有理映射

定义

${\displaystyle (U,f)\sim (U',g)\iff f|_{U\cap U'}=g|_{U\cap U'}}$

${\displaystyle f}$ 是从 ${\displaystyle U}$${\displaystyle V}$${\displaystyle g}$ 是从 ${\displaystyle V}$${\displaystyle W}$ 的有理映射，则一般并不能定义其合成 ${\displaystyle g\circ f}$。但是当 ${\displaystyle f}$ 的像（对某个，因而对每个代表元 ${\displaystyle (U_{0},f_{U_{0}})}$）在 ${\displaystyle V}$ 中稠密时，对每个 ${\displaystyle g}$ 的代表元 ${\displaystyle (V_{0},g_{V_{0}})}$${\displaystyle f_{U_{0}}(U_{0})\cap V_{0}}$ 皆非空，此时可以定义 ${\displaystyle g\circ f:=[f_{U_{0}}^{-1}(V_{0}),g_{V_{0}}\circ f_{U_{0}}]}$

例子

${\displaystyle k}$整环，设 ${\displaystyle V:=\mathbb {A} _{k}^{n}}$${\displaystyle W:=\mathbb {A} _{k}^{m}}$，则从 ${\displaystyle V}$${\displaystyle W}$ 的任何有理映射 ${\displaystyle f}$ 有唯一的表法：

${\displaystyle f=\left({\dfrac {f_{1}(x_{1},\ldots ,x_{n})}{g_{1}(x_{1},\ldots ,x_{n})}},\ldots ,{\dfrac {f_{m}(x_{1},\ldots ,x_{n})}{g_{m}(x_{1},\ldots ,x_{n})}}\right)}$

文献

• Grothendieck, Alexandre; Jean Dieudonné. Éléments de géométrie algébrique 2nd edition. Berlin; New York: Springer-Verlag. 1971. ISBN 978-3-540-05113-8 （法语）.
• Hartshorne, Robin. Algebraic Geoemtry. Berlin; New York: Springer-Verlag. 1977. ISBN 978-0-387-90244-9 （英语）.