The Stacks project

Lemma 15.86.2. 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 15.68.3 is an isomorphism if the following three conditions are satisfied

  1. $L, M$ have finite injective dimension,

  2. $R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)$ has finite tor dimension,

  3. 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 15.86.1 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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