Lemma 15.91.19. Let $I$ be a finitely generated ideal of a ring $A$. Let $K$ be a derived complete object of $D(A)$. If $K \otimes _ A^\mathbf {L} A/I = 0$, then $K = 0$.
Derived Nakayama
A related result is [Proposition 6.5, Dwyer-Greenlees]. The derived Nakayama lemma can for example be found in Bhatt's 3rd lecture on Prismatic cohomology at Columbia University in Fall 2018 as Section 2 property (2). Leonid Positselski proposed a proof in https://mathoverflow.net/a/331501. However, we follow the proof suggested by Anonymous in the comments.
Proof.
Choose generators $f_1, \ldots , f_ r$ of $I$. Denote $K_ n$ the Koszul complex on $f_1^ n, \ldots , f_ r^ n$ over $A$. Recall that $K_ n$ is bounded and that the cohomology modules of $K_ n$ are annihilated by $f_1^ n, \ldots , f_ r^ n$ and hence by $I^{nr}$. By Lemma 15.88.7 we see that $K \otimes _ A^\mathbf {L} K_ n = 0$. Since $K$ is derived complete by Lemma 15.91.18 we have $K = R\mathop{\mathrm{lim}}\nolimits K \otimes _ A^\mathbf {L} K_ n = 0$ as desired.
$\square$
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