Lemma 110.68.1. Let $R \to R'$ and $R \to A$ be ring maps. In general there does not exist a functor $T : D(A) \to D(A \otimes _ R R')$ of triangulated categories such that an $A$-module $M$ gives an object $T(M)$ of $D(A \otimes _ R R')$ which maps to $M \otimes _ R^\mathbf {L} R'$ under the map $D(A \otimes _ R R') \to D(R')$.
Proof. See discussion above. $\square$
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