Let $q : C \times X \rightarrow C$ be the projection, where $C$ is a curve and $X$ is a smooth projective variety $X$. Consider the associated derived pushforward $q_* : D^b(X \times C) \rightarrow D^b(C)$. Let $E$ be a sheaf on $X \times C$.
My question is: when is it true that the $k$-th exterior power commutes with derived pushforward, i.e. when is it true that $\Lambda^k q_*(E) \cong q_*(\Lambda^k E)$?
Thanks!