hcju/stacksqa huggingface dataset:
By Lemma \ref{lemma-fully-faithful} the functor $\epsilon^{-1}$ is fully faithful and commutes with pullback (restriction) to the strata. Hence $\epsilon^{-1}$ of a constructible \'etale sheaf is a constructible pro-\'etale sheaf. To finish the proof let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_\proetale$ as in Definition \ref{definition-constructible}. There is a canonical map $$ \epsilon^{-1}\epsilon_*\mathcal{F} \longrightarrow \mathcal{F} $$ We will show this map is an isomorphism. This will prove that $\mathcal{F}$ is in the essential image of $\epsilon^{-1}$ and finish the proof (details omitted). \medskip\noindent To prove this we may assume that $X$ is affine. In this case we have a finite partition $X = \coprod_i X_i$ by constructible locally closed strata such that $\mathcal{F}|_{X_i}$ is locally constant of finite type. Let $U \subset X$ be one of the open strata in the partition and let $Z \subset X$ be the reduced induced structure on the complement. By Lemma \ref{lemma-ses-associated-to-open} we have a short exact sequence $$ 0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0 $$ on $X_\proetale$. Functoriality gives a commutative diagram $$ \xymatrix{ 0 \ar[r] & \epsilon^{-1}\epsilon_*j_!j^{-1}\mathcal{F} \ar[r] \ar[d] & \epsilon^{-1}\epsilon_*\mathcal{F} \ar[r] \ar[d] & \epsilon^{-1}\epsilon_*i_*i^{-1}\mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & j_!j^{-1}\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & i_*i^{-1}\mathcal{F} \ar[r] & 0 } $$ By induction on the length of the partition we know that on the one hand $\epsilon^{-1}\epsilon_*i^{-1}\mathcal{F} \to i^{-1}\mathcal{F}$ and $\epsilon^{-1}\epsilon_*j^{-1}\mathcal{F} \to j^{-1}\mathcal{F}$ are isomorphisms and on the other that $i^{-1}\mathcal{F} = \epsilon^{-1}\mathcal{A}$ and $j^{-1}\mathcal{F} = \epsilon^{-1}\mathcal{B}$ for some constructible sheaves of $\Lambda$-modules $\mathcal{A}$ on $Z_\etale$ and $\mathcal{B}$ on $U_\etale$. Then $$ \epsilon^{-1}\epsilon_*j_!j^{-1}\mathcal{F} = \epsilon^{-1}\epsilon_*j_!\epsilon^{-1}\mathcal{B} = \epsilon^{-1}\epsilon_*\epsilon^{-1}j_!\mathcal{B} = \epsilon^{-1}j_!\mathcal{B} = j_!\epsilon^{-1}\mathcal{B} = j_!j^{-1}\mathcal{F} $$ the second equality by Lemma \ref{lemma-jshriek-comparison}, the third equality by Lemma \ref{lemma-fully-faithful}, and the fourth equality by Lemma \ref{lemma-jshriek-comparison} again. Similarly, we have $$ \epsilon^{-1}\epsilon_*i_*i^{-1}\mathcal{F} = \epsilon^{-1}\epsilon_*i_*\epsilon^{-1}\mathcal{A} = \epsilon^{-1}\epsilon_*\epsilon^{-1}i_*\mathcal{A} = \epsilon^{-1}i_*\mathcal{A} = i_*\epsilon^{-1}\mathcal{A} = i_*i^{-1}\mathcal{F} $$ this time using Lemma \ref{lemma-morphism-comparison}. By the five lemma we conclude the vertical map in the middle of the big diagram is an isomorphism.
The categories $D_c(X_\etale, \Lambda)$ and $D_c(X_\proetale, \Lambda)$ are strictly full, saturated, triangulated subcategories of $D(X_\etale, \Lambda)$ and $D(X_\proetale, \Lambda)$ by \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-constructible-abelian} and Lemma \ref{lemma-constructible-serre} and Derived Categories, Section \ref{derived-section-triangulated-sub}. The statement of the lemma follows by combining Lemmas \ref{lemma-compare-derived} and \ref{lemma-compare-constructible}.
Note that $H^i(K \otimes_\Lambda^\mathbf{L} L)$ is the same as $H^i(\tau_{\geq i - 1}K \otimes_\Lambda^\mathbf{L} \tau_{\geq i - 1}L)$. Thus we may assume $K$ and $L$ are bounded. In this case we can apply Lemma \ref{lemma-compare-constructible-derived} to reduce to the case of the \'etale site, see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-tensor-c}.
