Let $X$ be a irreducible curve which has a unique node, $\pi \colon X^{\prime}\rightarrow X$ be the normalization, and $\mathcal{F}$ be a torsion free sheaf of rank $1$ on $X$.
Then,$\pi^{*}\mathcal{F}/\mbox{torsion}$ is a line bundle on $X^{\prime}$.
I want to show the following lemma,
The natural map $\mathcal{F}\rightarrow \pi_{*}(\pi^{*}\mathcal{F}/\mbox{torsion})$ is injective and has a torsion cokernel.
$\pi$ is an isomorphism outside the node of $X$. Let $x\in X$ be the node, and I want to show $\mathcal{F}_{x}\rightarrow (\pi_{*}(\pi^{*}\mathcal{F}/\mbox{torsion}))_{x}$ is injective, but I don't know how the pushforward $\pi_{*}$ works on the stalk.
I think this may be a dumb question, but I'm stuck. Thanks in advance.