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Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$.

Question. Is $i^*\mathcal{I}$ still an injective sheaf on $Z$?

The difficulty seems to lie in the fact that $Z$ is closed: I do not know whether $i^*$ has a left-adjoint (and if yes, whether it is an extension by zero).

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    $\begingroup$ This is not true. Take $X=\operatorname{Spec}(A) $, with $A$ a Gorenstein ring of Krull dimension $0$; then $\mathscr{O}_X$ is injective. Now take $Z= \operatorname{Spec}(B) $, where $B$ is a quotient of $A$ which is not Gorenstein; then $\mathscr{O}_Z=i^*\mathscr{O}_X$ is not injective. $\endgroup$
    – abx
    Commented Apr 12, 2021 at 14:12
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    $\begingroup$ And you do know that there is no left adjoint already in the affine case: math.stackexchange.com/questions/3055802/… $\endgroup$
    – Bugs Bunny
    Commented Apr 12, 2021 at 14:14
  • $\begingroup$ Outch! Thank you two very much, for both comments. $\endgroup$
    – Stabilo
    Commented Apr 13, 2021 at 7:14

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