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 \ref{lemma-tensor-c} 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 \ref{lemma-constructible-serre} we see that each $K_n$ has constructible cohomology sheaves.
Let $R = \bigoplus I^n/I^{n + 1}$. Observe that $R$ is a Noetherian ring. Since each of the sheaves $\mathcal{F}/I^n\mathcal{F}$ is a constructible sheaf of $\Lambda/I^n\Lambda$-modules also $I^n\mathcal{F}/I^{n + 1}\mathcal{F}$ is a constructible sheaf of $\Lambda/I$-modules and hence the pullback of a constructible sheaf $\mathcal{G}_n$ on $X_\etale$ by Lemma \ref{lemma-compare-constructible}. Set $\mathcal{G} = \bigoplus \mathcal{G}_n$. This is a sheaf of $R$-modules on $X_\etale$ and the map $$ \mathcal{G}_0 \otimes_{\Lambda/I} \underline{R} \longrightarrow \mathcal{G} $$ is surjective because the maps $$ \mathcal{F}/I\mathcal{F} \otimes \underline{I^n/I^{n + 1}} \to I^n\mathcal{F}/I^{n + 1}\mathcal{F} $$ are surjective. Hence $\mathcal{G}$ is a constructible sheaf of $R$-modules by \'Etale Cohomology, Proposition \ref{etale-cohomology-proposition-constructible-over-noetherian}. Choose a partition $X = \coprod X_i$ such that $\mathcal{G}|_{X_i}$ is a locally constant sheaf of $R$-modules of finite type (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-constructible-quasi-compact-quasi-separated}). We claim this is a partition as in the lemma. Namely, replacing $X$ by $X_i$ we may assume $\mathcal{G}$ is locally constant. It follows that each of the sheaves  $I^n\mathcal{F}/I^{n + 1}\mathcal{F}$ is locally constant. Using the short exact sequences $$ 0 \to I^n\mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^n\mathcal{F} \to 0 $$ induction and Modules on Sites, Lemma \ref{sites-modules-lemma-kernel-finite-locally-constant} the lemma follows.
Pick a $\Lambda/I^n$-module $M_n$ such that $\mathcal{F}/I^n\mathcal{F} \cong \underline{M_n}$. Since we have the surjections $\mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^n\mathcal{F}$ we conclude that there exist surjections $M_{n + 1} \to M_n$ inducing isomorphisms $M_{n + 1}/I^nM_{n + 1} \to M_n$. Fix a choice of such surjections and set $M = \lim M_n$. Then $M$ is an $I$-adically complete $\Lambda$-module with $M/I^nM = M_n$, see Algebra, Lemma \ref{algebra-lemma-limit-complete}. Since $M_1$ is a finite type $\Lambda$-module (Modules on Sites, Lemma \ref{sites-modules-lemma-locally-constant-finite-type}) we see that $M$ is a finite $\Lambda^\wedge$-module. Consider the sheaves $$ \mathcal{I}_n = \mathit{Isom}(\underline{M_n}, \mathcal{F}/I^n\mathcal{F}) $$ on $X_\proetale$. Modding out by $I^n$ defines a transition map $$ \mathcal{I}_{n + 1} \longrightarrow \mathcal{I}_n $$ By our choice of $M_n$ the sheaf $\mathcal{I}_n$ is a torsor under $$ \mathit{Isom}(\underline{M_n}, \underline{M_n}) = \underline{\text{Isom}_\Lambda(M_n, M_n)} $$ (Modules on Sites, Lemma \ref{sites-modules-lemma-locally-constant}) since $\mathcal{F}/I^n\mathcal{F}$ is (\'etale) locally isomorphic to $\underline{M_n}$. It follows from More on Algebra, Lemma \ref{more-algebra-lemma-hom-systems-ML} that the system of sheaves $(\mathcal{I}_n)$ is Mittag-Leffler. For each $n$ let $\mathcal{I}'_n \subset \mathcal{I}_n$ be the image of $\mathcal{I}_N \to \mathcal{I}_n$ for all $N \gg n$. Then $$ \ldots \to \mathcal{I}'_3 \to \mathcal{I}'_2 \to \mathcal{I}'_1 \to * $$ is a sequence of sheaves of sets on $X_\proetale$ with surjective transition maps. Since $*(X)$ is a singleton (not empty) and since evaluating at $X$ transforms surjective maps of sheaves of sets into surjections of sets, we can pick $s \in \lim \mathcal{I}'_n(X)$. The sections define isomorphisms $\underline{M}^\wedge \to \lim \mathcal{F}/I^n\mathcal{F} = \mathcal{F}$ and the proof is done.
