The Stacks project

Lemma 24.35.2. If $(N_ n)$ and $(N'_ n)$ are pro-isomorphic in the derived category as defined above, then for every object $(M_ n)$ of $D(\mathbf{N}, \mathcal{A})$ we have

\[ R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{A_ n}^\mathbf {L} N_ n) = R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{A_ n}^\mathbf {L} N'_ n) \]

in $D(R)$.

Proof. The assumption implies that the inverse system $(M_ n \otimes _{A_ n}^\mathbf {L} N_ n)$ of $D(R)$ is pro-isomorphic (in the usual sense) to the inverse system $(M_ n \otimes _{A_ n}^\mathbf {L} N'_ n)$ of $D(R)$. Hence the result follows from the fact that taking $R\mathop{\mathrm{lim}}\nolimits $ is well defined for inverse systems in the derived category, see discussion in More on Algebra, Section 15.87. $\square$

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