at zero. *Taking the function field ''k''(''V'') of an irreducible
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
''V'', the functions ''F'' in the function field may all be realised as morphisms from ''V'' to the projective line over ''k''. (cf. #Properties) The image will either be a single point, or the whole projective line (this is a consequence of the completeness of projective varieties). That is, unless ''F'' is actually constant, we have to attribute to ''F'' the value ∞ at some points of ''V''. *For any algebraic varieties ''X'', ''Y'', the projection p: X \times Y \to X, \, (x, y) \mapsto x is a morphism of varieties. If ''X'' and ''Y'' are affine, then the corresponding ring homomorphism is p^: k \to k \times Y= k \otimes_k k \, f \mapsto f \otimes 1 where (f \otimes 1)(x, y) = f(p(x, y)) = f(x).


Properties

A morphism between varieties is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
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 \mathbf^2 \to \mathbf^2, \, (x, y) \mapsto (x, xy) 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 ''f'':''X''→''Y'' of algebraic varieties is said to be ''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 ''f''(''U'') ⊂ ''V'' and then f^: k \to k /math> is injective. Thus, the dominant map ''f'' induces an injection on the level of function fields: :k(Y) = \varinjlim k \hookrightarrow k(X), \, g \mapsto g \circ f 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 k(Y) \hookrightarrow k(X) 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 In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functio ...
. There is also a relative version of this fact; se

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 In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphi ...
t \mapsto t^p.) 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 In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers. Parshin, Alexei N., and Igor Rostislavovich Shafarevich, eds. ''Algebraic Geometry III: Complex Algeb ...
is a
holomorphic map In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
. (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 :f: X \to \mathbf^m 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 :f: U \to \mathbf^m - \ is a morphism, where ''y''''i'' are the homogeneous coordinates. Note the target space is the affine space A''m'' through the identification (a_0 : \dots : a_m) = (1 : a_1 / a_0 : \dots : a_m / a_0) \sim (a_1 / a_0, \dots, a_m / a_0). Thus, by definition, the restriction ''f'' , ''U'' is given by :f, _U(x) = (g_1(x), \dots, g_m(x)) 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, :f(x) = (f_0(x) : f_1(x) : \dots : f_m(x)) 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 \overline; the difference being that ''f''''i'''s are in the homogeneous coordinate ring of \overline. 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 y^2 = xz in P2. Then two maps (x : y : z) \mapsto (x : y) and (x : y : z) \mapsto (y : z) agree on the open subset \ of ''X'' (since (x : y) = (xy : y^2) = (xy: xz) = (y : z)) and so defines a morphism f: X \to \mathbf^1.


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 In mathematics, a commutative ring ''R'' is catenary if for any pair of prime ideals :''p'', ''q'', any two strictly increasing chains :''p''=''p''0 ⊂''p''1 ... ⊂''p'n''= ''q'' of prime ideals are contained in maximal strictly incre ...
" 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...
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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
, then for any coherent sheaf ''F'' on ''Y'', writing χ for the Euler characteristic, :\chi(f^* F) = \deg(f) \chi (F). (The
Riemann–Hurwitz formula In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramifi ...
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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
and ''F'' a coherent sheaf on ''Y'', then from the Leray spectral sequence \operatorname^p(Y, R^q f_* f^* F) \Rightarrow \operatorname^(X, f^* F), one gets: :\chi(f^* F) = \sum_^ (-1)^ \chi(R^q f_* f^* F). In particular, if ''F'' is a tensor power L^ of a line bundle, then R^q f_*(f^* F) = R^q f_* \mathcal_X \otimes L^ and since the support of R^q f_* \mathcal_X has positive codimension if ''q'' is positive, comparing the leading terms, one has: :\operatorname(f^* L) = \operatorname(f) \operatorname(L) (since the generic rank of f_* \mathcal_X 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.


See also

*
Algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addit ...
* Smooth morphism * Étale morphisms – The algebraic analogue of
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Formal ...
s. * Resolution of singularities *
contraction morphism In algebraic geometry, a contraction morphism is a surjective projective morphism f: X \to Y between normal projective varieties (or projective schemes) such that f_* \mathcal_X = \mathcal_Y or, equivalently, the geometric fibers are all connected ( ...


Notes


Citations


References

* * * *Milne
Algebraic geometry
old version v. 5.xx. * * * {{refend Algebraic varieties Types of functions Functions and mappings