For a complex algebraic variety $X$ (reduced and of finite type), consider $D^b(Coh(X))$, the bounded derived category of coherent sheaves on $X$.
Question:
(1) Is it true that $F\in D^b(Coh(X))$ is a line bundle (in degree $0$) if and only if $RHom(F, \mathbb{C}_p)\cong \mathbb{C}$ for every closed point $p$, where $\mathbb{C}_p$ is the skyscraper sheaf at $p$? If this is not true in general, what conditions on $X$ can ensure the statement (e.g. $X$ is smooth)?
(2) What happens when the ground field is not $\mathbb{C}$?
