Let $F$ be a vector bundle over a smooth curve $X$, and consider $V$ be the extension : $$0\rightarrow F\rightarrow V \rightarrow \mathcal O_X\rightarrow 0$$ assosciated to an element $e\in H^1(F)$. Consider a quotient $Q$ of $F$, i.e $p:F\rightarrow Q\rightarrow 0$. $p$ induces a map $\tilde p: H^1(F)\rightarrow H^1(Q)$.
Let $f=\tilde p(e)$ and let $W$ the extension associated to $f$, i.e $$0\rightarrow Q\rightarrow W\rightarrow \mathcal O_X\rightarrow 0$$ Question: Is $W$ a quotient of $V$?
Thanks.
Yes: As $\text{H}^1(X;-)\cong\text{Ext}^1_{\text{QCoh}(X)}({\mathscr O}_X,-)$, this is a consequence of the general, explicit description of the action on morphisms of the (Yoneda-)Ext functor $\text{Ext}^1_{\mathscr A}(X,-): {\mathscr A}^{\text{op}}\times{\mathscr A}\to\textsf{Ab}$ for an abelian category ${\mathscr A}$: Namely, if $f: Y\to Y^{\prime}$, then $\text{Ext}_{\mathscr A}^1(X,Y)\to\text{Ext}_{\mathscr A}^1(X,Y^{\prime})$ sends the equivalence class of a short exact sequence $0\to Y\to Q\to X\to 0$ to its pushout along $Y\to Y^{\prime}$. Since epimorphisms are stable under pushout, the resulting exact sequence $0\to Y^{\prime}\to Q^{\prime}\to X\to 0$ is indeed a termwise quotient of $0\to Y\to Q\to X\to 0$ if $Y\to Y^{\prime}$ was an epimorphism.