By Lemma \ref{lemma-compare-locally-constant} we have $\mathcal{F}/I^n\mathcal{F} = \epsilon^{-1}\mathcal{G}_n$ for some locally constant sheaf $\mathcal{G}_n$ of $\Lambda/I^n$-modules. By \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-connected-locally-constant} there exists a finite $\Lambda/I^n$-module $M_n$ such that $\mathcal{G}_n$ is locally isomorphic to $\underline{M_n}$. Choose a covering $\{W_t \to X\}_{t \in T}$ with each $W_t$ affine and weakly contractible. Then $\mathcal{F}|_{W_t}$ satisfies the assumptions of Lemma \ref{lemma-weird} and hence $\mathcal{F}|_{W_t} \cong \underline{N_t}^\wedge$ for some finite $\Lambda^\wedge$-module $N_t$. Note that $N_t/I^nN_t \cong M_n$ for all $t$ and $n$. Hence $N_t \cong N_{t'}$ for all $t, t' \in T$, see More on Algebra, Lemma \ref{more-algebra-lemma-isomorphic-completions}. This proves that $\mathcal{F}$ is adic lisse.
Assume that $\mathcal{F}/I\mathcal{F}$ is zero. Let $I = (f_1, \ldots, f_r)$. Let $i < r$ be the largest integer such that $\mathcal{G} = \mathcal{F}/(f_1, \ldots, f_i)\mathcal{F}$ is nonzero. If $i$ does not exist, then $\mathcal{F} = 0$ which is what we want to show. Then $\mathcal{G}$ is derived complete as a cokernel of a map between derived complete modules, see Proposition \ref{proposition-enough-weakly-contractibles}. By our choice of $i$ we have that $f_{i + 1} : \mathcal{G} \to \mathcal{G}$ is surjective. Hence $$ \lim (\ldots \to \mathcal{G} \xrightarrow{f_{i + 1}} \mathcal{G} \xrightarrow{f_{i + 1}} \mathcal{G}) $$ is nonzero, contradicting the derived completeness of $\mathcal{G}$.
Since $X$ is weakly contractible, there exists a finite disjoint open covering $X = \coprod U_i$ such that $\mathcal{F}/I\mathcal{F}|_{U_i}$ is isomorphic to the constant sheaf associated to a finite $\Lambda/I$-module $M_i$. Choose finitely many generators $m_{ij}$ of $M_i$. We can find sections $s_{ij} \in \mathcal{F}(X)$ restricting to $m_{ij}$ viewed as a section of $\mathcal{F}/I\mathcal{F}$ over $U_i$.  Let $t$ be the total number of $s_{ij}$. Then we obtain a map $$ \alpha : \underline{\Lambda}^{\oplus t} \longrightarrow \mathcal{F} $$ which is surjective modulo $I$ by construction. By Lemma \ref{lemma-naive-completion} the derived completion of $\underline{\Lambda}^{\oplus t}$ is the sheaf $(\underline{\Lambda}^\wedge)^{\oplus t}$. Since $\mathcal{F}$ is derived complete we see that $\alpha$ factors through a map $$ \alpha^\wedge : (\underline{\Lambda}^\wedge)^{\oplus t} \longrightarrow \mathcal{F} $$ Then $\mathcal{Q} = \Coker(\alpha^\wedge)$ is a derived complete sheaf of $\Lambda$-modules by Proposition \ref{proposition-enough-weakly-contractibles}. By construction $\mathcal{Q}/I\mathcal{Q} = 0$. It follows from Lemma \ref{lemma-derived-complete-zero} that $\mathcal{Q} = 0$ which is what we wanted to show.
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 \ref{sites-cohomology-lemma-bounded}. Then (1) follows as $K$ is derived complete by the description of limits in Cohomology on Sites, Proposition \ref{sites-cohomology-proposition-enough-weakly-contractibles} 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 \ref{lemma-compare-truncations-constructible}, Part (4) follows from Lemma \ref{lemma-compare-constructible-derived} and the fact that $L_n$ has finite tor dimension because $K_n$ does (small argument omitted).
