In what condition $\operatorname{Hom}(E,F)$ is of finite dimension?

Question

Let $X$ be a nice (say, proper and smooth) variety over $\mathbb C$, and $E,F$ be coherent sheaves on $X$. Is it true that $\operatorname{Hom}(E,F)$ is always a finite-dimensional vecter space over $\mathbb C$?

If $E,F$ are vector bundles, this follows from $\operatorname{Hom}(E,F)=H^0(E^\vee \otimes F)$. Does this also work for $E,F\in \operatorname{Coh}(X)$?

Answer 1

If $E$ and $F$ are coherent sheaves, then so is the sheaf Hom $\mathcal{Hom}(E,F)$. Since $\operatorname{Hom}(E,F)$ is just the global sections of $\mathcal{Hom}(E,F)$, this implies $\operatorname{Hom}(E,F)$ is finite-dimensional.