hcju/stacksqa huggingface dataset:
Let $I$ be an injective object of $\mathcal{A}$. Choose an embedding $I \to \underline{M}$ for some $A$-module $M$. As $I$ is injective we see that $\underline{M} = I \oplus F$ for some module-valued functor $F$. Then $M = I(A) \oplus F(A)$ and it follows that $I = \underline{I(A)}$. Thus we see that any injective object is of the form $\underline{I}$ for some $A$-module $I$. It is clear that the module $I$ has to be pure injective since any universally exact sequence $0 \to M \to N \to L \to 0$ gives rise to an exact sequence $0 \to \underline{M} \to \underline{N} \to \underline{L} \to 0$ of $\mathcal{A}$. \medskip\noindent Finally, suppose that $I$ is a pure injective $A$-module. Choose an embedding $\underline{I} \to J$ into an injective object of $\mathcal{A}$ (see Lemma \ref{lemma-enough-injectives}). We have seen above that $J = \underline{I'}$ for some $A$-module $I'$ which is pure injective. As $\underline{I} \to \underline{I'}$ is injective the map $I \to I'$ is universally injective. By assumption on $I$ it splits. Hence $\underline{I}$ is a summand of $J = \underline{I'}$ whence an injective object of the category $\mathcal{A}$.
Let us construct a canonical arrow from right to left. Namely, if $N \to I^\bullet$ is a pure injective resolution, then $M^a \to (I^\bullet)^a$ is an exact complex of (adequate) $\mathcal{O}$-modules. Hence any element of $\text{Pext}^i_A(M, N)$ gives rise to a map $N^a \to M^a[i]$ in $D(\mathcal{O})$, i.e., an element of the group on the left. \medskip\noindent To prove this map is an isomorphism, note that we may replace $\Ext^i_{\textit{Mod}(\mathcal{O})}(M^a, N^a)$ by $\Ext^i_{\textit{Adeq}(\mathcal{O})}(M^a, N^a)$, see Lemma \ref{lemma-ext-adequate}. Let $\mathcal{A}$ be the category of adequate functors on $\textit{Alg}_A$. We have seen that $\mathcal{A}$ is equivalent to $\textit{Adeq}(\mathcal{O})$, see Lemma \ref{lemma-adequate-affine}; see also the proof of Lemma \ref{lemma-right-adjoint-adequate}. Hence now it suffices to prove that $$ \Ext^i_\mathcal{A}(\underline{M}, \underline{N}) = \text{Pext}^i_A(M, N) $$ However, this is clear from Lemma \ref{lemma-pure-injective-injective-adequate} as a pure injective resolution $N \to I^\bullet$ exactly corresponds to an injective resolution of $\underline{N}$ in $\mathcal{A}$.
First choose a scheme $U$ and a surjective \'etale morphism $U \to X$. For each $i$ choose a scheme $W_i$ and a surjective \'etale morphism $W_i \to X_i$. Then $\{W_i \to X\}_{i \in I}$ is a smooth covering which refines $\{X_i \to X\}_{i \in I}$. Hence $\{W_i \times_X U \to U\}_{i \in I}$ is a smooth covering of schemes. By More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-dominates-smooth} we can choose an \'etale covering $\{U_j \to U\}$ which refines $\{W_i \times_X U \to U\}$. Then $\{U_j \to X\}_{j \in J}$ is an \'etale covering refining $\{X_i \to X\}_{i \in I}$.
Denote the functor $u : Y_{spaces, \etale} \to (\textit{Spaces}/X)_\etale$. In other words, given an \'etale morphism $j : U \to Y$ corresponding to an object of $Y_{spaces, \etale}$ we set $u(U \to T) = (f \circ j : U \to S)$. The category $Y_{spaces, \etale}$ has fibre products and equalizers and $u$ commutes with them. It is immediate that $u$ cocontinuous. The functor $u$ is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the Lemma follows from Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers}.
