hcju/stacksqa huggingface dataset:
Let $(I_\lambda)$ be a fundamental system of weakly admissible ideals of definition in $A$. Consider $Y$ as in (2). Then $Y \times_X \Spec(A/I_\lambda)$ is affine (Definition \ref{definition-finite-type} and Lemma \ref{lemma-affine-representable-by-algebraic-spaces}). Say $Y \times_X \Spec(A/I_\lambda) = \Spec(B_\lambda)$. The ring map $A/I_\lambda \to B_\lambda$ is of finite type because $\Spec(B_\lambda) \to \Spec(A/I_\lambda)$ is of finite type (by Definition \ref{definition-finite-type}). Then $(B_\lambda)$ is an object of $\mathcal{C}$. \medskip\noindent Conversely, given an object $(B_\lambda)$ of $\mathcal{C}$ we can set $Y = \colim \Spec(B_\lambda)$. This is an affine formal algebraic space. We claim that $$ Y \times_X \Spec(A/I_\lambda) = \left(\colim_\mu \Spec(B_\mu)\right) \times_X \Spec(A/I_\lambda) = \Spec(B_\lambda) $$ To show this it suffices we get the same values if we evaluate on a quasi-compact scheme $U$. A morphism $U \to \left(\colim_\mu \Spec(B_\mu)\right) \times_X \Spec(A/I_\lambda)$ comes from a morphism $U \to \Spec(B_\mu) \times_{\Spec(A/I_\mu)} \Spec(A/I_\lambda)$ for some $\mu \geq \lambda$ (use Lemma \ref{lemma-factor-through-thickening} two times). Since $\Spec(B_\mu) \times_{\Spec(A/I_\mu)} \Spec(A/I_\lambda) = \Spec(B_\lambda)$ by our second assumption on objects of $\mathcal{C}$ this proves what we want. Using this we can show the morphism $Y \to X$ is of finite type. Namely, we note that for any morphism $U \to X$ with $U$ a quasi-compact scheme, we get a factorization $U \to \Spec(A/I_\lambda) \to X$ for some $\lambda$ (see lemma cited above). Hence $$ Y \times_X U = Y \times_X \Spec(A/I_\lambda)) \times_{\Spec(A/I_\lambda)} U = \Spec(B_\lambda) \times_{\Spec(A/I_\lambda)} U $$ is a scheme of finite type over $U$ as desired. Thus the construction $(B_\lambda) \mapsto \colim \Spec(B_\lambda)$ does give a functor from category (1) to category (2). \medskip\noindent To finish the proof we show that the above constructions define quasi-inverse functors between the categories (1) and (2). In one direction you have to show that $$ \left(\colim_\mu \Spec(B_\mu)\right) \times_X \Spec(A/I_\lambda) = \Spec(B_\lambda) $$ for any object $(B_\lambda)$ in the category $\mathcal{C}$. This we proved above. For the other direction you have to show that $$ Y = \colim (Y \times_X \Spec(A/I_\lambda)) $$ given $Y$ in the category (2). Again this is true by evaluating on quasi-compact test objects and because $X = \colim \Spec(A/I_\lambda)$.
By Lemma \ref{lemma-countably-indexed} we see that $X = \text{Spf}(A)$ where $A$ is an object of $\textit{WAdm}^{count}$. Since a closed immersion is representable and affine, we conclude by Lemma \ref{lemma-property-goes-up-affine-morphism} that $Y$ is an affine formal algebraic space and countably index. Thus applying Lemma \ref{lemma-countably-indexed} again we see that $Y = \text{Spf}(B)$ with $B$ an object of $\textit{WAdm}^{count}$. By Lemma \ref{lemma-closed-immersion-into-McQuillan} we conclude that $f$ is given by a morphism $A \to B$ of $\textit{WAdm}^{count}$ which is taut and has dense image. To finish the proof we apply Lemma \ref{lemma-dense-image-surjective}.
