Suppose $C$ is an integral nodal curve with one node. It is claimed in the arxiv version of a paper by Bogomolov, Hassett, Tschinkel that the dualizing sheaf and the sheaf of differentials are related by the formula $$\Omega_C \simeq \omega_C \otimes I_p,$$ where $I_p$ is the ideal sheaf of a node: (see p.10). Is this true? I was under the impression that $\Omega_C$ has torsion at the point $p$, and that there was an exact sequence $$ 0 \to T \to \Omega_C \to \mu_* \omega_{C'} \to 0,$$ where $\mu: C' \to C$ is the normalization morphism and $T$ is a sky-scraper sheaf supported at $p$ (proof: argue locally analytically and assume $C$ is given by $st=0$). Yet we have $\mu_* \omega_{C'} \simeq \omega_C \otimes I_p$, so this would appear to be a contradiction.

What am I doing wrong?