In this case the functor $u : X_{spaces, \etale} \to (\textit{Spaces}/X)_\etale$, in addition to the properties seen in the proof of Lemma \ref{lemma-put-in-T-etale} above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma \ref{sites-lemma-bigger-site}.
The functor $u$ is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma \ref{lemma-put-in-T-etale}). Hence Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers} apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/Y$ and $V/X$ we have $\Mor_X(u(U), V) = \Mor_Y(U, V \times_X Y)$ as desired. Thus we may apply Sites, Lemmas \ref{sites-lemma-have-functor-other-way} and \ref{sites-lemma-have-functor-other-way-morphism} to get the formula for $f_{big, *}$.
The equality $i_f = f_{big} \circ i_Y$ follows from the equality $i_f^{-1} = i_T^{-1} \circ f_{big}^{-1}$ which is clear from the descriptions of these functors above. Thus we see (1). \medskip\noindent The functor $u : X_{spaces, \etale} \to Y_{spaces, \etale}$, $u(U \to X) = (U \times_X Y \to Y)$ was shown to give rise to a morphism of sites and correspong morphism of small \'etale topoi in Properties of Spaces, Lemma \ref{spaces-properties-lemma-functoriality-etale-site}. The description of the pushforward is clear. \medskip\noindent Part (3) follows because $\pi_X$ and $\pi_Y$ are given by the inclusion functors and $f_{spaces, \etale}$ and $f_{big}$ by the base change functors $U \mapsto U \times_X Y$. \medskip\noindent Statement (4) follows from (3) by precomposing with $i_Y$.
Since the diagram is cartesian, we have for $U'/X'$ that $U' \times_{X'} Y' = U' \times_X Y$. Hence both $i_g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\textit{Spaces}/Y)_\etale$ to the sheaf $U' \mapsto \mathcal{F}(U' \times_{X'} Y')$ on $X'_\etale$ (use Lemmas \ref{lemma-put-in-T-etale} and \ref{lemma-morphism-big-etale}). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma \ref{sites-lemma-localize-morphism}.
The functor $u$ is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers} apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/X$ we have $\Mor_X(u(U), V) = \Mor_Y(U, V \times_X Y)$ as desired. Thus we may apply Sites, Lemmas \ref{sites-lemma-have-functor-other-way} and \ref{sites-lemma-have-functor-other-way-morphism} to get the formula for $f_{big, *}$.
We will show that an fppf covering is a ph covering, and then the rest follows from Lemma \ref{lemma-zariski-etale-smooth-syntomic-fppf}. Let $\{X_i \to X\}_{i \in I}$ be an fppf covering of algebraic spaces over a base scheme $S$. Let $U$ be an affine scheme and let $U \to X$ be a morphism. We can refine the fppf covering $\{X_i \times_U U \to U\}_{i \in I}$ by an fppf covering $\{T_i \to U\}_{i \in I}$ where $T_i$ is a scheme (Lemma \ref{lemma-refine-fppf-schemes}). Then we can find a standard ph covering $\{U_j \to U\}_{j = 1, \ldots, m}$ refining $\{T_i \to U\}_{i \in I}$ by More on Morphisms, Lemma \ref{more-morphisms-lemma-fppf-ph} (and the definition of ph coverings for schemes). Thus $\{X_i \to X\}_{i \in I}$ is a ph covering by definition.
Let $U \to X$ be a morphism with $U$ affine. By Chow's lemma (in the weak form given as Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}) we see that there is a surjective proper morphism of schemes $V \to U$ which factors through $Y \times_X U \to U$. Taking any finite affine open cover of $V$ we obtain a standard ph covering of $U$ refining $\{X \times_Y U \to U\}$ as desired.
