Lemma 15.89.7 (0H82)—The Stacks project

Lemma 15.89.7. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $K$ be an object of $D(R)$ such that $H^ i(K \otimes _ R^\mathbf {L} R/I) = 0$ for $i > 0$. Then

$H^ i(K \otimes _ R^\mathbf {L} R/I^ n) = 0$ for all $n \geq 1$ and $i > 0$,
$H^ i(K \otimes _ R^\mathbf {L} N) = 0$ for any $I$-power torsion $R$-module $N$ and $i > 0$, and
for any $M \in D^ b(R)$ whose cohomology modules $H^ i(M)$ are $I$-power torsion and $0$ for $i > 0$ we have $H^ i(K \otimes _ R^\mathbf {L} M) = 0$ for $i > 0$.

Proof. Proof of (2). We can write $N = \bigcup N[I^ n]$. We have $K \otimes _ R^\mathbf {L} N = \text{hocolim}_ n K \otimes _ R^\mathbf {L} N[I^ n]$ as tensor products commute with colimits (details omitted; hint: represent $K$ by a K-flat complex and compute directly). Hence we may assume $N$ is annihilated by $I^ n$. Consider the $R$-algebra $R' = R/I^ n \oplus N$ where $N$ is an ideal of square zero. It suffices to show that the object $K' = K \otimes _ R^\mathbf {L} R'$ of $D(R')$ has vanishing cohomology in positive degrees. We have a surjection $R' \to R/I$ of $R$-algebras whose kernel $J$ is nilpotent (any product of $n$ elements in the kernel is zero). We have

\[ K \otimes _ R^\mathbf {L} R/I = (K \otimes _ R^\mathbf {L} R') \otimes _{R'}^\mathbf {L} R/I = K' \otimes _{R'}^\mathbf {L} R/I \]

By assumption the complex $K \otimes _ R^\mathbf {L} R/I$ has tor-amplitude in $[-\infty , 0]$. Thus the conclusion by Lemma 15.67.20.

Part (1) follows trivially from part (2). Part (3) follows from part (2), induction on the number of nonzero cohomology modules of $M$, and the distinguished triangles of truncation from Derived Categories, Remark 13.12.4. Details omitted. $\square$

The Stacks project

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