Lemma 52.6.17. Let \mathcal{C} be a site. Let \mathcal{O} \to \mathcal{O}' be a homomorphism of sheaves of rings. Let \mathcal{I} \subset \mathcal{O} be a finite type sheaf of ideals. If \mathcal{O} \to \mathcal{O}' is flat and \mathcal{O}/\mathcal{I} \cong \mathcal{O}'/\mathcal{I}\mathcal{O}', then the restriction functor D(\mathcal{O}') \to D(\mathcal{O}) induces an equivalence D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}') \to D_{comp}(\mathcal{O}, \mathcal{I}).
Proof. Lemma 52.6.7 implies restriction r : D(\mathcal{O}') \to D(\mathcal{O}) sends D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}') into D_{comp}(\mathcal{O}, \mathcal{I}). We will construct a quasi-inverse E \mapsto E'.
Let K \to \mathcal{O} be the morphism of D(\mathcal{O}) constructed in Lemma 52.6.11. Set K' = K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}' in D(\mathcal{O}'). Then K' \to \mathcal{O}' is a map in D(\mathcal{O}') which satisfies the conclusions of Lemma 52.6.11 with respect to \mathcal{I}' = \mathcal{I}\mathcal{O}'. The map K \to r(K') is a quasi-isomorphism by Lemma 52.6.16. Now, for E \in D_{comp}(\mathcal{O}, \mathcal{I}) we set
viewed as an object in D(\mathcal{O}') using the \mathcal{O}'-module structure on K'. Since E is derived complete we have E = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, E), see proof of Proposition 52.6.12. On the other hand, since K \to r(K') is an isomorphism in we see that there is an isomorphism E \to r(E') in D(\mathcal{O}). To finish the proof we have to show that, if E = r(M') for an object M' of D_{comp}(\mathcal{O}', \mathcal{I}'), then E' \cong M'. To get a map we use
where the second arrow uses the map K' \to \mathcal{O}'. To see that this is an isomorphism, one shows that r applied to this arrow is the same as the isomorphism E \to r(E') above. Details omitted. \square
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