Lemma 15.102.3. Let R be a ring. Let K \in D(R) be pseudo-coherent. Let (M_ n) be an inverse system of R-modules. Then R\mathop{\mathrm{lim}}\nolimits K \otimes _ R^\mathbf {L} M_ n = K \otimes _ R^\mathbf {L} R\mathop{\mathrm{lim}}\nolimits M_ n.
Proof. Consider the defining distinguished triangle
R\mathop{\mathrm{lim}}\nolimits M_ n \to \prod M_ n \to \prod M_ n \to R\mathop{\mathrm{lim}}\nolimits M_ n[1]
and apply Lemma 15.65.5. \square
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