Lemma 56.3.9. Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. If

$F : \text{Mod}_ A \longrightarrow \text{Mod}_ B$

is an $R$-linear equivalence of categories, then there exists an isomorphism $A \to B$ of $R$-algebras and an invertible $B$-module $L$ such that $F$ is isomorphic to the functor $M \mapsto (M \otimes _ A B) \otimes _ B L$.

Proof. We get $A \to B$ and $L$ from Lemma 56.3.8. To finish the proof, we need to show that the $R$-linearity of $F$ forces $A \to B$ to be an $R$-algebra map. We omit the details. $\square$

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