# Morphism (variety)

In algebraic geometry, a morphism of varieties is a mapping of varieties with certain regularity properties. An affine variety morphism is a polynomial map. Morphisms of affine varieties clearly correspond to homomorphisms of their coordinate rings . The definition can be generalized to quasi-affine, projective and quasi-projective varieties by locally defining morphisms with the help of regular functions.

Abstract variety morphisms are local sheaf morphisms.

(Note: The designation is not uniform in the literature. Sometimes the term regular mapping is also used for a morphism, not to be confused with regular functions .)

## Definitions

### Affine varieties

${\ displaystyle \ mathbb {A} _ {k} ^ {n}}$denote the n- dimensional affine space over a body k.

A subset is an algebraic set if it is determined by an ideal : ${\ displaystyle V \ subset \ mathbb {A} _ {k} ^ {n}}$${\ displaystyle I \ subset k [x_ {1}, \ ldots, x_ {n}]}$

${\ displaystyle V = \ {(x_ {1}, \ ldots, x_ {n}) | f (x_ {1}, \ ldots, x_ {n}) = 0 {\ text {for all}} f \ in I \}}$

An algebraic set is an affine variety if it cannot be written as a true union of two algebraic sets.

If and are algebraic sets or affine varieties, then a mapping is called ${\ displaystyle V \ subset \ mathbb {A} _ {k} ^ {n}}$${\ displaystyle W \ subset \ mathbb {A} _ {k} ^ {m}}$

${\ displaystyle f \ colon V \ to W}$

Morphism, if there are polynomials , so for the mapping ${\ displaystyle f_ {1}, \ ldots, f_ {m} \ in k [x_ {1}, \ ldots, x_ {n}]}$

${\ displaystyle F \ colon \ mathbb {A} _ {k} ^ {n} \ to \ mathbb {A} _ {k} ^ {m}}$
${\ displaystyle F \ colon (x_ {1}, \ ldots, x_ {n}) \ mapsto (f_ {1} (x_ {1}, \ ldots, x_ {n}), \ ldots, f_ {m} ( x_ {1}, \ ldots, x_ {n}))}$

holds that

${\ displaystyle F | _ {V} = f}$

An isomorphism is a bijective morphism whose inverse mapping is also a morphism. There are bijective morphisms that are not isomorphisms.

The morphisms from to form an algebra, the coordinate ring, which is denoted by. There is a canonical isomorphism ${\ displaystyle V}$${\ displaystyle k}$${\ displaystyle k}$${\ displaystyle k [V]}$

${\ displaystyle k (x_ {1}, \ ldots, x_ {n}) / I (V) \ to k [V]}$

where the vanishing ideal of is: ${\ displaystyle I (V)}$${\ displaystyle V}$

${\ displaystyle I (V): = \ {f \ in k [x_ {1}, \ ldots, x_ {n}] | f (x_ {1}, \ ldots, x_ {n}) = 0 {\ text {for all}} (x_ {1}, \ ldots, x_ {n}) \ in V \}}$

### Relation to algebra homomorphisms

Is a morphism ${\ displaystyle \ alpha}$

${\ displaystyle \ alpha \ colon V \ to W}$

then ${\ displaystyle \ alpha ^ {*}}$

${\ displaystyle \ alpha ^ {*} \ colon k [W] \ to k [V]}$

defined by

${\ displaystyle \ alpha ^ {*} (f) = f \ circ \ alpha}$

a homomorphism of algebras. ${\ displaystyle k}$

This assignment is a contravariant functor from the category of algebraic sets to the category of reduced -algebras of finite type. Every reduced algebra is isomorphic to one . The functor is an equivalence of categories . ${\ displaystyle k}$${\ displaystyle k}$${\ displaystyle k [V]}$

The assignment is also a contravariant functor from the category of affine varieties to the category of zero- divisor - algebras of finite type. This is also an equivalence of categories. ${\ displaystyle k}$

### Affine, quasi-affine, projective and quasi-projective varieties

In order to extend the definition to quasi-affine, projective and quasi-projective varieties, regular functions are first defined in order to then define a morphism locally.

