Let $C$ be a curve and $f:C \to \mathbb{P}^1$ a finite morphism. I know that the induced sheaf $\mathcal{A}= f_*(\mathcal{O}_C)$ is coherent.
I want to know why following statements are equivalent:
1) $\mathcal{A}= f_*(\mathcal{O}_C)$ is locally free; therefore therefore for every $u \in \mathbb{P}^1$ there exist an open affine neghboerhood $U= Spec(R) \subset \mathbb{P}^1$ such that $\Gamma(U, f_*{O}_C) \cong R^n$.
2) $C$ don't has any embedded components, where the embedded components are defined as $Emb(\mathcal{O}_C):= \{c \in C \ | \ m_c \in Ass_{\mathcal{O}_{C,c}}(\mathcal{O}_{C,c}) $and $c$ not generic$\} $