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 :
![{\ displaystyle V \ subset \ mathbb {A} _ {k} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/055cf73e9f5c17af46731aaf3d24503912bc5abf)
![{\ displaystyle I \ subset k [x_ {1}, \ ldots, x_ {n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adadfaa2ece8cb5229e383d8170d88a24039cbb7)
![V = \ {(x_1, \ ldots, x_n) | f (x_1, \ ldots, x_n) = 0 \ text {for all} f \ in I \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca45ae62f98b743657640702ef09fb4f6522a42f)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/055cf73e9f5c17af46731aaf3d24503912bc5abf)
![{\ displaystyle W \ subset \ mathbb {A} _ {k} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/617f03a990eda0de13a0ea1fe2a9006dcf88e63d)
![f \ colon V \ to W](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa58648b9aa9476995b2b41de27a9a591e631f56)
Morphism, if there are polynomials , so for the mapping
![f_1, \ ldots, f_m \ in k [x_1, \ ldots, x_n]](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea74fc730c5b75358788778ea6b5152695c4974)
![{\ displaystyle F \ colon \ mathbb {A} _ {k} ^ {n} \ to \ mathbb {A} _ {k} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbe184954b9f435149b55e87aefc01010b7cc69)
![F \ colon (x_1, \ ldots, x_n) \ mapsto (f_1 (x_1, \ ldots, x_n), \ ldots, f_m (x_1, \ ldots, x_n))](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f932b2f6fb73ebc200979ebf2021fce5bc6fe67)
holds that
![F | _V = f](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5895396be28d8707609a70b32bdfed521af0547)
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
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![k [V]](https://wikimedia.org/api/rest_v1/media/math/render/svg/1741d36e6904bb3d9bb000011dfe16bfe78cc0a1)
![{\ displaystyle k (x_ {1}, \ ldots, x_ {n}) / I (V) \ to k [V]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bec8ab10c021772e7c3a27be283d9cc51ef44021)
where the vanishing ideal of is:
![I (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b795ac49e67e8d9f329c21b929192d8158881d3c)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![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 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942c2a782def9795cdb5c07aa4bd80deed2a3806)
Relation to algebra homomorphisms
Is a morphism
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![\ alpha \ colon V \ to W](https://wikimedia.org/api/rest_v1/media/math/render/svg/2847f26288b1e2986065636a988be6e7e52c56b0)
then
![\ alpha ^ * \ colon k [W] \ to k [V]](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7b96d887cd822fbdbb3ad6e9b3e5b74b38a995)
defined by
![\ alpha ^ * (f) = f \ circ \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f200c21315a8f16cdebdb11ba98eaba1198f571)
a homomorphism of algebras.
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
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 .
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![k [V]](https://wikimedia.org/api/rest_v1/media/math/render/svg/1741d36e6904bb3d9bb000011dfe16bfe78cc0a1)
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.
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
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![Y \ subset \ mathbb A ^ n_k](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2df993d19932d18be54e1e89d1d34efec15530)
![f \ colon Y \ to k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c763329d084fc279a7d5fd5f91eaacd13c3e6adb)
![P \ in Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/f93a7bcc1e270131e4fe42db6837dba44989c1ba)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![P \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/46ad21a5b4042d77a9c21a30af60cf30efbc5d5a)
![g, h \ in k [x_1, \ ldots, x_n]](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5e89e95909ab10eb059305c3a5368212a970f8)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
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![Y \ subset \ mathbb P ^ n_k](https://wikimedia.org/api/rest_v1/media/math/render/svg/73c37c9d38bed442fdfb0b806a7db7af493fb2ed)
![f \ colon Y \ to k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c763329d084fc279a7d5fd5f91eaacd13c3e6adb)
![P \ in Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/f93a7bcc1e270131e4fe42db6837dba44989c1ba)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![P \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/46ad21a5b4042d77a9c21a30af60cf30efbc5d5a)
![g, h \ in k [x_0, x_1, \ ldots, x_n]](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d30b842fb80b4ce3dc3a582465948ba2ca871d8)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
and are not functions on that , but is a well-defined function since and are homogeneous of the same degree.
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
![\ mathbb P ^ n_k](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e85fa8c924852a17194f87d8fd940c9037730a)
![\ frac gh](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3f842bf6d1263b70a383c580137f5022393b4d)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
If a quasi-affine or a quasi-projective variety, then a function is regular if it is regular at every point in .
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![f \ colon Y \ to k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c763329d084fc279a7d5fd5f91eaacd13c3e6adb)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
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.)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![\ mathbb A ^ 1_k](https://wikimedia.org/api/rest_v1/media/math/render/svg/007dfe3406d318b38c0277226847af4c75b7e15d)
Morphisms
In the following, and are affine, quasi-affine, projective or quasi-projective varieties.
