The Stacks project

Lemma 96.17.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $.

  1. $f_*\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{Y}_\tau )$ for $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_\tau )$, and

  2. $f_*\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y})$ for $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$.

Proof. This follows formally from the fact that $f^{-1}$ is an exact left adjoint of $f_*$, see Homology, Lemma 12.29.1. $\square$

Comments (2)

Comment #8686 by Anonymous on

Does part (2) of this lemma need a flatness hypothesis as in Tag 20.11.11?

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