= Local splitting when a quotient sheaf is locally free
In a <short exact sequence of sheaves> $0\to\mathcal F\to\mathcal G\to\mathcal H\to0$, if $\mathcal H$ is <locally free> of finite rank, the sequence splits near every point. Trivialize $\mathcal H$, lift its finitely many basis sections locally, and shrink to one neighbourhood on which all lifts exist. The lifts define a splitting. If $\mathcal F$ is also locally free, then so is $\mathcal G$. Conversely locally free kernel and middle term need not give a locally free quotient: multiplication by $t$ on $\mathcal O_{\operatorname{Spec}k[t]}$ has quotient supported at the origin.
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