Lemma 15.92.19. Let $I = (f_1, \ldots , f_ r)$ be a finitely generated ideal of a ring $A$. Let $K$ be a derived complete object of $D(A)$. The following are equivalent
$H^ i(K) = 0$ for $i > 0$,
$H^ i(K \otimes _ A^\mathbf {L} A/I) = 0$ for $i > 0$,
$H^ i(K \otimes _ A^\mathbf {L} K_1^\bullet ) = 0$ for $i > 0$ where $K_1^\bullet $ is as in Situation 15.92.15.
Proof. The implication (1) $\Rightarrow $ (2) is always true. The implication (2) $\Rightarrow $ (3) follows from Lemma 15.89.7. Assume (3). For $s = 0, \ldots , r$ consider the complex
where $K_\bullet (A, f_1, \ldots , f_ s)$ is the Koszul complex placed in cohomological degrees $-s, \ldots , 0$. We have a distinguished triangles
by Lemma 15.28.8. Since $K(0) = K$ is derived complete, it follows by ascending induction that each $K(s)$ is a derived complete object of $D(A)$. By descending induction we'll show that $K(s)$ has no nonzero cohomology in degrees $> 0$. Namely, this holds for $s = r$ by our assumption (3). If it holds for $K(s)$ and $s > 0$, then $f_ s : H^ i(K(s - 1)) \to H^ i(K(s - 1))$ is surjective for $i > 0$ and we conclude that $H^ i(K(s - 1))$ vanishes from Lemmas 15.92.6 and 15.92.7. $\square$
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