The Stacks project

Lemma 21.28.5. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. 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 $\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$.

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. By assumptions (2) and (3) and Lemmas 21.28.4 and 21.28.1 we see that $f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal {A}^+(\mathcal{O})$ is fully faithful. Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. Then we can write $\mathcal{F} = f^*\mathcal{F}'$. Then $Rf_*\mathcal{F} = Rf_* f^*\mathcal{F}' = \mathcal{F}'$. In particular, we have $R^ pf_*\mathcal{F} = 0$ for $p > 0$ and $f_*\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$. Thus for any $K \in D^+_\mathcal {A}(\mathcal{O})$ we see, using the spectral sequence $E_2^{p, q} = R^ pf_*H^ q(K)$ converging to $R^{p + q}f_*K$, that $Rf_*K$ is in $D^+_{\mathcal{A}'}(\mathcal{O}')$. Of course, it also follows from Lemmas 21.28.3 and 21.28.2 that $Rf_* : D_\mathcal {A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')$ is fully faithful. Since $f^*$ and $Rf_*$ are adjoint we then get the result of the lemma, for example by Categories, Lemma 4.24.4. $\square$

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