Suppose $(S',Q)$ is the germ of a singular algebraic curve curve with normalisation $S \to S'$. Let $\mathcal O_Q'$ be the local ring of $S'$ at $Q$, let $\mathcal O_Q$ be its normalization, and let $\mathfrak c_Q = \operatorname{Ann}(\mathcal O_Q' / \mathcal O_Q) \subset \mathcal O_Q'$ be the conductor.
Let $\Omega_Q'$ be the module of regular differentials (see this question for a definition). Let $\Omega_Q$ be the module of differentials which are regular at every point $P \mapsto Q$ of $S$, i.e. $\Omega_Q' \supset \Omega_Q = \bigcap_{P \mapsto Q} \Omega_P$. Let $\omega \in \Omega_Q'$ be a differential with maximal pole order at every $P \mapsto Q$ (such a thing exists, as Serre proves).
Now consider the map \begin{align} \varphi: \mathcal O_Q' & \to \Omega_Q' \\ f & \mapsto f \cdot \omega, \end{align} and suppose $\overline \varphi: \mathcal O_Q' / \mathfrak c_Q \to \Omega_Q' / \Omega_Q$ is surjective. Serre then claims that any $\beta \in \Omega_Q$ can be written as $g \cdot \omega$ for some $g \in \mathfrak c_Q$.
Q: Why is that last step true? Showing that $\varphi$ is surjective is the whole point of the argument...