Lemma 15.90.7. Let $I$ be a finitely generated ideal of a ring $A$. Let $M$ be a derived complete $A$-module. If $M/IM = 0$, then $M = 0$.

Proof. Assume that $M/IM$ is zero. Let $I = (f_1, \ldots , f_ r)$. Let $i < r$ be the largest integer such that $N = M/(f_1, \ldots , f_ i)M$ is nonzero. If $i$ does not exist, then $M = 0$ which is what we want to show. Then $N$ is derived complete as a cokernel of a map between derived complete modules, see Lemma 15.90.6. By our choice of $i$ we have that $f_{i + 1} : N \to N$ is surjective. Hence

$\mathop{\mathrm{lim}}\nolimits (\ldots \to N \xrightarrow {f_{i + 1}} N \xrightarrow {f_{i + 1}} N)$

is nonzero, contradicting the derived completeness of $N$. $\square$

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