Say $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}$ is isomorphic to $\underline{E_n}$ for some object $E_n$ of $D(\Lambda/I^n)$. Since $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}$ has finite tor dimension and has finite type cohomology sheaves we see that $E_1$ is perfect (see More on Algebra, Lemma \ref{more-algebra-lemma-perfect}). The transition maps $$ K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^{n + 1}} \to K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n} $$ locally come from (possibly many distinct) maps of complexes $E_{n + 1} \to E_n$ in $D(\Lambda/I^{n + 1})$ see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-locally-constant-map}. For each $n$ choose one such map and observe that it induces an isomorphism $E_{n + 1} \otimes_{\Lambda/I^{n + 1}}^\mathbf{L} \Lambda/I^n \to E_n$ in $D(\Lambda/I^n)$. By More on Algebra, Lemma \ref{more-algebra-lemma-Rlim-perfect-gives-complete} we can find a finite complex $M^\bullet$ of finite projective $\Lambda^\wedge$-modules and isomorphisms $M^\bullet/I^nM^\bullet \to E_n$ in $D(\Lambda/I^n)$ compatible with the transition maps. \medskip\noindent Now observe that for each finite collection of indices $n > m > k$ the triple of maps $$ H^0(X, K \otimes_\Lambda^\mathbf{L} \Lambda/I^n) \to H^0(X, K \otimes_\Lambda^\mathbf{L} \Lambda/I^m) \to H^0(X, K \otimes_\Lambda^\mathbf{L} \Lambda/I^k) $$ is isomorphic to $$ H^0(X, \underline{M^\bullet/I^nM^\bullet}) \to H^0(X, \underline{M^\bullet/I^mM^\bullet}) \to H^0(X, \underline{M^\bullet/I^kM^\bullet}) $$ Namely, choose any isomorphism $$ \underline{M^\bullet/I^nM^\bullet} \to K \otimes_\Lambda^\mathbf{L} \Lambda/I^n $$ induces similar isomorphisms module $I^m$ and $I^k$ and we see that the assertion is true. Thus to prove the lemma it suffices to show that the system $H^0(X, \underline{M^\bullet/I^nM^\bullet})$ has Mittag-Leffler. Since taking sections over $X$ is exact, it suffices to prove that the system of $\Lambda$-modules $$ H^0(M^\bullet/I^nM^\bullet) $$ has Mittag-Leffler. Set $A = \Lambda^\wedge$ and consider the spectral sequence $$ \text{Tor}_{-p}^A(H^q(M^\bullet), A/I^nA) \Rightarrow H^{p + q}(M^\bullet/I^nM^\bullet) $$ By More on Algebra, Lemma \ref{more-algebra-lemma-tor-strictly-pro-zero} the pro-systems $\{\text{Tor}_{-p}^A(H^q(M^\bullet), A/I^nA)\}$ are zero for $p > 0$. Thus the pro-system $\{H^0(M^\bullet/I^nM^\bullet)\}$ is equal to the pro-system $\{H^0(M^\bullet)/I^nH^0(M^\bullet)\}$ and the lemma is proved.
Write $K_n = K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}$. We will use the results of Lemma \ref{lemma-describe-constructible-complexes} without further mention. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-locally-constant-bounded} 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 $D_{ctf}(X_\etale, \Lambda/I^n)$ with locally constant cohomology sheaves. By \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-connected-ctf-locally-constant} there exist perfect $M_n \in D(\Lambda/I^n)$ such that $L_n$ is \'etale 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 \ref{sites-cohomology-lemma-locally-constant-map}. Fix a choice of such maps. By More on Algebra, Lemma \ref{more-algebra-lemma-Rlim-perfect-gives-complete} 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 $$ \underline{M^\bullet}^\wedge = \lim \underline{M^\bullet/I^nM^\bullet} = R\lim \underline{M^\bullet/I^nM^\bullet} $$ Let $E^\bullet$ be the dual complex to $M^\bullet$, see More on Algebra, Lemma \ref{more-algebra-lemma-dual-perfect-complex} and its proof. Consider the objects $$ H_n = R\SheafHom_{\Lambda/I^n}(\underline{M^\bullet/I^nM^\bullet}, K_n) = \underline{E^\bullet/I^nE^\bullet} \otimes_{\Lambda/I^n}^\mathbf{L} K_n $$ of $D(X_\proetale, \Lambda/I^n)$. Modding out by $I^n$ defines a transition map $H_{n + 1} \to H_n$. Set $H = R\lim H_n$. 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\SheafHom_{\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 \ref{lemma-weakly-contractible-locally-constant-ML} to $H = R\lim H_n$. 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 $$ H^0(W_t, H_1) = \Hom(\underline{M^\bullet/IM^\bullet}, K_1) $$ we find an element $(\varphi_{t, n})$ in the inverse limit which produces an isomorphism mod $I$. Then $$ R\lim \varphi_{t, n} : \underline{M^\bullet}^\wedge|_{W_t} = R\lim \underline{M^\bullet/I^nM^\bullet}|_{W_t} \longrightarrow R\lim K_n|_{W_t} = K|_{W_t} $$ is an isomorphism. This finishes the proof.
