= Affine principal-parts interpolation for a locally free sheaf
Let $X$ be an irreducible <affine curve>, $A=k[X]$, $K=\operatorname{Frac}A$, and $\mathcal F=\widetilde M$ be <locally free>. Given finitely many classes in $M_K/M_{\mathfrak m_P}$, choose one nonzero $a\in A$ clearing denominators of representatives. The ring $A/(a)$ is <Artinian> and is the product of its finitely many local factors. Hence $M/aM\cong\bigoplus_{P\in V(a)}(M/aM)_{\mathfrak m_P}$. Prescribe $av_P$ modulo $aM_{\mathfrak m_P}$ at the requested points and zero at the other zeros of $a$. Lift this tuple to $w\in M$; the section $w/a$ has exactly the requested rational principal parts and no others. This proves surjectivity without assuming the curve is nonsingular.
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