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
denote the n- dimensional affine space over a body k.
A subset is an algebraic set if it is determined by an ideal :
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
Morphism, if there are polynomials , so for the mapping
holds that
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
where the vanishing ideal of is:
Relation to algebra homomorphisms
Is a morphism
then
defined by
a homomorphism of algebras.
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 .
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.
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
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
and are not functions on that , but is a well-defined function since and are homogeneous of the same degree.
If a quasi-affine or a quasi-projective variety, then a function is regular if it is regular at every point in .
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.)
Morphisms
In the following, and are affine, quasi-affine, projective or quasi-projective varieties.
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:
- A continuous function is a morphism if it holds for all open subsets that, if is a regular function, then is also regular on .
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.
Examples
Neil's parable
An isomorphism is bijective and a homeomorphism , but a bijective homeomorphism is not necessarily an isomorphism: Is the semicubical parabola ,
so is the picture
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:
is an isomorphism for which there is no morphism with
It applies
- and
For morphism with , so and , on the other hand, applies .
An isomorphism from to an affine variety can be given. It is an irreducible polynomial in general and
the corresponding quasi-affine variety, also the hypersurface
so is the picture
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
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.
Individual evidence
- ^ 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
- ^ Joe Harris : Algebraic Geometry. A first course. Springer, New Your 1992, ISBN 3-540-97716-3 , Theorem 3.16.
literature
- Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
- Klaus Hulek: Elementary Algebraic Geometry . Vieweg, Braunschweig / Wiesbaden 2000, ISBN 3-528-03156-5 .
- Robin Hartshorne : Algebraic Geometry , Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9