Namely, we first let $\{V_k \to T\}$ be a covering as in Lemma \ref{lemma-get-many-weakly-contractible}. Then the pro-\'etale coverings $\{T_i \times_T V_k \to V_k\}$ can be refined by a finite disjoint open covering $V_k = V_{k, 1} \amalg \ldots \amalg V_{k, n_k}$, see Lemma \ref{lemma-w-contractible-proetale-cover}. Then $\{V_{k, i} \to T\}$ is a covering of $\Sch_\proetale$ which refines $\{T_i \to T\}_{i \in I}$.
The inclusion functor is fully faithful and continuous. We have seen that $S_\proetale$ and $S_\proetale'$ have fibre products and final objects and that our functor commutes with these (Lemma \ref{lemma-fibre-products-proetale}). It follows from Lemma \ref{lemma-proetale-induced} that the inclusion functor is cocontinuous. Hence the existence of $f$ and $g$ follows from Sites, Lemma \ref{sites-lemma-bigger-site}. The equality in (1) is Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}. Part (2) follows from (1) as $\mathcal{F} = g^{-1}g_*\mathcal{F} = g^{-1}f^{-1}\mathcal{F}$.
Assumptions (b), (c), and (e) of Sites, Lemma \ref{sites-lemma-bigger-site} are immediate for the functors $\Sch_\proetale \to \Sch'_\proetale$ and $(\Sch/S)_\proetale \to (\Sch'/S)_\proetale$. Property (a) holds by Lemma \ref{lemma-proetale-induced}. Property (d) holds because fibre products in the categories $\Sch_\proetale$, $\Sch'_\proetale$ exist and are compatible with fibre products in the category of schemes.
By Lemma \ref{lemma-change-alpha} the inclusion functor $(\Sch/S)_\proetale \to (\Sch'/S)_\proetale$ satisfies the assumptions of Sites, Lemma \ref{sites-lemma-bigger-site}. This implies (2) and (1) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}.
Let $A \to A_w$, resp. $B \to B_w$ be the faithfully flat, ind-Zariski ring map constructed in Lemma \ref{lemma-make-w-local} for $A$, resp.\ $B$. Since $\Spec(B_w)$ is w-local, there exists a unique factorization $A \to A_w \to B_w$ such that $\Spec(B_w) \to \Spec(A_w)$ is w-local by Lemma \ref{lemma-universal}. Note that $A_w \to B_w$ identifies local rings, see Lemma \ref{lemma-local-isomorphism-permanence}. By Lemma \ref{lemma-w-local-morphism-equal-stalks-is-ind-zariski} this means $A_w \to B_w$ is ind-Zariski. Since $B \to B_w$ is faithfully flat, ind-Zariski (Lemma \ref{lemma-make-w-local}) and the composition $A \to B \to B_w$ is ind-Zariski (Lemma \ref{lemma-composition-ind-zariski}) the proposition is proved.
The ring map $A \to B$ induces (separable) algebraic extensions of residue fields, see More on Algebra, Lemma \ref{more-algebra-lemma-weakly-etale-residue-field-extensions}. Thus we may apply Lemma \ref{lemma-get-w-local-algebraic-residue-field-extensions} and choose a diagram $$ \xymatrix{ B \ar[r] & D \\ A \ar[r] \ar[u] & C \ar[u] } $$ with the properties as listed in the lemma. Note that $C \to D$ is weakly \'etale by More on Algebra, Lemma \ref{more-algebra-lemma-weakly-etale-permanence}. Pick a maximal ideal $\mathfrak m \subset D$. By construction this lies over a maximal ideal $\mathfrak m' \subset C$. By More on Algebra, Theorem \ref{more-algebra-theorem-olivier} the ring map $C_{\mathfrak m'} \to D_\mathfrak m$ is an isomorphism. As every point of $\Spec(C)$ specializes to a closed point we conclude that $C \to D$ identifies local rings. Thus Proposition \ref{proposition-maps-wich-identify-local-rings} applies to the ring map $C \to D$. Pick $D \to D'$ faithfully flat and ind-Zariski such that $C \to D'$ is ind-Zariski. Then $B \to D'$ is a solution to the problem posed in the proposition.
