Lemma 15.99.1. Let $R \to R'$ be a ring map. For $K \in D(R)$ and $M \in D(R')$ there is a canonical isomorphism

**Proof.**
Choose a K-injective complex of $R'$-modules $J^\bullet $ representing $M$. Choose a quasi-isomorphism $J^\bullet \to I^\bullet $ where $I^\bullet $ is a K-injective complex of $R$-modules. Choose a K-flat complex $K^\bullet $ of $R$-modules representing $K$. Consider the map

The map on degree $n$ terms is given by the map

coming from precomposing by $K^{-q} \to K^{-q} \otimes _ R R'$ and postcomposing by $J^ p \to I^ p$. To finish the proof it suffices to show that we get isomorphisms on cohomology groups:

which is true because base change $- \otimes _ R^\mathbf {L} R' : D(R) \to D(R')$ is left adjoint to the restriction functor $D(R') \to D(R)$ by Lemma 15.60.3. $\square$

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