# Example for pullback of stable sheaf not stable

Suppose $C$ is a complete algebraic curve.

Define a coherent locally free sheaf $\mathcal{F}$ over $C$ to be stable if $\mu(\mathcal{E})<\mu(\mathcal{F})$ for any subsheaf $\mathcal{E}$, where $\mu(\mathcal{E})=\text{deg}(\mathcal{E})/\text{rank}(\mathcal{E})$.

Suppose $\tilde{C}\to C$is a degree $d$ covering, I read about that the pull back of a semi-stable sheaf is semi-stable, but the case is not true for stable, is there any examples?

Thanks for comments and reference books!

Let $f \colon \tilde{E} \to E$ be an isogeny of degree $r$ between two elliptic curves, and let $\mathscr{L}$ be a degree $d$ line bundle on $\tilde{E}$, where $(r, \, d)=1$. Then $\mathscr{E} = f_* \mathscr L$ is stable vector bundle on $E$ of rank $r$ and degree $d$. Moreover $$f^* \mathscr{E} = \bigoplus_{\sigma \in G} t_\sigma^* \mathscr{L},$$
where $G \subset \tilde{E}$ is the kernel of $f$ and $t_\sigma \colon \tilde E \to \tilde E$ denotes the translation by $\sigma \in G$.
In particular, $f^* \mathscr{E}$ is semistable but not stable.