Extensions of sheaves with isomorphic middle terms

Question

Let $\mathcal{F}$ and $\mathcal{G}$ be two coherent sheaves on a variety $X/k$. If I know that $\dim_k \operatorname{Ext}^1(\mathcal{F}, \mathcal{G})=1$, and I have two different nontrivial (not splitting) extensions $$ 0 \to \mathcal{G} \to \mathcal{E} \to \mathcal{F} \to 0, $$ and $$ 0 \to \mathcal{G} \to \mathcal{E'} \to \mathcal{F} \to 0, $$ could I conclude that $\mathcal{E} \cong \mathcal{E'}$? Is it true if I additionally assume that $\mathcal{F}$ and $\mathcal{G}$ are line bundles?

Answer 1

Yes, it's true.

Suppose the first extension corresponds to an element $\xi\in\operatorname{Ext}^1(\mathcal{F}, \mathcal{G})$. Then if $\lambda\in k$, the extension corresponding to $\lambda\xi$ can be constructed by taking the pullback along the map $\mathcal{F}\to\mathcal{F}$ given by multiplication by $\lambda$. But since, if $\lambda\neq0$, this is an isomorphism, the resulting extension will be isomorphic, as a short exact sequence, to the original one.