The Stacks project

This is analogous to [Theorem 2.2.3, six-I].

Lemma 21.28.7. Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ be a morphism of ringed sites. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$ be weak Serre subcategories. Assume

  1. $f$ is flat,

  2. $f^*$ induces an equivalence of categories $\mathcal{A}' \to \mathcal{A}$,

  3. $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism for $\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$,

  4. $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the assumption of Situation 21.25.1,

  5. $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$ and $\mathcal{A}$ satisfy the assumption of Situation 21.25.5.

Then $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal {A}(\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\mathcal {A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$.

Proof. The proof of this lemma is exactly the same as the proof of Lemma 21.28.6 except the reference to Lemma 21.25.4 is replaced by a reference to Lemma 21.25.6. $\square$

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