Part (1) is clear. Consider $g : X' \to X$ and $\{X_i \to X\}_{i\in I}$ a ph covering as in (3). By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-type} the morphisms $X' \times_X X_i \to X'$ are locally of finite type. If $h' : Z \to X'$ is a morphism from an affine scheme towards $X'$, then set $h = g \circ h' : Z \to X$. The assumption on $\{X_i \to X\}_{i\in I}$ means there exists a standard ph covering $\{Z_j \to Z\}_{j = 1, \ldots, n}$ and morphisms $Z_j \to X_{i(j)}$ covering $h$ for certain $i(j) \in I$. By the universal property of the fibre product we obtain morphisms $Z_j \to X' \times_X X_{i(j)}$ over $h'$ also. Hence $\{X' \times_X X_i \to X'\}_{i\in I}$ is a ph covering. This proves (3). \medskip\noindent Let $\{X_i \to X\}_{i\in I}$ and $\{X_{ij} \to X_i\}_{j\in J_i}$ be as in (2). Let $h : Z \to X$ be a morphism from an affine scheme towards $X$. By assumption there exists a standard ph covering $\{Z_j \to Z\}_{j = 1, \ldots, n}$ and morphisms $h_j : Z_j \to X_{i(j)}$ covering $h$ for some indices $i(j) \in I$. By assumption there exist standard ph coverings $\{Z_{j, l} \to Z_j\}_{l = 1, \ldots, n(j)}$ and morphisms $Z_{j, l} \to X_{i(j)j(l)}$ covering $h_j$ for some indices $j(l) \in J_{i(j)}$. By Topologies, Lemma \ref{topologies-lemma-refine-by-standard-ph} the family $\{Z_{j, l} \to Z\}$ can be refined by a standard ph covering. Hence we conclude that $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a ph covering.
We will show that an fppf covering is an fpqc covering, and then the rest follows from Lemma \ref{lemma-zariski-etale-smooth-syntomic-fppf}. Let $\{f_i : U_i \to U\}_{i \in I}$ be an fppf covering of algebraic spaces over $S$. By definition this means that the $f_i$ are flat which checks the first condition of Definition \ref{definition-fpqc-covering}. To check the second, let $V \to U$ be a morphism with $V$ affine. We may choose an \'etale covering $\{V_{ij} \to V \times_U U_i\}$ with $V_{ij}$ affine. Then the compositions $f_{ij} : V_{ij} \to V \times_U U_i \to V$ are flat and locally of finite presentation as compositions of such (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-composition-finite-presentation}, \ref{spaces-morphisms-lemma-composition-flat}, \ref{spaces-morphisms-lemma-etale-flat}, and \ref{spaces-morphisms-lemma-etale-locally-finite-presentation}). Hence these morphisms are open (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fppf-open}) and we see that $|V| = \bigcup_{i \in I} \bigcup_{j \in J_i} f_{ij}(|V_{ij}|)$ is an open covering of $|V|$. Since $|V|$ is quasi-compact, this covering has a finite refinement. Say $V_{i_1j_1}, \ldots, V_{i_Nj_N}$ do the job. Then $\{V_{i_kj_k} \to V\}_{k = 1, \ldots, N}$ is a standard fpqc covering of $V$ refinining the family $\{U_i \times_U V \to V\}$. This finishes the proof.
