Lemma 61.29.6. Let X be a connected scheme. Let \Lambda be a Noetherian ring and let I \subset \Lambda be an ideal. If K is in D_{cons}(X, \Lambda ) such that K \otimes _\Lambda \underline{\Lambda /I} has locally constant cohomology sheaves, then K is adic lisse (Definition 61.29.4).
Proof. Write K_ n = K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}. We will use the results of Lemma 61.29.2 without further mention. By Cohomology on Sites, Lemma 21.53.5 we see that K_ n has locally constant cohomology sheaves for all n. We have K_ n = \epsilon ^{-1}L_ n some L_ n in D_{ctf}(X_{\acute{e}tale}, \Lambda /I^ n) with locally constant cohomology sheaves. By Étale Cohomology, Lemma 59.77.7 there exist perfect M_ n \in D(\Lambda /I^ n) such that L_ n is étale locally isomorphic to \underline{M_ n}. The maps L_{n + 1} \to L_ n corresponding to K_{n + 1} \to K_ n induces isomorphisms L_{n + 1} \otimes _{\Lambda /I^{n + 1}}^\mathbf {L} \underline{\Lambda /I^ n} \to L_ n. Looking locally on X we conclude that there exist maps M_{n + 1} \to M_ n in D(\Lambda /I^{n + 1}) inducing isomorphisms M_{n + 1} \otimes _{\Lambda /I^{n + 1}} \Lambda /I^ n \to M_ n, see Cohomology on Sites, Lemma 21.53.3. Fix a choice of such maps. By More on Algebra, Lemma 15.97.4 we can find a finite complex M^\bullet of finite projective \Lambda ^\wedge -modules and isomorphisms M^\bullet /I^ nM^\bullet \to M_ n in D(\Lambda /I^ n) compatible with the transition maps. To finish the proof we will show that K is locally isomorphic to
\underline{M^\bullet }^\wedge = \mathop{\mathrm{lim}}\nolimits \underline{M^\bullet /I^ nM^\bullet } = R\mathop{\mathrm{lim}}\nolimits \underline{M^\bullet /I^ nM^\bullet }
Let E^\bullet be the dual complex to M^\bullet , see More on Algebra, Lemma 15.74.15 and its proof. Consider the objects
H_ n = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\Lambda /I^ n}(\underline{M^\bullet /I^ nM^\bullet }, K_ n) = \underline{E^\bullet /I^ nE^\bullet } \otimes _{\Lambda /I^ n}^\mathbf {L} K_ n
of D(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n). Modding out by I^ n defines a transition map H_{n + 1} \to H_ n. Set H = R\mathop{\mathrm{lim}}\nolimits H_ n. Then H is an object of D_{cons}(X, \Lambda ) (details omitted) with H \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n} = H_ n. Choose a covering \{ W_ t \to X\} _{t \in T} with each W_ t affine and weakly contractible. By our choice of M^\bullet we see that
\begin{align*} H_ n|_{W_ t} & \cong R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\Lambda /I^ n}(\underline{M^\bullet /I^ nM^\bullet }, \underline{M^\bullet /I^ nM^\bullet }) \\ & = \underline{ \text{Tot}(E^\bullet /I^ nE^\bullet \otimes _{\Lambda /I^ n} M^\bullet /I^ nM^\bullet ) } \end{align*}
Thus we may apply Lemma 61.29.5 to H = R\mathop{\mathrm{lim}}\nolimits H_ n. We conclude the system H^0(W_ t, H_ n) satisfies Mittag-Leffler. Since for all n \gg 1 there is an element of H^0(W_ t, H_ n) which maps to an isomorphism in
H^0(W_ t, H_1) = \mathop{\mathrm{Hom}}\nolimits (\underline{M^\bullet /IM^\bullet }, K_1)
we find an element (\varphi _{t, n}) in the inverse limit which produces an isomorphism mod I. Then
R\mathop{\mathrm{lim}}\nolimits \varphi _{t, n} : \underline{M^\bullet }^\wedge |_{W_ t} = R\mathop{\mathrm{lim}}\nolimits \underline{M^\bullet /I^ nM^\bullet }|_{W_ t} \longrightarrow R\mathop{\mathrm{lim}}\nolimits K_ n|_{W_ t} = K|_{W_ t}
is an isomorphism. This finishes the proof. \square
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