Algebraic geometry: an exercise Let X be the Riemann sphere with local coordinate $z$ in one chart and $w=1/z$ in the other chart. Let $\omega$ be a meromorphic 1-form on X. Show that if $\omega=f(z)dz$ in the coordinate z, then f must be a rational function of z.
EDIT: I'm sorry but I didn't know the rules of this site and I'm sorry for my awful English. However, I have to solve the following exercise: "Miranda - Algebraic Curves and Riemann Surfaces" pag. 111 ex. IV.1 A, that's to say
Let X be the Riemann sphere with local coordinate $z$ in one chart and $w=1/z$ in the other chart. Let $\omega$ be a meromorphic 1-form on X. Show that if $\omega=f(z)dz$ in the coordinate $z$, then f must be a rational function of z.
Now, my idea is: let $\omega=\{\omega_{i}\}_{i=1,2}$ such that $\omega_{1}=f(z)dz$ and $\omega_{2}=g(w)dw$ where $z$ and $w$ are the local coordinate with respect to the chart $(U_1,h_1)$ and $(U_2,h_2)$ [where $U_1=\mathbb{C}$ and $U_2=\mathbb{C}_{\infty} \setminus\{0\}$]. We know that $\omega_1$ transforms under $h_2 h_1 ^{-1}$ into $\omega_2$, therefore $g(w)=f(1/w)(-1/w^{2})$ is a meromorphic function.
But now I can't conclude the exercise. Can you help me, please, using Miranda's notations?
 A: Hint 1:
$$dw=-\frac1{z^2}\,dz.$$
You seem to have absorbed this part.
Hint 2: Meromorphic means that all the singularities are poles, and thus isolated. In particular any compact set contains a finite number of poles (if any). Can you cover the Riemann sphere by two compact sets? Can you conclude that $f$ must have a finite number of poles? This implies that there exists a polynomial $p(z)$ such that $f(z)p(z)$ is entire. Then show that $f(1/w)p(1/w)$ has a pole at $w=0$. What does that tell you about the Taylor series of $f(z)p(z)$? Can you show that it must be a polynomial?
A: In the context of the book, the answer of Jyrki Lahtonen is kind of reinventing the wheel.

*

*For $\omega$ meromorphic,  we have for $\omega|_{\mathbb C} = f(z) dz$ and $\omega|_{\mathbb C_{\infty} \setminus 0} = g(w) dw$, where I guess $g(w) = f(\frac 1 w) \frac{-1}{w^2}$ on $\mathbb C \setminus 0 (= \mathbb C \cap (\mathbb C_{\infty} \setminus 0$)), that the functions $f: \mathbb C \to \mathbb C$ and $g: \mathbb C_{\infty} \setminus 0 \to \mathbb C$ are meromorphic.


*We can show that for $f: \mathbb C \to \mathbb C$ meromorphic, $f$ extends to $\tilde f: \mathbb C_{\infty} \to \mathbb C$ meromorphic.


*It's proved earlier in Theorem II.2.1 and Example II.1.18 that any rational function $h: \mathbb C \setminus \{\text{zeroes of denom}\} \to \mathbb C$ is equivalent to meromorphic $h_m: \mathbb C_{\infty} \to \mathbb C$.


*Actually, $h_m = \tilde f$ iff $h=f$ such that for $f: \mathbb C  \to \mathbb C$ meromorphic, we have $f: \mathbb C \setminus \{\text{poles(f)}\}  \to \mathbb C$ holomorphic with $\{\text{poles(f)}\} = \{\text{zeroes of denom}\}$