It is enough to show that (e) implies (a). Let $J' \subset B$ be a weak ideal of definition and let $I' \subset A$ be the closure of $J'A$. We have to show that $B/J' \to A/I'$ is of finite type. If the corresponding statement holds for the smaller weak ideal of definition $J'' = J' \cap J$, then it holds for $J'$. Thus we may assume $J' \subset J$. As $J$ is an ideal of definition (and not just a weak ideal of definition), we get $J^n \subset J'$ for some $n \geq 1$. Thus we can consider the diagram $$ \xymatrix{ 0 \ar[r] & I/I' \ar[r] & A/I' \ar[r] & A/I \ar[r] & 0 \\ 0 \ar[r] & J/J' \ar[r] \ar[u] & B/J' \ar[r] \ar[u] & B/J \ar[r] \ar[u] & 0 } $$ with exact rows. Since $I' \subset A$ is open and since $I$ is the closure of $J A$ we see that $I/I' = (J/J') \cdot A/I'$. Because $J/J'$ is a nilpotent ideal and as $B/J \to A/I$ is of finite type, we conclude from Algebra, Lemma \ref{algebra-lemma-finite-type-mod-nilpotent} that $A/I'$ is of finite type over $B/J'$ as desired.
Since $X$ and $Y$ are affine it is clear that conditions (1) and (2) are equivalent. In cases (1) and (2) the morphism $f$ is representable by algebraic spaces by definition, hence affine by Lemma \ref{lemma-affine-representable-by-algebraic-spaces}. Thus if (1) or (2) holds we see that $X$ is countably indexed by Lemma \ref{lemma-property-goes-up-affine-morphism}. Write $X = \text{Spf}(A)$ and $Y = \text{Spf}(B)$ for topological $S$-algebras $A$ and $B$ in $\textit{WAdm}^{count}$, see Lemma \ref{lemma-countably-indexed}. By Lemma \ref{lemma-morphism-between-formal-spectra} we see that $f$ corresponds to a continuous map $B \to A$. Hence now the result follows from Lemma \ref{lemma-topologically-finite-type-finite-type}.
By Lemma \ref{lemma-quotient-restricted-power-series} the property $P(\varphi)=$``$\varphi$ is taut and topologically of finite type'' is local on $\text{WAdm}^{count}$. Hence by Lemma \ref{lemma-property-defines-property-morphisms} we see that conditions (1), (2), and (3) are equivalent. On the other hand, by Lemma \ref{lemma-representable-affine-finite-type} the condition $P$ on morphisms of $\textit{WAdm}^{count}$ corresponds exactly to morphisms of countably indexed, affine formal algebraic spaces being locally of finite type. Thus the implication (1) $\Rightarrow$ (3) of Lemma \ref{lemma-finite-type-local} shows that (4) implies (1) of the current lemma. Similarly, the implication (4) $\Rightarrow$ (1) of Lemma \ref{lemma-finite-type-local} shows that (2) implies (4) of the current lemma.
Let $f : X \to Y$ and $Y' \to Y$ be morphisms of formal algebraic spaces. Let $f' : X' \to Y'$ be the base change of $f$ by $Y' \to Y$. Then $\Delta_{X'/Y'}$ is the base change of $\Delta_{X/Y}$ by the morphism $X' \times_{Y'} X' \to X \times_Y X$. Each of the properties of the diagonal used in Definition \ref{definition-separated-morphism} is stable under base change. Hence the lemma is true.
By general category theory the following diagram $$ \xymatrix{ X \times_Z Y \ar[r]_i \ar[d] & X \times_T Y \ar[d] \\ Z \ar[r]^-{\Delta_{Z/T}} \ar[r] & Z \times_T Z } $$ is a fibre product diagram. Hence $i$ is the base change of the diagonal morphism $\Delta_{Z/T}$. Thus the lemma follows from Lemma \ref{lemma-diagonal-morphism-formal-algebraic-spaces}.
Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of formal algebraic spaces to which the axiom in question applies. The diagonal $\Delta_{X/Z}$ is the composition $$ X \longrightarrow X \times_Y X \longrightarrow X \times_Z X. $$ Our separation axiom is defined by requiring the diagonal to have some property $\mathcal{P}$. By Lemma \ref{lemma-fibre-product-after-map} above we see that the second arrow also has this property. Hence the lemma follows since the composition of (representable) morphisms with property $\mathcal{P}$ also is a morphism with property $\mathcal{P}$.
