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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.

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    $\begingroup$ For homological algebra it is often enough to know that a rich class of injective objects is mapped to injective objects, where rich means that each object has a resolution in the class. For sheaves this role would be played by the class of products of injective skyscraper sheaves. $\endgroup$
    – user1688
    Commented Aug 4, 2016 at 6:20

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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.

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