The Stacks project

Lemma 61.29.2. In the situation above suppose $K$ is in $D_{cons}(X, \Lambda )$ and $X$ is quasi-compact. Set $K_ n = K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$. There exist $a, b$ such that

  1. $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ and $H^ i(K) = 0$ for $i \not\in [a, b]$,

  2. each $K_ n$ has tor amplitude in $[a, b]$,

  3. each $K_ n$ has constructible cohomology sheaves,

  4. each $K_ n = \epsilon ^{-1}L_ n$ for some $L_ n \in D_{ctf}(X_{\acute{e}tale}, \Lambda /I^ n)$ (√Čtale Cohomology, Definition 59.77.1).

Proof. By definition of local having finite tor dimension, we can find $a, b$ such that $K_1$ has tor amplitude in $[a, b]$. Part (2) follows from Cohomology on Sites, Lemma 21.46.9. Then (1) follows as $K$ is derived complete by the description of limits in Cohomology on Sites, Proposition 21.51.2 and the fact that $H^ b(K_{n + 1}) \to H^ b(K_ n)$ is surjective as $K_ n = K_{n + 1} \otimes ^\mathbf {L}_\Lambda \underline{\Lambda /I^ n}$. Part (3) follows from Lemma 61.27.6, Part (4) follows from Lemma 61.27.4 and the fact that $L_ n$ has finite tor dimension because $K_ n$ does (small argument omitted). $\square$


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