We will repeatedly use Lemma \ref{lemma-base-change-separated} without further mention. In particular, it is clear that (1) implies (2) and (2) implies (3). \medskip\noindent Assume (3) and let $Z \to Y$ be a morphism where $Z$ is an affine scheme. Let $U$, $V$ be affine schemes and let $a : U \to Z \times_Y X$ and $b : V \to Z \times_Y X$ be morphisms. Then $$ U \times_{Z \times_Y X} V = (Z \times_Y X) \times_{\Delta, (Z \times_Y X) \times_Z (Z \times_Y X)} (U \times_Z V) $$ and we see that this is quasi-compact if $\mathcal{P} =$``quasi-separated'' or an affine scheme equipped with a closed immersion into $U \times_Z V$ if $\mathcal{P} =$``separated''. Thus (4) holds. \medskip\noindent Assume (4) and let $Z \to Y$ be a morphism where $Z$ is an affine scheme. Let $U$, $V$ be affine schemes and let $a : U \to Z \times_Y X$ and $b : V \to Z \times_Y X$ be morphisms. Reading the argument above backwards, we see that $U \times_{Z \times_Y X} V \to U \times_Z V$ is quasi-compact if $\mathcal{P} =$``quasi-separated'' or a closed immersion if $\mathcal{P} =$``separated''. Since we can choose $U$ and $V$ as above such that $U$ varies through an \'etale covering of $Z \times_Y X$, we find that the corresponding morphisms $$ U \times_Z V \to (Z \times_Y X) \times_Z (Z \times_Y X) $$ form an \'etale covering by affines. Hence we conclude that $\Delta : (Z \times_Y X) \to (Z \times_Y X) \times_Z (Z \times_Y X)$ is quasi-compact, resp.\ a closed immersion. Thus (3) holds. \medskip\noindent Let us prove that (3) implies (5). Assume (3) and let $\{Y_j \to Y\}$ be as in Definition \ref{definition-formal-algebraic-space}. We have to show that the morphisms $$ \Delta_j : Y_j \times_Y X \longrightarrow (Y_j \times_Y X) \times_{Y_j} (Y_j \times_Y X) = Y_j \times_Y X \times_Y X $$ has the corresponding property (i.e., is quasi-compact or a closed immersion). Write $Y_j = \colim Y_{j, \lambda}$ as in Definition \ref{definition-affine-formal-algebraic-space}. Replacing $Y_j$ by $Y_{j, \lambda}$ in the formula above, we have the property by our assumption that (3) holds. Since the displayed arrow is the colimit of the arrows $\Delta_{j, \lambda}$ and since we can test whether $\Delta_j$ has the corresponding property by testing after base change by affine schemes mapping into $Y_j \times_Y X \times_Y X$, we conclude by Lemma \ref{lemma-factor-through-thickening}. \medskip\noindent Let us prove that (5) implies (1). Let $\{Y_j \to Y\}$ be as in (5). Then we have the fibre product diagram $$ \xymatrix{ \coprod Y_j \times_Y X \ar[r] \ar[d] & X \ar[d] \\ \coprod Y_j \times_Y X \times_Y X \ar[r] & X \times_Y X } $$ By assumption the left vertical arrow is quasi-compact or a closed immersion. It follows from Spaces, Lemma \ref{spaces-lemma-descent-representable-transformations-property} that also the right vertical arrow is quasi-compact or a closed immersion.
If $X$ is affine, say $X = \Spec(B)$, then we see from Lemma \ref{lemma-morphism-between-formal-spectra} that morphisms $\text{Spf}(A) \to \Spec(B)$ correspond to continuous $S$-algebra maps $B \to A$ where $B$ has the discrete topology. These are just $S$-algebra maps, which correspond to morphisms $\Spec(A) \to \Spec(B)$.
Let $\varphi : \text{Spf}(A) \to X$ be a morphism. Since $\Spec(A/I)$ is local we see that $\varphi$ maps $\Spec(A/I)$ into an affine open $U \subset X$. However, this then implies that $\Spec(A/J)$ maps into $U$ for every ideal of definition $J$. Hence we may apply Lemma \ref{lemma-map-into-affine} to see that $\varphi$ comes from a morphism $\Spec(A) \to X$. This proves surjectivity of the map. We omit the proof of injectivity.
