Lemma 15.91.14. Let $A$ be a ring and let $I \subset A$ be an ideal. Let $(K_ n)$ be an inverse system of objects of $D(A)$ such that for all $f \in I$ and $n$ there exists an $e = e(n, f)$ such that $f^ e$ is zero on $K_ n$. Then for $K \in D(A)$ the object $K' = R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} K_ n)$ is derived complete with respect to $I$.

Proof. Since the category of derived complete objects is preserved under $R\mathop{\mathrm{lim}}\nolimits$ it suffices to show that each $K \otimes _ A^\mathbf {L} K_ n$ is derived complete. By assumption for all $f \in I$ there is an $e$ such that $f^ e$ is zero on $K \otimes _ A^\mathbf {L} K_ n$. Of course this implies that $T(K \otimes _ A^\mathbf {L} K_ n, f) = 0$ and we win. $\square$

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