Part (1) is clear. Consider $g : X' \to X$ and $\{X_i \to X\}_{i\in I}$ an fpqc covering as in (3). By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-flat} the morphisms $X' \times_X X_i \to X'$ are flat. If $h' : Z \to X'$ is a morphism from an affine scheme towards $X'$, then set $h = g \circ h' : Z \to X$. The assumption on $\{X_i \to X\}_{i\in I}$ means there exists a standard fpqc covering $\{Z_j \to Z\}_{j = 1, \ldots, n}$ and morphisms $Z_j \to X_{i(j)}$ covering $h$ for certain $i(j) \in I$. By the universal property of the fibre product we obtain morphisms $Z_j \to X' \times_X X_{i(j)}$ over $h'$ also. Hence $\{X' \times_X X_i \to X'\}_{i\in I}$ is an fpqc covering. This proves (3). \medskip\noindent Let $\{X_i \to X\}_{i\in I}$ and $\{X_{ij} \to X_i\}_{j\in J_i}$ be as in (2). Let $h : Z \to X$ be a morphism from an affine scheme towards $X$. By assumption there exists a standard fpqc covering $\{Z_j \to Z\}_{j = 1, \ldots, n}$ and morphisms $h_j : Z_j \to X_{i(j)}$ covering $h$ for some indices $i(j) \in I$. By assumption there exist standard fpqc coverings $\{Z_{j, l} \to Z_j\}_{l = 1, \ldots, n(j)}$ and morphisms $Z_{j, l} \to X_{i(j)j(l)}$ covering $h_j$ for some indices $j(l) \in J_{i(j)}$. By Topologies, Lemma \ref{topologies-lemma-fpqc-affine-axioms} the family $\{Z_{j, l} \to Z\}$ is a standard fpqc covering. Hence we conclude that $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is an fpqc covering.
If $\{X_i \to X\}_{i \in I}$ is an fpqc covering, then so is $\{U \times_X X_i \to U\}_{i \in I}$ by Lemma \ref{lemma-fpqc}. Assume that $\{U \times_X X_i \to U\}_{i \in I}$ is an fpqc covering. Let $h : Z \to X$ be a morphism from an affine scheme towards $X$. Then we see that $U \times_X Z \to Z$ is a surjective \'etale morphism of schemes, in particular open. Hence we can find finitely many affine opens $W_1, \ldots, W_t$ of $U \times_X Z$ whose images cover $Z$. For each $j$ we may apply the condition that $\{U \times_X X_i \to U\}_{i \in I}$ is an fpqc covering to the morphism $W_j \to U$, and obtain a standard fpqc covering $\{W_{jl} \to W_j\}$ which refines $\{W_j \times_X X_i \to W_j\}_{i \in I}$. Hence $\{W_{jl} \to Z\}$ is a standard fpqc covering of $Z$ (see Topologies, Lemma \ref{topologies-lemma-fpqc-affine-axioms}) which refines $\{Z \times_X X_i \to X\}$ and we win.
We omit the construction of the base change map (this uses only formal properties of derived pushforward and completed derived pullback, compare with Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}). Write $K_n = K \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n}$. By Lemma \ref{lemma-naive-completion} we have $K = R\lim K_n$ because $K$ is derived complete. By Lemmas \ref{lemma-pushforward-Noetherian-case} and \ref{lemma-naive-completion} we can unwind the left hand side $$ Lg_{comp}^* Rf_* K = R\lim Lg^*(Rf_*K)\otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n} = R\lim Lg^* Rf_* K_n $$ the last equality because $\Lambda/I^n$ is a perfect module and the projection formula (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-projection-formula}). Using Lemma \ref{lemma-pushforward-Noetherian-case} we can unwind the right hand side $$ Rf'_* L(g')^*_{comp} K = Rf'_* R\lim L(g')^* K_n  = R\lim Rf'_* L(g')^* K_n $$ the last equality because $Rf'_*$ commutes with $R\lim$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Rf-commutes-with-Rlim}). Thus it suffices to show the maps $$ Lg^* Rf_* K_n \longrightarrow Rf'_* L(g')^* K_n $$ are isomorphisms. By Lemma \ref{lemma-compare-derived} and our second condition we can write $K_n = \epsilon^{-1}L_n$ for some $L_n \in D^+(X_\etale, \Lambda/I^n)$. By Lemma \ref{lemma-morphism-comparison} and the fact that $\epsilon^{-1}$ commutes with pullbacks we obtain $$ Lg^* Rf_* K_n = Lg^* Rf_* \epsilon^*L_n = Lg^* \epsilon^{-1} Rf_* L_n = \epsilon^{-1} Lg^* Rf_* L_n $$ and $$ Rf'_* L(g')^* K_n = Rf'_* L(g')^* \epsilon^{-1} L_n = Rf'_* \epsilon^{-1} L(g')^* L_n = \epsilon^{-1} Rf'_* L(g')^* L_n $$ (this also uses that $L_n$ is bounded below). Finally, by the proper base change theorem for \'etale cohomology (\'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-proper-base-change}) we have $$ Lg^* Rf_* L_n = Rf'_* L(g')^* L_n $$ (again using that $L_n$ is bounded below) and the theorem is proved.
