Lemma 15.98.2. Let R be a ring. Let K, L, M be objects of D(R). the map
R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(R\mathop{\mathrm{Hom}}\nolimits _ R(K, L), M)
of Lemma 15.73.3 is an isomorphism if the following three conditions are satisfied
L, M have finite injective dimension,
R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) has finite tor dimension,
for every n \in \mathbf{Z} the truncation \tau _{\leq n}K is pseudo-coherent
Proof.
Pick an integer n and consider the distinguished triangle
\tau _{\leq n}K \to K \to \tau _{\geq n + 1}K \to \tau _{\leq n}K[1]
see Derived Categories, Remark 13.12.4. By assumption (3) and Lemma 15.98.1 the map is an isomorphism for \tau _{\leq n}K. Hence it suffices to show that both
R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} \tau _{\geq n + 1}K \quad \text{and}\quad R\mathop{\mathrm{Hom}}\nolimits _ R(R\mathop{\mathrm{Hom}}\nolimits _ R(\tau _{\geq n + 1}K, L), M)
have vanishing cohomology in degrees \leq n - c for some c. This follows immediately from assumptions (2) and (1).
\square
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