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OR:

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 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. M ...
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
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 In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal ...
and birational maps are frequently used as well.

# Definition

If ''X'' and ''Y'' are closed subvarieties 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 In algebra, a polynomial map or polynomial mapping P: V \to W between vector spaces over an infinite field ''k'' is a polynomial in linear functionals with coefficients in ''k''; i.e., it can be written as :P(v) = \sum_ \lambda_(v) \cdots \lambda_ ...
$\mathbb^n\to \mathbb^m$. Explicitly, it has the form: :$f = \left(f_1, \dots, f_m\right)$ where the $f_i$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 idea ...
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 considere ...
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. 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(x ...
s between the coordinate rings: if ''f'':''X''→''Y'' is a morphism of affine varieties, then it defines the algebra homomorphism : where

# 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 function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s studied in differential geometry. The
ring of regular functions 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. ...
(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 idea ...
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 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 ...
is constant (this can be viewed as an algebraic analogue of Liouville's theorem in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
). 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 In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
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 Example may refer to: * '' exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, e ...
. If ''X'' is a
quasi-projective variety In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...
; 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 An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a '' vinculum'', a notation for grouping symbols which is expressed in m ...
/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 An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a '' vinculum'', a notation for grouping symbols which is expressed in m ...
/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 th ...
s, then each ring homomorphism determines a morphism :$\phi^a: X \to Y, \, \mathfrak \mapsto \phi^\left(\mathfrak\right)$ by taking the pre-images of
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s. 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 ...
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 : $\mathfrak_ = \phi^\left(\mathfrak_x\right)$ where $\mathfrak_x, \mathfrak_$ are the
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals con ...
s 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 $g\left(x\right) = \left(x, x^2\right)$. 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 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 ''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 $f\left(x, y\right) =$. *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_\left(x\right) = \mathbf^2 - \$ and so is in
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 In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
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 where $\left(f \otimes 1\right)\left(x, y\right) = f\left(p\left(x, y\right)\right) = f\left(x\right)$.

# 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 $\mathbf^2 \to \mathbf^2, \, \left(x, y\right) \mapsto \left(x, xy\right)$ 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
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is ...
of the
generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic ...
of ''Y'' to that of ''X''.) Conversely, every inclusion of fields $k\left(Y\right) \hookrightarrow k\left(X\right)$ is induced by a dominant
rational map In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal ...
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 In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a sm ...
), 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; 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 $t \mapsto t^p$.) 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 In algebraic geometry, Zariski's main theorem, proved by , is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness ...
.) 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 Alge ...
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 deriv ...
. (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 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 deriv ...
(complex-analytic function).

# Morphisms to a projective space

Let :$f: X \to \mathbf^m$ be a morphism from a
projective variety 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 ...
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 $\left(a_0 : \dots : a_m\right) = \left(1 : a_1 / a_0 : \dots : a_m / a_0\right) \sim \left(a_1 / a_0, \dots, a_m / a_0\right)$. Thus, by definition, the restriction ''f'' , ''U'' is given by :$f, _U\left(x\right) = \left(g_1\left(x\right), \dots, g_m\left(x\right)\right)$ 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, :$f\left(x\right) = \left(f_0\left(x\right) : f_1\left(x\right) : \dots : f_m\left(x\right)\right)$ 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 In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...
''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 $\left(x : y : z\right) \mapsto \left(x : y\right)$ and $\left(x : y : z\right) \mapsto \left(y : z\right)$ agree on the open subset $\$ of ''X'' (since $\left(x : y\right) = \left(xy : y^2\right) = \left(xy: xz\right) = \left(y : z\right)$) 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 incr ...
" 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 In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
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, :$\chi\left(f^* F\right) = \deg\left(f\right) \chi \left(F\right).$ (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 $\operatorname^p\left(Y, R^q f_* f^* F\right) \Rightarrow \operatorname^\left(X, f^* F\right)$, one gets: :$\chi\left(f^* F\right) = \sum_^ \left(-1\right)^ \chi\left(R^q f_* f^* F\right).$ In particular, if ''F'' is a tensor power $L^$ of a line bundle, then $R^q f_*\left(f^* F\right) = R^q f_* \mathcal_X \otimes L^$ and since the support of $R^q f_* \mathcal_X$ has positive codimension if ''q'' is positive, comparing the leading terms, one has: :$\operatorname\left(f^* L\right) = \operatorname\left(f\right) \operatorname\left(L\right)$ (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.

*
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 additio ...
*
Smooth morphism In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) me ...
*
Étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy th ...
s – The algebraic analogue of local diffeomorphisms. *
Resolution of singularities In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characterist ...
* contraction morphism

# References

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