Note that $Y$ is a closed subspace of $X \times T$ as $\pi_0(X)$ is a profinite space hence Hausdorff (use Topology, Lemmas \ref{topology-lemma-spectral-pi0} and \ref{topology-lemma-fibre-product-closed}). Since $X \times T$ is spectral (Topology, Lemma \ref{topology-lemma-product-spectral-spaces}) it follows that $Y$ is spectral (Topology, Lemma \ref{topology-lemma-spectral-sub}). Let $Y \to \pi_0(Y) \to T$ be the canonical factorization (Topology, Lemma \ref{topology-lemma-space-connected-components}). It is clear that $\pi_0(Y) \to T$ is surjective. The fibres of $Y \to T$ are homeomorphic to the fibres of $X \to \pi_0(X)$. Hence these fibres are connected. It follows that $\pi_0(Y) \to T$ is injective. We conclude that $\pi_0(Y) \to T$ is a homeomorphism by Topology, Lemma \ref{topology-lemma-bijective-map}. \medskip\noindent Next, assume that $X$ is w-local and let $X_0 \subset X$ be the set of closed points. The inverse image $Y_0 \subset Y$ of $X_0$ in $Y$ maps bijectively onto $T$ as $X_0 \to \pi_0(X)$ is a bijection by Lemma \ref{lemma-spectral-split}. Moreover, $Y_0$ is quasi-compact as a closed subset of the spectral space $Y$. Hence $Y_0 \to \pi_0(Y) = T$ is a homeomorphism by Topology, Lemma \ref{topology-lemma-bijective-map}. It follows that all points of $Y_0$ are closed in $Y$. Conversely, if $y \in Y$ is a closed point, then it is closed in the fibre of $Y \to \pi_0(Y) = T$ and hence its image $x$ in $X$ is closed in the (homeomorphic) fibre of $X \to \pi_0(X)$. This implies $x \in X_0$ and hence $y \in Y_0$. Thus $Y_0$ is the collection of closed points of $Y$ and for each $y \in Y_0$ the set of generalizations of $y$ is the fibre of $Y \to \pi_0(Y)$. The lemma follows.
The map $A \to A_w$ is ind-Zariski by construction. For every $E$ the morphism $Z_E \to X$ is a bijection, hence (2). As $Z \subset X_w$ we conclude $X_w \to X$ is surjective and $A \to A_w$ is faithfully flat by Algebra, Lemma \ref{algebra-lemma-ff-rings}. This proves (1). \medskip\noindent Suppose that $y \in X_w$, $y \not \in Z$. Then there exists an $E$ such that the image of $y$ in $X_E$ is not contained in $Z_E$. Then for all $E \subset E'$ also $y$ maps to an element of $X_{E'}$ not contained in $Z_{E'}$. Let $T_{E'} \subset X_{E'}$ be the reduced closed subscheme which is the closure of the image of $y$. It is clear that $T = \lim_{E \subset E'} T_{E'}$ is the closure of $y$ in $X_w$. For every $E \subset E'$ the scheme $T_{E'} \cap Z_{E'}$ is nonempty by construction of $X_{E'}$. Hence $\lim T_{E'} \cap Z_{E'}$ is nonempty and we conclude that $T \cap Z$ is nonempty. Thus $y$ is not a closed point. It follows that every closed point of $X_w$ is in $Z$. \medskip\noindent Suppose that $y \in X_w$ specializes to $z, z' \in Z$. We will show that $z = z'$ which will finish the proof of (3) and will imply (5). Let $x, x' \in X$ be the images of $z$ and $z'$. Since $Z \to X$ is bijective it suffices to show that $x = x'$. If $x \not = x'$, then there exists an $f \in A$ such that $x \in D(f)$ and $x' \in V(f)$ (or vice versa). Set $E = \{f\}$ so that $$ X_E = \Spec(A_f) \amalg \Spec(A_{V(f)}^\sim) $$ Then we see that $z$ and $z'$ map $x_E$ and $x'_E$ which are in different parts of the given decomposition of $X_E$ above. But then it impossible for $x_E$ and $x'_E$ to be specializations of a common point. This is the desired contradiction. \medskip\noindent Recall that given a finite subset $E \subset A$ we have $Z_E$ is a disjoint union of the locally closed subschemes $Z(E', E'')$ each isomorphic to the spectrum of $(A/I)_f$ where $I$ is the ideal generated by $E''$ and $f$ the product of the elements of $E'$. Any nilpotent element $b$ of $(A/I)_f$ is the class of $g/f^n$ for some $g \in A$. Then setting $E' = E \cup \{g\}$ the reader verifies that $b$ is pulls back to zero under the transition map $Z_{E'} \to Z_E$ of the system. This proves (4).