The statement makes sense because $X_{/T}$ is adic* by Lemma \ref{lemma-formal-completion-types} (and hence we're allowed to use the terminology adic for morphisms, see Definition \ref{definition-adic-morphism}). Let $p$ be given. By Lemma \ref{lemma-map-into-algebraic-space} we get a unique morphism $g : \Spec(R) \to X$ corresponding to the composition $\text{Spf}(R) \to X_{/T} \to X$. Let $Z \subset X$ be the reduced induced closed subspace structure on $T$. The incusion morphism $Z \to X$ corresponds to a morphism $Z \to X_{/T}$. Since $p$ is adic it is representable by algebraic spaces and we find $$ \text{Spf}(R) \times_{X_{/T}} Z = \text{Spf}(R) \times_X Z $$ is an algebraic space endowed with a closed immersion to $\text{Spf}(R)$. (Equality holds because $X_{/T} \to X$ is a monomorphism.) Thus this fibre product is equal to $\Spec(R/J)$ for some ideal $J \subset R$ wich contains $\mathfrak m_R^{n_0}$ for some $n_0 \geq 1$. This implies that $\Spec(R) \times_X Z$ is a closed subscheme of $\Spec(R)$, say $\Spec(R) \times_X Z = \Spec(R/I)$, whose intersection with $\Spec(R/\mathfrak m_R^n)$ for $n \geq n_0$ is equal to $\Spec(R/J)$. In algebraic terms this says $I + \mathfrak m_R^n = J + \mathfrak m_R^n = J$ for all $n \geq n_0$. By Krull's intersection theorem this implies $I = J$ and we conclude.
This will follow immediately from the case of algebraic spaces (Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-morphism-topoi}) if we can show that the induced morphism $X_{red} \to Y_{red}$ is \'etale. Observe that $X \times_Y Y_{red}$ is an algebraic space, \'etale over the reduced algebraic space $Y_{red}$, and hence reduced itself (by our definition of reduced algebraic spaces in Properties of Spaces, Section \ref{spaces-properties-section-types-properties}. Hence $X_{red} = X \times_Y Y_{red}$ as desired.
Denote $\mathcal{C}$ the category introduced in the lemma. Observe that for $\varphi : U \to X$ in $\mathcal{C}$ the morphism $\varphi$ is representable (by schemes) and affine, see Lemma \ref{lemma-affine-representable-by-algebraic-spaces}. Recall that $X_{affine, \etale} = X_{red, affine, \etale}$. Hence we can define a functor $$ \mathcal{C} \longrightarrow X_{affine, \etale},\quad (U \to X) \longmapsto U \times_X X_{red} $$ because $U \times_X X_{red}$ is an affine scheme. \medskip\noindent To finish the proof we will construct a quasi-inverse. Namely, write $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. For each $\lambda$ we have $X_{red} \subset X_\lambda$ is a thickening. Thus for every $\lambda$ we have an equivalence $$ X_{red, affine, \etale} = X_{\lambda, affine, \etale} $$ for example by More on Algebra, Lemma \ref{more-algebra-lemma-locally-nilpotent-henselian}. Hence if $U_{red} \to X_{red}$ is an \'etale morphism with $U_{red}$ affine, then we obtain a system of \'etale morphisms $U_\lambda \to X_\lambda$ of affine schemes compatible with the transition morphisms in the system defining $X$. Hence we can take $$ U = \colim U_\lambda $$ as our affine formal algebraic space over $X$. The construction gives that $U \times_X X_\lambda = U_\lambda$. This shows that $U \to X$ is representable and \'etale. We omit the verification that the constructions are mutually inverse to each other.
