The Stacks project

Lemma 15.93.9. Let $A$ be a ring derived complete with respect to an ideal $I$. Set $J = \bigcap I^ n$. If $I$ can be generated by $r$ elements then $J^ N = 0$ where $N = 2^ r$.

Proof. When $r = 1$ this is Lemma 15.93.7. Say $I = (f_1, \ldots , f_ r)$ with $r > 1$. By Lemma 15.91.6 the ring $A_ t = A/f_ r^ tA$ is derived complete with respect to $I$ and hence a fortiori derived complete with respect to $I_ t = (f_1, \ldots , f_{r - 1})A_ t$. Observe that $A \to A_ t$ sends $J$ into $J_ t = \bigcap I_ t^ n$. By induction $J_ t^{N/2} = 0$ with $N = 2^ r$. The ideal $\bigcap \mathop{\mathrm{Ker}}(A \to A_ t) = \bigcap f_ r^ t A$ has square zero by the case $r = 1$. This finishes the proof. $\square$

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