Let $X$ be a Gorenstein (not necessarily smooth) projective $\mathbb{C}$-scheme and $S$ another $k$-scheme. Let $I$ be an injective sheaf on $X$. Denote by $p:X \times\_k S \to X$ the natural projection map. Is there any known condition, under which $p^\*I$ is an injective sheaf on $X \times\_k S$? 


If there is no general condition, it would be very helpful if someone could suggest a reference for examples when this phenomenon happens.



When is the pullback of an injective sheaf injective?
Let's say that $X={\rm Spec\,} k$ for a field $k$. Then it is certainly Gorenstein and $k$ is an injective sheaf on $X$. For any $S$ and $p$ as defined in the question, $p^\*k\simeq \mathscr O\_S$. If this is injective, then the injective dimension of $S$ is $0$, and hence $\dim S=0$. 


I am pretty sure that this can be adopted for an arbitrary $X$ to prove something to the effect that what you are asking may only hold if $\dim S=0$ and clearly not even always in that case. 


Sorry. 