Denote $\mathcal{C}$ the category introduced in the lemma. Recall that $X_{spaces, \etale} = X_{red, spaces, \etale}$. Hence we can define a functor $$ \mathcal{C} \longrightarrow X_{spaces, \etale},\quad (U \to X) \longmapsto U \times_X X_{red} $$ because $U \times_X X_{red}$ is an algebraic space \'etale over $X_{red}$. \medskip\noindent To finish the proof we will construct a quasi-inverse. Choose an object $\psi : V \to X_{red}$ of $X_{red, spaces, \etale}$. Consider the functor $U_{V, \psi} : (\Sch/S)_{fppf} \to \textit{Sets}$ given by $$ U_{V, \psi}(T) = \{(a, b) \mid a : T \to X, \ b : T \times_{a, X} X_{red} \to V, \ \psi \circ b = a|_{T \times_{a, X} X_{red}}\} $$ We claim that the transformation $U_{V, \psi} \to X$, $(a, b) \mapsto a$ defines an object of the category $\mathcal{C}$. First, let's prove that $U_{V, \psi}$ is a formal algebraic space. Observe that $U_{V, \psi}$ is a sheaf for the fppf topology (some details omitted). Next, suppose that $X_i \to X$ is an \'etale covering by affine formal algebraic spaces as in Definition \ref{definition-formal-algebraic-space}. Set $V_i = V \times_{X_{red}} X_{i, red}$ and denote $\psi_i : V_i \to X_{i, red}$ the projection. Then  we have $$ U_{V, \psi} \times_X X_i = U_{V_i, \psi_i} $$ by a formal argument because $X_{i, red} = X_i \times_X X_{red}$ (as $X_i \to X$ is representable by algebraic spaces and \'etale). Hence it suffices to show that $U_{V_i, \psi_i}$ is an affine formal algebraic space, because then we will have a covering $U_{V_i, \psi_i} \to U_{V, \psi}$ as in Definition \ref{definition-formal-algebraic-space}. On the other hand, we have seen in the proof of Lemma \ref{lemma-etale-morphism-topoi} that $\psi_i : V_i \to X_i$ is the base change of a representable and \'etale morphism $U_i \to X_i$ of affine formal algebraic spaces. Then it is not hard to see that $U_i = U_{V_i, \psi_i}$ as desired. \medskip\noindent We omit the verification that $U_{V, \psi} \to X$ is representable by algebraic spaces and \'etale. Thus we obtain our functor $(V, \psi) \mapsto (U_{V, \psi} \to X)$ in the other direction. We omit the verification that the constructions are mutually inverse to each other.
Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. By (\ref{equation-etale-topos}) it suffices to construct $\mathcal{O}_X$ as a sheaf of topological rings on $X_{affine, \etale}$. Denote $\mathcal{C}$ the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that $U$ is an affine formal algebraic space and $\varphi$ is representable by algebraic spaces and \'etale. By Lemma \ref{lemma-identify-affine-etale} the functor $U \mapsto U_{red}$ is an equivalence of categories $\mathcal{C} \to X_{affine, \etale}$. Hence by the rule given above the lemma, we already have $\mathcal{O}_X$ as a presheaf of topological rings on $X_{affine, \etale}$. Thus it suffices to check the sheaf condition. \medskip\noindent By definition of $X_{affine, \etale}$ a covering corresponds to a finite family $\{g_i : U_i \to U\}_{i = 1, \ldots, n}$ of morphisms of $\mathcal{C}$ such that $\{U_{i, red} \to U_{red}\}$ is an \'etale covering. The morphisms $g_i$ are representably by algebraic spaces (Lemma \ref{lemma-permanence-representable}) hence affine (Lemma \ref{lemma-affine-representable-by-algebraic-spaces}). Then $g_i$ is \'etale (follows formally from Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-permanence} as $U_i$ and $U$ are \'etale over $X$ in the sense of Bootstrap, Section \ref{bootstrap-section-representable-by-spaces-properties}). Finally, write $U = \colim U_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. \medskip\noindent With these preparations out of the way, we can prove the sheaf property as follows. For each $\lambda$ we set $U_{i, \lambda} = U_i \times_U U_\lambda$ and $U_{ij, \lambda} = (U_i \times_U U_j) \times_U U_\lambda$. By the above, these are affine schemes, $\{U_{i, \lambda} \to U_\lambda\}$ is an \'etale covering, and $U_{ij, \lambda} = U_{i, \lambda} \times_{U_\lambda} U_{j, \lambda}$. Also we have $U_i = \colim U_{i, \lambda}$ and $U_i \times_U U_j = \colim U_{ij, \lambda}$. For each $\lambda$ we have an exact sequence $$ 0 \to \Gamma(U_\lambda, \mathcal{O}_{U_\lambda}) \to \prod\nolimits_i \Gamma(U_{i, \lambda}, \mathcal{O}_{U_{i, \lambda}}) \to \prod\nolimits_{i, j} \Gamma(U_{ij, \lambda}, \mathcal{O}_{U_{ij, \lambda}}) $$ as we have the sheaf condition for the structure sheaf on $U_\lambda$ and the \'etale topology (see \'Etale Cohomology, Proposition \ref{etale-cohomology-proposition-quasi-coherent-sheaf-fpqc}). Since limits commute with limits, the inverse limit of these exact sequences is an exact sequence $$ 0 \to \lim \Gamma(U_\lambda, \mathcal{O}_{U_\lambda}) \to \prod\nolimits_i \lim \Gamma(U_{i, \lambda}, \mathcal{O}_{U_{i, \lambda}}) \to \prod\nolimits_{i, j} \lim \Gamma(U_{ij, \lambda}, \mathcal{O}_{U_{ij, \lambda}}) $$ which exactly means that $$ 0 \to \mathcal{O}_X(U_{red}) \to \prod\nolimits_i \mathcal{O}_X(U_{i, red}) \to \prod\nolimits_{i, j} \mathcal{O}_X((U_i \times_U U_j)_{red}) $$ is exact and hence the sheaf propery holds as desired.
