I am working on Problem XI.4. E of Miranda’s book “Algebraic Curves and Riemann Surfaces”.
Let $X=\mathbb{P}^1, p = \infty, \mathcal{O}[n]=\mathcal{O}[n\cdot p]$. Write down a nontrivial extension of $\mathcal{O}[0]$ by $\mathcal{O}[-2]$.
Here’s my (wrong) solution:
By Riemann-Roch theorem, $\dim H^1(-2\cdot p)=1$, thus a nontrivial extension exists.
Denote $\mathcal{F}=\mathcal{O}[-2], \mathcal{G}=\mathcal{O}[0]$, let $\{U_i\}$ be an open cover of $X$ on which $\mathcal{F}$ and $\mathcal{G}$ both trivialize. Assume on $U_i$, the local automorphism is given by \begin{pmatrix} 1 & b_i \\ 0 & 1 \end{pmatrix} where $b_i\in\mathcal{O}(U_i)$.
In particular, choose $U_1=X\setminus\{\infty\}, U_2=X\setminus\{0\}$ and let $z, w$ be local coordinates on $U_1, U_2$ respectively. Then a local generator of $\mathcal{F}$ can be given by $f_1(z)=1$ on $U_1$ and $f_2(w)=w^2$ on $U_2$. A local generator of $\mathcal{G}$ can be given by $g_1=g_2=1$.
The transition functions are $t_{12}=f_1/f_2=1/w^2$ and $s_{12}=1$. Now according to page 363, we know $\{b_i\}$ should satisfy the compatibility condition, i.e., $b_i=s_{ij}b_j/t_{ij}$. In our problem, that is, $b_1(z)=w^2b_2(w)$. However, I don’t think there are $b_i\in\mathcal{O}(U_i)$ which can satisfy this equation.
Could you help me spot what’s wrong in my solution? Many thanks!