# The Stacks Project

## Tag 09AW

Lemma 15.82.4. 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 $K = R\mathop{\rm lim}\nolimits K_n$ 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 15.82.3 and 15.82.2 (to get derived completeness). $\square$

The code snippet corresponding to this tag is a part of the file more-algebra.tex and is located in lines 22262–22275 (see updates for more information).

\begin{lemma}
\label{lemma-Rlim-perfect-gives-complete}
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
\begin{enumerate}
\item $A$ is $I$-adically complete,
\item $K_1$ is a perfect object, and
\item the maps induce isomorphisms
$K_{n + 1} \otimes_{A/I^{n + 1}}^\mathbf{L} A/I^n \to K_n$.
\end{enumerate}
Then $K = R\lim K_n$ 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 $n$.
\end{lemma}

\begin{proof}
Combine Lemmas \ref{lemma-Rlim-perfect-gives-perfect} and
\ref{lemma-Rlim-pseudo-coherent-gives-complete-pseudo-coherent}
(to get derived completeness).
\end{proof}

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