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
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:
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![\ phi \ colon X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/980fd1d353c89457722d77bfa1d61d63443990fb)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
- A continuous function is a morphism if it holds for all open subsets that, if is a regular function, then is also regular on .
![\ phi \ colon X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/980fd1d353c89457722d77bfa1d61d63443990fb)
![V \ subset Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/7069d59c5d6883930f887f01f1864e80f2da76df)
![f \ colon V \ to k](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd9585a25f2c87957ba9ef4c2f183278dcbf38d)
![f \ circ \ phi \ colon \ phi ^ {- 1} (V) \ to k](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f0e0183f4ee276b6efa35182f6d55c63c43ed9)
![\ phi ^ {- 1} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4769992755a23ff8e3c0bd8bf92584dac8a1a82)
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.
![\ phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4)
![U \ subset X](https://wikimedia.org/api/rest_v1/media/math/render/svg/c01cf5893c47ae0bfe4df06f73175c8d35bd68fa)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![\ phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![x \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906)
![\ phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
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 ,
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![X: = \ {(x, y) \ in \ mathbb A ^ 2_k | \ y ^ 2 = x ^ 3 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37192b9dc4a6903011359b756068d70256a5d14c)
so is the picture
![f \ colon \ mathbb A ^ 1_k \ to X](https://wikimedia.org/api/rest_v1/media/math/render/svg/eac8b0e1730f00c46b956e487c4bc48d7802d4e2)
![f \ colon (x) \ mapsto (x ^ 2, x ^ 3)](https://wikimedia.org/api/rest_v1/media/math/render/svg/943e6b64efc1cd5cc7808d5b0514f238fd4c7bc3)
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:
![Y = A ^ 1_k \ setminus \ {0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e250429289b6ebd16f6d471f6178b819d74db279)
![f \ colon Y \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/edd65568cb205c538f237f52c9d781810abb7232)
![f \ colon x \ mapsto \ frac 1 x](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9cf37fef945717888ce2442bee8485933c97487)
is an isomorphism for which there is no morphism with![g \ colon A ^ 1_k \ to A ^ 1_k](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ff94b85300d9ee9fa1faf1de39d21e3769fa31)
It applies
-
and
![Y \ cup \ {0 \} = A ^ 1_k](https://wikimedia.org/api/rest_v1/media/math/render/svg/40997ff275d988f993352d33ac2b757305e3848e)
![A ^ 1_k \ cup \ {\ infty \} = P ^ 1_k](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc3a52d6176a776f317331271fc16c8eafbed2c)
For morphism with , so and , on the other hand, applies .
![g \ colon P ^ 1_k \ to P ^ 1_k](https://wikimedia.org/api/rest_v1/media/math/render/svg/860d78c9269fe8b32ebcd7387db76eae73fff033)
![(x: y) \ mapsto (y: x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/59b5f3b64102b474ff6a04de25b076853fd293d9)
![0 \ mapsto \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbdf00f78e6897416f0797036534c66fb1938310)
![\ infty \ mapsto 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/72c4d651853778c45bf02226a1b125c0b5edd61c)
![g_ {| Y} = f](https://wikimedia.org/api/rest_v1/media/math/render/svg/dca7655a474b2d25eb369a7136c449ecf7b16a7c)
An isomorphism from to an affine variety can be given. It is an irreducible polynomial in general and
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![h \ in k [x_1, \ ldots, x_n]](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd01c8b493e947c1c4bc2941f2b576e3d7816915)
![A ^ n_k \ setminus V (f) = \ {(x_1, \ ldots, x_n) \ in A ^ n_k | f (x_1, \ ldots, x_n) \ ne 0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c154038a66fec6165c682a21ec464d15515305)
the corresponding quasi-affine variety, also the hypersurface
![X \ subset A ^ {n + 1} _k](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7435892d038dd1303fb0ae00d4504dce51313a2)
![X: = \ {(x_1, \ ldots, x_ {n + 1}) \ in A ^ {n + 1} _k | x_ {n + 1} f (x_1, \ ldots, x_n) = 1 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c50b86e33080e4f8262d1262b78f87d959d5ce27)
so is the picture
![\ phi \ colon X \ to A ^ n_k \ setminus V (f)](https://wikimedia.org/api/rest_v1/media/math/render/svg/587858a2a61a2a833c370d84e3a3b2677031aee1)
![\ phi \ colon (x_1, \ ldots, x_ {n + 1}) \ mapsto (x_1, \ ldots, x_n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/742eec109007e13c50a97d29065e42d069ce67e1)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/731a06537d05172e3d0bca94f0b1292e38115c38)
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 \ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7cfca7406315505076b56e4b691560c32a4b24)
![{\ displaystyle A_ {k} ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47bed01ddcbf79251cf00c00726f2460f4ab645b)
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