Denote $Y = \Spec(B)$ and $Y_0 \subset Y$ the set of closed points. Denote $f : Y \to X$ the given morphism. Recall that $Y_0$ is profinite, in particular every constructible subset of $Y_0$ is open and closed. Let $E \subset A$ be a finite subset. Recall that $A_w = \colim A_E$ and that the set of closed points of $\Spec(A_w)$ is the limit of the closed subsets $Z_E \subset X_E = \Spec(A_E)$. Thus it suffices to show there is a unique factorization $A \to A_E \to B$ such that $Y \to X_E$ maps $Y_0$ into $Z_E$. Since $Z_E \to X = \Spec(A)$ is bijective, and since the strata $Z(E', E'')$ are constructible we see that $$ Y_0 = \coprod f^{-1}(Z(E', E'')) \cap Y_0 $$ is a disjoint union decomposition into open and closed subsets. As $Y_0 = \pi_0(Y)$ we obtain a corresponding decomposition of $Y$ into open and closed pieces. Thus it suffices to construct the factorization in case $f(Y_0) \subset Z(E', E'')$ for some decomposition $E = E' \amalg E''$. In this case $f(Y)$ is contained in the set of points of $X$ specializing to $Z(E', E'')$ which is homeomorphic to $X_{E', E''}$. Thus we obtain a unique continuous map $Y \to X_{E', E''}$ over $X$. By Lemma \ref{lemma-fully-faithful-spaces-over-X} this corresponds to a unique morphism of schemes $Y \to X_{E', E''}$ over $X$. This finishes the proof.
By the references mentioned above (Algebra, Lemma \ref{algebra-lemma-ring-with-only-minimal-primes} or Topology, Lemma \ref{topology-lemma-characterize-profinite-spectral}) there are no specializations between distinct points of $\Spec(A)$ and $\Spec(B)$ is profinite if and only if there are no specializations between distinct points of $\Spec(B)$. These specializations can only happen in the fibres of $\Spec(B) \to \Spec(A)$. In this way we see that (1) is true. \medskip\noindent The assumption in (2) implies all primes of $B$ are maximal by Algebra, Lemma \ref{algebra-lemma-finite-residue-extension-closed}. Thus (2) holds. If $A \to B$ is a local isomorphism or identifies local rings, then the residue field extensions are trivial, so (3) and (4) follow from (2). If $A \to B$ is weakly \'etale, then More on Algebra, Lemma \ref{more-algebra-lemma-weakly-etale-residue-field-extensions} tells us it induces separable algebraic residue field extensions, so (5) follows from (2). If $A \to B$ is quasi-finite, then the fibres are finite discrete topological spaces. Hence (6) follows from (1). Hence (3) follows from (1). Cases (7) and (8) follow from this as unramified and \'etale ring map are quasi-finite (Algebra, Lemmas \ref{algebra-lemma-unramified-quasi-finite} and \ref{algebra-lemma-etale-quasi-finite}). If $B = \colim B_i$ is a filtered colimit of $A$-algebras, then $\Spec(B) = \lim \Spec(B_i)$ in the category of topological spaces by Limits, Lemma \ref{limits-lemma-inverse-limit-top}. Hence if each $\Spec(B_i)$ is profinite, so is $\Spec(B)$ by Topology, Lemma \ref{topology-lemma-directed-inverse-limit-profinite}. This proves (9).