Applying Lemma \ref{lemma-get-w-local-algebraic-residue-field-extensions} to $\text{id}_A : A \to A$ we find a faithfully flat, ind-\'etale ring map $A \to C$ such that $C$ is w-local and such that every local ring at a maximal ideal of $C$ is strictly henselian. Choose an extremally disconnected space $T$ and a surjective continuous map $T \to \pi_0(\Spec(C))$, see Topology, Lemma \ref{topology-lemma-existence-projective-cover}. Note that $T$ is profinite. Apply Lemma \ref{lemma-construct-profinite} to find an ind-Zariski ring map $C \to D$ such that $\pi_0(\Spec(D)) \to \pi_0(\Spec(C))$ realizes $T \to \pi_0(\Spec(C))$ and such that $$ \xymatrix{ \Spec(D) \ar[r] \ar[d] & \pi_0(\Spec(D)) \ar[d] \\ \Spec(C) \ar[r] & \pi_0(\Spec(C)) } $$ is cartesian in the category of topological spaces. Note that $\Spec(D)$ is w-local, that $\Spec(D) \to \Spec(C)$ is w-local, and that the set of closed points of $\Spec(D)$ is the inverse image of the set of closed points of $\Spec(C)$, see Lemma \ref{lemma-silly}. Thus it is still true that the local rings of $D$ at its maximal ideals are strictly henselian (as they are isomorphic to the local rings at the corresponding maximal ideals of $C$). It follows from Lemma \ref{lemma-w-local-strictly-henselian-extremally-disconnected} that $D$ is w-contractible.
Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset such that every $U \in \mathcal{B}$ is weakly contractible and every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$. We will use the results of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-w-contractible} and Proposition \ref{sites-cohomology-proposition-enough-weakly-contractibles} without further mention. \medskip\noindent Recall that $R\lim$ commutes with $R\Gamma(U, -)$, see Injectives, Lemma \ref{injectives-lemma-RF-commutes-with-Rlim}. Let $f \in I$. Recall that $T(K, f)$ is the homotopy limit of the system $$ \ldots \xrightarrow{f} K \xrightarrow{f} K \xrightarrow{f} K $$ in $D(\mathcal{C}, \Lambda)$. Thus $$ R\Gamma(U, T(K, f)) = T(R\Gamma(U, K), f). $$ Since we can test isomorphisms of maps between objects of $D(\mathcal{C}, \Lambda)$ by evaluating at $U \in \mathcal{B}$ we conclude an object $K$ of $D(\mathcal{C}, \Lambda)$ is derived complete if and only if for every $U \in \mathcal{B}$ the object $R\Gamma(U, K)$ is derived complete as an object of $D(\Lambda)$. \medskip\noindent The remark above implies that items (1), (5) follow from the corresponding results for modules over rings, see More on Algebra, Lemmas \ref{more-algebra-lemma-hom-from-Af} and \ref{more-algebra-lemma-serre-subcategory}. In the same way (2) can be deduced from More on Algebra, Proposition \ref{more-algebra-proposition-derived-complete-modules} as $(I^n\mathcal{F})(U) = I^n \cdot \mathcal{F}(U)$ for $U \in \mathcal{B}$ (by exactness of evaluating at $U$). \medskip\noindent Proof of (4). The homotopy limit $R\lim \mathcal{F}_n$ is in $D_{comp}(X, \Lambda)$ (see discussion following Algebraic and Formal Geometry, Definition \ref{algebraization-definition-derived-complete}). By part (5) just proved we conclude that $\lim \mathcal{F}_n = H^0(R\lim \mathcal{F}_n)$ is derived complete. Part (3) is a special case of (4). \medskip\noindent Proof of (6) and (7). Follows from Lemma \ref{lemma-naive-completion} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-bounded} and the computation of homotopy limits in Cohomology on Sites, Proposition \ref{sites-cohomology-proposition-enough-weakly-contractibles}.