Pushforward and sheaf-hom

Question

If $S$ is a surface over the complex numbers $\mathbb{C}$, and $C$ is a curve in $S$, and $i:C\longrightarrow S$ is the inclusion morphism. If $A$ is a line bundle over $C$, then is it true that $Hom_{\mathcal{O}_S}(i_*A,i_*A)=Hom_{\mathcal{O}_C}(A,A)$?

Clearly they are both supported on $C$. But are the sheaves the same too?

Answer 1

If your curve is a local complete intersection, then one can show in general:

Assume $Z$ is a local complete intersection of codimension $m$ in an algebraic variety $Y$. Let $F$ and $G$ be coherent sheaves on $Z$ and assume that $F$ is locally free, then one has:

• $\mathcal{E}xt^k(i_{*}F,i_{*}G)=i_{*}(\Lambda^k N_{Z/Y}\otimes F^{\vee}\otimes G)$ for $0\leq k \leq m$
• $\mathcal{E}xt^k(i_{*}F,i_{*}G)=0$ otherwise

Here $N_{Z/Y}$ is the normal bundle of $Z$ in $Y$.