We may work with $X_{affine, \etale}$ as this gives the same topos. We will apply Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection} to show we have vanishing. Since $X_{affine, \etale}$ has finite disjoint unions, this reduces us to the {\v C}ech complex of a covering given by a single arrow $\{U_{red} \to V_{red}\}$ in $X_{affine, \etale} = X_{red, affine, \etale}$ (see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-cech-complex}). Thus we have to show that $$ 0 \to \mathcal{O}_X(V_{red}) \to \mathcal{O}_X(U_{red}) \to \mathcal{O}_X(U_{red} \times_{V_{red}} U_{red}) \to \ldots $$ is exact. We will do this below in the case $V_{red} = X_{red}$. The general case is proven in exactly the same way. \medskip\noindent Recall that $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological ring having a countable fundamental system of weak ideals of definition. We have seen in Lemmas \ref{lemma-affine-identify-affine-etale} and \ref{lemma-affine-etale-mcquillan} that the object $U_{red}$ in $X_{affine, \etale}$ corresponds to a morphism $U \to X$ of affine formal algebraic spaces which is representable by algebraic space and \'etale and $U = \text{Spf}(B^\wedge)$ where $B$ is an \'etale $A$-algebra. By our rule for the structure sheaf we see $$ \mathcal{O}_X(U_{red}) = B^\wedge $$ We recall that $B^\wedge = \lim B/JB$ where the limit is over weak ideals of definition $J \subset A$. Working through the definitions we obtain $$ \mathcal{O}_X(U_{red} \times_{X_{red}} U_{red}) = (B \otimes_A B)^\wedge $$ and so on. Since $U \to X$ is a covering the map $A \to B$ is faithfully flat, see Lemma \ref{lemma-etale-surjective}. Hence the complex $$ 0 \to A \to B \to B \otimes_A B \to B \otimes_A B \otimes_A B \to \ldots $$ is {\bf universally} exact, see Descent, Lemma \ref{descent-lemma-ff-exact}. Our goal is to show that $$ H^n(0 \to A^\wedge \to B^\wedge \to (B \otimes_A B)^\wedge \to (B \otimes_A B \otimes_A B)^\wedge \to \ldots) $$ is zero for $p > 0$. To see what is going on, let's split our exact complex (before completion) into short exact sequences $$ 0 \to A \to B \to M_1 \to 0,\quad 0 \to M_i \to B^{\otimes_A i + 1} \to M_{i + 1} \to 0 $$ By what we said above, these are universally exact short exact sequences. Hence $JM_i = M_i \cap J(B^{\otimes_A i + 1})$ for every ideal $J$ of $A$. In particular, the topology on $M_i$ as a submodule of $B^{\otimes_A i + 1}$ is the same as the topology on $M_i$ as a quotient module of $B^{\otimes_A i}$. Therefore, since there exists a countable fundamental system of weak ideals of definition in $A$, the sequences $$ 0 \to A^\wedge \to B^\wedge \to M_1^\wedge \to 0,\quad 0 \to M_i^\wedge \to (B^{\otimes_A i + 1})^\wedge \to M_{i + 1}^\wedge \to 0 $$ remain exact by Lemma \ref{lemma-ses}. This proves the lemma.
Part (1) follows from Lemma \ref{lemma-finite-etale-connected-galois-category} and Proposition \ref{proposition-galois}. Part (2) is a special case of Lemma \ref{lemma-functoriality-galois}. For part (3) observe that the diagram $$ \xymatrix{ \textit{F\'Et}_Y \ar[r] \ar[d]_{F_{\overline{y}}} & \textit{F\'Et}_X \ar[d]^{F_{\overline{x}}} \\ \textit{Sets} \ar@{=}[r] & \textit{Sets} } $$ is commutative (actually commutative, not just $2$-commutative) because $\overline{y} = f \circ \overline{x}$. Hence we can apply Lemma \ref{lemma-functoriality-galois} with the implied transformation of functors to get (3).