#### Regular functions

If it is a quasi-affine variety, then a function is regular at a point if there is an open neighborhood with and polynomials such that nowhere has zeros and${\ displaystyle Y \ subset \ mathbb {A} _ {k} ^ {n}}$${\ displaystyle f \ colon Y \ to k}$${\ displaystyle P \ in Y}$${\ displaystyle U}$${\ displaystyle P \ in U}$${\ displaystyle g, h \ in k [x_ {1}, \ ldots, x_ {n}]}$${\ displaystyle h}$${\ displaystyle U}$${\ displaystyle f | _ {U} = {\ frac {g} {h}}}$

Is a quasi-projective variety, then a function is regular at a point if there is an open neighborhood with and there are homogeneous polynomials with the same degree, so that nowhere has zeros and${\ displaystyle Y \ subset \ mathbb {P} _ {k} ^ {n}}$${\ displaystyle f \ colon Y \ to k}$${\ displaystyle P \ in Y}$${\ displaystyle U}$${\ displaystyle P \ in U}$${\ displaystyle g, h \ in k [x_ {0}, x_ {1}, \ ldots, x_ {n}]}$${\ displaystyle h}$${\ displaystyle U}$${\ displaystyle f | _ {U} = {\ frac {g} {h}}}$

${\ displaystyle g}$and are not functions on that , but is a well-defined function since and are homogeneous of the same degree. ${\ displaystyle h}$${\ displaystyle \ mathbb {P} _ {k} ^ {n}}$${\ displaystyle {\ frac {g} {h}}}$${\ displaystyle g}$${\ displaystyle h}$

If a quasi-affine or a quasi-projective variety, then a function is regular if it is regular at every point in . ${\ displaystyle Y}$${\ displaystyle f \ colon Y \ to k}$${\ displaystyle Y}$

If the body is identified with the affine space , then a regular function is continuous in the Zariski topology . (Conversely, however, not every continuous mapping is a regular function.) ${\ displaystyle k}$${\ displaystyle \ mathbb {A} _ {k} ^ {1}}$

### Morphisms

In the following, and are affine, quasi-affine, projective or quasi-projective varieties. ${\ displaystyle X}$${\ displaystyle Y}$

These objects naturally carry a topology , namely the Zariski topology , in which the closed sets are exactly the algebraic sets.

A morphism from to is a continuous function that retrieves regular functions from to regular functions from . More accurate: ${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle \ phi \ colon X \ to Y}$${\ displaystyle Y}$${\ displaystyle X}$

• A continuous function is a morphism if it holds for all open subsets that, if is a regular function, then is also regular on .${\ displaystyle \ phi \ colon X \ to Y}$${\ displaystyle V \ subset Y}$${\ displaystyle f \ colon V \ to k}$${\ displaystyle f \ circ \ phi \ colon \ phi ^ {- 1} (V) \ to k}$${\ displaystyle \ phi ^ {- 1} (V)}$

## Rational illustration

A rational mapping is a morphism from an open set to , so that has no continuation on a proper superset of . Is , is called in regularly . A morphism is therefore also called a regular mapping. ${\ displaystyle \ phi}$${\ displaystyle U \ subset X}$${\ displaystyle Y}$${\ displaystyle \ phi}$${\ displaystyle Y}$${\ displaystyle x \ in U}$${\ displaystyle \ phi}$${\ displaystyle x}$

## Examples

### Neil's parable

Neil's parabola in the affine real plane

An isomorphism is bijective and a homeomorphism , but a bijective homeomorphism is not necessarily an isomorphism: Is the semicubical parabola , ${\ displaystyle X}$

${\ displaystyle X: = \ {(x, y) \ in \ mathbb {A} _ {k} ^ {2} | \ y ^ {2} = x ^ {3} \}}$

so is the picture

${\ displaystyle f \ colon \ mathbb {A} _ {k} ^ {1} \ to X}$
${\ displaystyle f \ colon (x) \ mapsto (x ^ {2}, x ^ {3})}$

a bijective homeomorphism that is not an isomorphism since the inverse mapping is not a morphism.

