Lemma 15.97.4. Let $A$ be a ring and $I \subset A$ an ideal. Suppose given $K_ n \in D(A/I^ n)$ and maps $K_{n + 1} \to K_ n$ in $D(A/I^{n + 1})$. Assume

1. $A$ is $I$-adically complete,

2. $K_1$ is a perfect object, and

3. the maps induce isomorphisms $K_{n + 1} \otimes _{A/I^{n + 1}}^\mathbf {L} A/I^ n \to K_ n$.

Then $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is a perfect, derived complete object of $D(A)$ and $K \otimes _ A^\mathbf {L} A/I^ n \to K_ n$ is an isomorphism for all $n$.

Proof. Combine Lemmas 15.97.3 and 15.97.2 (to get derived completeness). $\square$

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