Lemma 15.99.2. Let R \to R' be a ring map. Let K, M \in D(R). The map (15.99.1.1)
R\mathop{\mathrm{Hom}}\nolimits _ R(K, M) \otimes _ R^\mathbf {L} R' \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _{R'}(K \otimes _ R^\mathbf {L} R', M \otimes _ R^\mathbf {L} R')
is an isomorphism in D(R') in the following cases
K is perfect,
R' is perfect as an R-module,
R \to R' is flat, K is pseudo-coherent, and M \in D^{+}(R), or
R' has finite tor dimension as an R-module, K is pseudo-coherent, and M \in D^{+}(R)
Proof.
We may check the map is an isomorphism after applying the restriction functor D(R') \to D(R). After applying this functor our map becomes the map
R\mathop{\mathrm{Hom}}\nolimits _ R(K, L) \otimes _ R^\mathbf {L} R' \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(K, L \otimes _ R^\mathbf {L} R')
of Lemma 15.73.5. See discussion above the lemma to match the left and right hand sides; in particular, this uses Lemma 15.99.1. Thus we conclude by Lemma 15.98.3.
\square
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