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Show that $\mathcal{O}_{\mathbb{P^1}}(-1)$ has no nonzero global sections.

I can cover $\mathbb{P}^1$ by two open sets $U_0[1: x_1 / x_0]= [1 :y_1]$ and $U_1[x_0/x_1 : 1] = [z_0 : 1]$

The transition function of the sheaf is given by $g_1 (z_0) = x_1 / x_0 f_o(y_0)$

To show that $\mathcal{O}_{\mathbb{P}^1}(-1)$ as no nonzero global sections, I need to show that there do not exists sections on $g_0 \in \mathcal{O}_{\mathbb{P}^1}(U_0)$ and $g_1 \in \mathcal{O}_{\mathbb{P}^1}(U_1)$ such that

$$ g_0 = x_1/x_0 g_1$$

How would one efficiently go about proving such a statement?

(There is a related question, however, the solutions to this question again states that this sheaf as no nonzero global sections but does next explain why.)

user7090
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1 Answers1

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Note that $\mathcal{O}_{\mathbb{P}^1}(U_0)=R[y_1]=R[\frac{x_1}{x_0}]$ and $\mathcal{O}_{\mathbb{P}^1}(U_1)=R[z_0]=R[\frac{x_0}{x_1}]$. A section $g_0$ is therefore a polynomial in $w:=\frac{x_1}{x_0}$ while a section $g_1$ is a polynomial in $\frac{x_0}{x_1}=w^{-1}$.

Plugging in to the equation $g_0=\frac{x_0}{x_1}g_1$, (note: this is the correct version, your post as of the time of writing of this answer has an incorrect version) we get $\sum_{i=0}^n a_iw^i = w^{-1}\sum_{j=0}^m b_iw^{-i}$. Since every power of $w$ is strictly negative on the RHS and non-negative on the LHS, it is clear that $a_i=b_i=0$.

KReiser
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