On The Arithmetic of Elliptic Curves, Joseph H. Silverman, the definition of rational map is given by:
Let $V_1$ and $V_2 \subseteq \mathbb{P}^n$ be projective varieties. A rational map from $V_1$ to $V_2$ is a map of the form $$\varphi : V_1 \rightarrow V_2, \qquad \varphi = [f_0,\ldots,f_n]$$
where the functions $f_0,…,f_n ∈ K(V_1)$ have the property that for every point $P ∈ V_1$ at which $f_0,…,f_n$ are all defined, $$\varphi (P) = f_0(P),…,f_n(P).$$
And Hartshorne, the definition is:
Let $X$ and $Y$ be varieties. A rational map $\phi: X \to Y$ is an equivalence of pairs $(U, \phi_U)$ where $U$ is a nonempty open subset of $X$, and $\phi_U$ is a morphism of $U$ to $Y$, and where $(U, \phi_U)$ and $(V, \phi_V)$ are considered equivalent if $\phi_U$ and $\phi_V$ agree on $U \cap V$.
I wonder if these two definitions are equivalent? So far I can see the first definition satisfies Hartshorne's definition, but how to see if Hartshorne's also agrees with the first definition?
Best Answer
Let's get a handle on the assumptions baked in to each text's use of the word "varieties". Silverman assumes projective and integral, while Hartshorne assumes just integral. So if you fail to enforce "projective" on Hartshorne's definition, then the definitions are not equivalent.
If one assumes that Hartshorne's varieties are in fact projective, then the two definitions coincide. Starting from Hartshorne's definition, we may find a maximal domain of definition $U$ for our morphism: take the union of the open sets in all the pairs. Now I claim that there is a unique (honest) map $f:U\to V_2$ which represents our rational map $\phi:X\to Y$. Consider two pairs $(V,g),(V',g')$ which are representatives of $\phi$. Then $g,g'$ agree on a dense open subset $W\subset V\cap V'$, and as $V_1$ is reduced and $V_2$ separated, we get that they actually agree on all of $V\cap V'$. This means we can glue together $g,g'$ along $W$ to get an honest morphism $f:U\to V_2$. Now we can pick an affine open subsets $W_2\subset V_2$ and $W_1\subset f^{-1}(W_2) \subset V_1$ which gives us a map of coordinate algebras $k[W_2]\to k[W_1]$ which we can then extend to a map on fraction fields $k(W_2)\to k(W_1)$ and recover the $f_i$ from Silverman's definition.
Warning: in general, you can't pick a single formula with which to write down the $f_i$ on the whole domain of definition. Consider the example from Ted's answer here: on the variety $V(xz-yw)\subset\Bbb P^3$, there's a rational map given by $\frac xy$ when $y\neq 0$ and $\frac wz$ when $z\neq 0$. So this rational map is defined away from $V(y,z)$, and it's represented by $f_i=\frac xy = \frac wz$ because these terms are equal in $k(X)$, but they're not the same formula.