TITLE: Different definitions of rational mappings.
QUESTION [8 upvotes]: In the book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by David A. Cox, John Little, Donal O'Shea, the rational mapping is defined as follows.

In the book "Algebraic Geometry: A First Course" by Joe Harris, the rational mapping is defined as follows.

What are differences between these two definitions? What are the regular maps in Definition 4? Thank you very much.

REPLY [3 votes]: Definition 4 only make sense for affine varieties, but if $X$ and $Y$ are affine, then both definitions are (basically) the same.
Starting with $\phi= (\tfrac{f_1}{g_1},\dots,\tfrac{f_n}{g_n})$ in definition 4, then the associated pair $(U,\varphi)$ of definition 7.3 is the following : $U$ is the non-empty open set of $X$ where $g_i \neq 0$ ($\forall i$), and $\varphi$ is the obvious map.
In the other directtion, let $(U, \varphi : U \rightarrow Y)$ as in definition 7.3. Denote $(\varphi_1,\dots,\varphi_n)$ the components of $\varphi$. The functions $\phi_i : U \rightarrow Y$ are regular functions of an open subset of an affine variety. This implies that $\varphi_i$ can be written $\varphi_i=\tfrac{f_i}{g_i}$ (see lemma 2.1 of Harris' book).