Let $A \to A_w$ and $Z \subset Y = \Spec(A_w)$ as in Lemma \ref{lemma-make-w-local}. Let $T \subset Z$ be the inverse image of $V(I)$. Then $T \to V(I)$ is a homeomorphism by Topology, Lemma \ref{topology-lemma-bijective-map}. Let $B = (A_w)_T^\sim$, see Lemma \ref{lemma-localization}. It is clear that $B$ is w-local with closed points $V(IB)$. The ring map $A/I \to B/IB$ is ind-Zariski and induces a homeomorphism on underlying topological spaces. Hence it is an isomorphism by Lemma \ref{lemma-local-isomorphism-fully-faithful}.
By Lemma \ref{lemma-profinite-goes-up} applied to $A/I \to B/IB$ all points of $Z = V(IB) = \Spec(B/IB)$ are closed, in fact $\Spec(B/IB)$ is a profinite space. To finish the proof we apply Lemma \ref{lemma-localize-along-closed-profinite} to $IB \subset B$.
Let $Z \subset X$ be the inverse image of $T$. Then $Z$ is the intersection $Z = \bigcap Z_\alpha$ of the open and closed subsets of $X$ containing $Z$, see Topology, Lemma \ref{topology-lemma-closed-union-connected-components}. For each $\alpha$ we have $Z_\alpha = \Spec(A_\alpha)$ where $A \to A_\alpha$ is a local isomorphism (a localization at an idempotent). Setting $B = \colim A_\alpha$ proves the lemma.
Namely, write $T = \lim T_i$ as the limit of an inverse system finite discrete spaces over a directed set (see Topology, Lemma \ref{topology-lemma-profinite}). For each $i$ let $Z_i = \Im(T \to \pi_0(X) \times T_i)$. This is a closed subset. Observe that $X \times T_i$ is the spectrum of $A_i = \prod_{t \in T_i} A$ and that $A \to A_i$ is a local isomorphism. By Lemma \ref{lemma-construct} we see that $Z_i \subset \pi_0(X \times T_i) = \pi_0(X) \times T_i$ corresponds to a surjection $A_i \to B_i$ which is ind-Zariski such that $\Spec(B_i) = X \times_{\pi_0(X)} Z_i$ as subsets of $X \times T_i$. The transition maps $T_i \to T_{i'}$ induce maps $Z_i \to Z_{i'}$ and $X \times_{\pi_0(X)} Z_i \to X \times_{\pi_0(X)} Z_{i'}$. Hence ring maps $B_{i'} \to B_i$ (Lemmas \ref{lemma-local-isomorphism-fully-faithful} and \ref{lemma-ind-zariski-implies}). Set $B = \colim B_i$. Because $T = \lim Z_i$ we have $X \times_{\pi_0(X)} T = \lim  X \times_{\pi_0(X)} Z_i$ and hence $Y = \Spec(B) = \lim \Spec(B_i)$ fits into the cartesian diagram $$ \xymatrix{ Y \ar[r] \ar[d] & T \ar[d] \\ X \ar[r] & \pi_0(X) } $$ of topological spaces. By Lemma \ref{lemma-silly} we conclude that $T = \pi_0(Y)$.