The Stacks project

Lemma 61.19.11. 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

  1. in the essential image of $\epsilon ^{-1} :D(X_{\acute{e}tale}, \Lambda /I) \to D(X_{pro\text{-}\acute{e}tale}, \Lambda /I)$, and

  2. has tor amplitude in $[a,\infty )$ for some $a \in \mathbf{Z}$,

then (1) and (2) hold for $K_ n$ as an object of $D(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n)$.

Proof. Assertion (2) for $K_ n$ follows from the more general Cohomology on Sites, Lemma 21.46.9. Assertion (1) for $K_ n$ follows by induction on $n$ from 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 isomorphism

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

and the fact proven in Lemma 61.19.8 that the essential image of $\epsilon ^{-1}$ is a triangulated subcategory of $D^+(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n)$. $\square$

Comments (2)

Comment #6338 by Owen on

'The second assertion' is actually the first assertion.

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