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 $X$. 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).

Many thanks!

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).

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 $X$. 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 the left-adjoint of $i^*$ (is it an extension by zero?).

Many thanks!

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 $X$. 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).

Many thanks!