exact: $$\label{conor}
\xymatrix{ 0 \ar[r] & {\mathcal C}_f \ar[r] & f^* \Omega ^1 _X \ar[r]^{df} & 
\Omega ^1 _C \ar[r] & {\mathcal Q}_f \ar[r] & 0 }$$ Since the dualizing sheaf $\omega _C$ is locally free, tensoring by $\omega _C$ is exact and taking global sections is left exact. From these remarks we deduce that $${\rm H}^0 \bigl( C, {\mathcal C}_f \otimes \omega _C \bigr) \simeq 
{\rm {\mathbb E}xt } ^2 \bigl( L _f ^\bullet , {\mathcal O}_C \bigr) ^\vee$$ and we conclude that in order to compute the obstruction space of $f$, it is enough to compute the global sections of the sheaf ${\mathcal C}_f \otimes \omega _C$.

The sheaf ${\mathcal C}_f$ defined in (\[conor\]) is the [conormal sheaf of $f$]{}.

We drop the subscript $f$, when the morphism is clear from the context.

A sheaf ${\mathcal F}$ on a scheme of pure dimension one is [pure]{} if the support of every non-zero section has pure dimension one.

It is clear that a locally free sheaf is pure. In fact, any subsheaf of a locally free sheaf is pure, and more generally any subsheaf of a pure sheaf is pure. In particular, the sheaves ${\mathcal C}_f$ defined in (\[conor\]) 