\longrightarrow {\mathcal F}\longrightarrow \nu _* \nu ^* {\mathcal F}\longrightarrow \iota _* {\mathcal F}|_{\{ p_1, \ldots , p_r \}} \longrightarrow 0$$

[*Remark*]{}. From now on, we may sometimes denote the sheaf $\iota _* {\mathcal F}|_{\{ p_1, \ldots , p_r \}}$ simply by $\oplus {\mathcal F}_{p_i}$, and similarly for the pushforwards of sheaves on irreducible components of a curve.

[*Proof.*]{} Consider the sequence defining the sheaf ${\mathcal Q}$: $$\label{defqu}
0 \longrightarrow {\mathcal O}_C \longrightarrow \nu _* {\mathcal O}_{\tilde C} \longrightarrow {\mathcal Q}\longrightarrow 0$$

Since $\nu $ is an isomorphism away from the inverse image of the $p_i$’s, it follows that ${\mathcal Q}$ is supported at the union of the $p_i$’s. We now want to prove that ${\mathcal Q}_{p_i}$ is a skyscraper sheaf, i.e. that it has length one. Since this is a local property, it is enough to check it when $C$ has a unique node. In this case, $C$ is the nodal union of two smooth ${\mathbb P}^1$’s, and $\tilde C$ is their disjoint union. Since the normalization map is finite, it is affine, and therefore ${\rm H}^j (\nu _* {\mathcal O}_{\tilde C}, C) \simeq {\rm H}^j 
({\mathcal O}_{\tilde C}, \tilde C)$. Therefore the long exact sequence defining ${\mathcal Q}$ is 