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. $\square$
Comments (0)