The Stacks project

Lemma 61.27.6. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. Let $K$ be an object of $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$. Set $K_ n = K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$. If $K_1$ is in $D^-_ c(X_{pro\text{-}\acute{e}tale}, \Lambda /I)$, then $K_ n$ is in $D^-_ c(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n)$ for all $n$.

Proof. Consider the distinguished triangles

\[ K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}} \to K_{n + 1} \to K_ n \to K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}}[1] \]

and the isomorphisms

\[ K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}} = K_1 \otimes _{\Lambda /I}^\mathbf {L} \underline{I^ n/I^{n + 1}} \]

By Lemma 61.27.5 we see that this tensor product has constructible cohomology sheaves (and vanishing when $K_1$ has vanishing cohomology). Hence by induction on $n$ using Lemma 61.27.3 we see that each $K_ n$ has constructible cohomology sheaves. $\square$

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