### Quasi-affine varieties

It is not always possible to define morphisms of quasi-affine varieties by restricting their affine major variety, since not every morphism of a quasi-affine variety is a restriction of a morphism of the major variety. The variety is quasi-affine. The morphism: ${\ displaystyle Y = A_ {k} ^ {1} \ setminus \ {0 \}}$

${\ displaystyle f \ colon Y \ to Y}$
${\ displaystyle f \ colon x \ mapsto {\ frac {1} {x}}}$

is an isomorphism for which there is no morphism with${\ displaystyle g \ colon A_ {k} ^ {1} \ to A_ {k} ^ {1}}$${\ displaystyle g_ {| Y} = f}$

It applies

${\ displaystyle Y \ subsetneq A_ {k} ^ {1} \ subsetneq P_ {k} ^ {1}}$ and
${\ displaystyle Y \ cup \ {0 \} = A_ {k} ^ {1}}$
${\ displaystyle A_ {k} ^ {1} \ cup \ {\ infty \} = P_ {k} ^ {1}}$

For morphism with , so and , on the other hand, applies . ${\ displaystyle g \ colon P_ {k} ^ {1} \ to P_ {k} ^ {1}}$${\ displaystyle (x: y) \ mapsto (y: x)}$${\ displaystyle 0 \ mapsto \ infty}$${\ displaystyle \ infty \ mapsto 0}$${\ displaystyle g_ {| Y} = f}$

An isomorphism from to an affine variety can be given. It is an irreducible polynomial in general and ${\ displaystyle Y}$${\ displaystyle h \ in k [x_ {1}, \ ldots, x_ {n}]}$

${\ displaystyle A_ {k} ^ {n} \ setminus V (f) = \ {(x_ {1}, \ ldots, x_ {n}) \ in A_ {k} ^ {n} | f (x_ {1 }, \ ldots, x_ {n}) \ neq 0 \}}$

the corresponding quasi-affine variety, also the hypersurface ${\ displaystyle X \ subset A_ {k} ^ {n + 1}}$

${\ displaystyle X: = \ {(x_ {1}, \ ldots, x_ {n + 1}) \ in A_ {k} ^ {n + 1} | x_ {n + 1} f (x_ {1}, \ ldots, x_ {n}) = 1 \}}$

so is the picture

${\ displaystyle \ phi \ colon X \ to A_ {k} ^ {n} \ setminus V (f)}$
${\ displaystyle \ phi \ colon (x_ {1}, \ ldots, x_ {n + 1}) \ mapsto (x_ {1}, \ ldots, x_ {n})}$

an isomorphism.

If, however, a sub-variety of codimension greater than 1 is removed from an affine variety, then this variety is not affine.

## Images of morphisms

Images of quasi-projective varieties under morphisms are generally not quasi-projective varieties. If one looks at the morphism, for example

${\ displaystyle f \ colon A_ {k} ^ {2} \ to A_ {k} ^ {2}, (x, y) \ mapsto (x, xy),}$

so you get as a picture . This is not a locally closed set in . However, the image is always a constructible set . In general, morphisms map constructible sets to constructible sets. ${\ displaystyle f (A_ {k} ^ {2}) = \ {(0,0) \} \ cup \ {(x, y) \ in A_ {k} ^ {2} \ mid x \ neq 0 \ }}$${\ displaystyle A_ {k} ^ {2}}$

## Individual evidence

1. ^ Harris, Joe: Algebraic geometry. A first course. Corrected reprint of the 1992 original. Graduate Texts in Mathematics, 133. Springer-Verlag, New York, 1995. ISBN 0-387-97716-3
2. ^ Joe Harris : Algebraic Geometry. A first course. Springer, New Your 1992, ISBN 3-540-97716-3 , Theorem 3.16.