proetale-lemma-local-isomorphism-fully-faithful

Let $A$ be a ring. Set $X = \Spec(A)$. The functor $$ B \longmapsto \Spec(B) $$ from the category of $A$-algebras $B$ such that $A \to B$ identifies local rings to the category of topological spaces over $X$ is fully faithful.


proetale-lemma-base-change-ind-zariski

Let $A \to B$ and $A \to A'$ be ring maps. Let $B' = B \otimes_A A'$ be the base change of $B$. If $A \to B$ is ind-Zariski, then $A' \to B'$ is ind-Zariski.


proetale-lemma-composition-ind-zariski

Let $A \to B$ and $B \to C$ be ring maps. If $A \to B$ and $B \to C$ are ind-Zariski, then $A \to C$ is ind-Zariski.


proetale-lemma-ind-zariski-permanence

Let $A$ be a ring. Let $B \to C$ be an $A$-algebra homomorphism. If $A \to B$ and $A \to C$ are ind-Zariski, then $B \to C$ is ind-Zariski.


proetale-lemma-ind-ind-zariski

A filtered colimit of ind-Zariski $A$-algebras is ind-Zariski over $A$.


proetale-lemma-ind-zariski-implies

Let $A \to B$ be ind-Zariski. Then $A \to B$ identifies local rings,


proetale-lemma-localization

Let $A$ be a ring. Set $X = \Spec(A)$. Let $Z \subset X$ be a locally closed subscheme which is of the form $D(f) \cap V(I)$ for some $f \in A$ and ideal $I \subset A$. Then \begin{enumerate} \item there exists a multiplicative subset $S \subset A$ such that $\Spec(S^{-1}A)$ maps by a homeomorphism to the set of points of $X$ specializing to $Z$, \item the $A$-algebra $A_Z^\sim = S^{-1}A$ depends only on the underlying locally closed subset $Z \subset X$, \item $Z$ is a closed subscheme of $\Spec(A_Z^\sim)$, \end{enumerate} If $A \to A'$ is a ring map and $Z' \subset X' = \Spec(A')$ is a locally closed subscheme of the same form which maps into $Z$, then there is a unique $A$-algebra map $A_Z^\sim \to (A')_{Z'}^\sim$.


proetale-lemma-refine

Let $X = \Spec(A)$ as above. Given any finite stratification $X = \coprod T_i$ by constructible subsets, there exists a finite subset $E \subset A$ such that the stratification (\ref{equation-stratify}) refines $X = \coprod T_i$.


proetale-lemma-make-w-local

Let $X = \Spec(A)$ be an affine scheme. With $A \to A_w$, $X_w = \Spec(A_w)$, and $Z \subset X_w$ as above. \begin{enumerate} \item $A \to A_w$ is ind-Zariski and faithfully flat, \item $X_w \to X$ induces a bijection $Z \to X$, \item $Z$ is the set of closed points of $X_w$, \item $Z$ is a reduced scheme, and \item every point of $X_w$ specializes to a unique point of $Z$. \end{enumerate} In particular, $X_w$ is w-local (Definition \ref{definition-w-local}).


proetale-lemma-universal

Let $A$ be a ring. Let $A \to A_w$ be the ring map constructed in Lemma \ref{lemma-make-w-local}. For any ring map $A \to B$ such that $\Spec(B)$ is w-local, there is a unique factorization $A \to A_w \to B$ such that $\Spec(B) \to \Spec(A_w)$ is w-local.


proetale-lemma-profinite-goes-up

Let $A$ be a ring such that $\Spec(A)$ is profinite. Let $A \to B$ be a ring map. Then $\Spec(B)$ is profinite in each of the following cases: \begin{enumerate} \item if $\mathfrak q,\mathfrak q' \subset B$ lie over the same prime of $A$, then neither $\mathfrak q \subset \mathfrak q'$, nor $\mathfrak q' \subset \mathfrak q$, \item $A \to B$ induces algebraic extensions of residue fields, \item $A \to B$ is a local isomorphism, \item $A \to B$ identifies local rings, \item $A \to B$ is weakly \'etale, \item $A \to B$ is quasi-finite, \item $A \to B$ is unramified, \item $A \to B$ is \'etale, \item $B$ is a filtered colimit of $A$-algebras as in (1) -- (8), \item etc. \end{enumerate}


proetale-lemma-localize-along-closed-profinite

Let $A$ be a ring. Let $V(I) \subset \Spec(A)$ be a closed subset which is a profinite topological space. Then there exists an ind-Zariski ring map $A \to B$ such that $\Spec(B)$ is w-local, the set of closed points is $V(IB)$, and $A/I \cong B/IB$.


proetale-lemma-w-local-algebraic-residue-field-extensions

Let $A$ be a ring such that $X = \Spec(A)$ is w-local. Let $I \subset A$ be the radical ideal cutting out the set $X_0$ of closed points in $X$. Let $A \to B$ be a ring map inducing algebraic extensions on residue fields at primes. Then \begin{enumerate} \item every point of $Z = V(IB)$ is a closed point of $\Spec(B)$, \item there exists an ind-Zariski ring map $B \to C$ such that \begin{enumerate} \item $B/IB \to C/IC$ is an isomorphism, \item the space $Y = \Spec(C)$ is w-local, \item the induced map $p : Y \to X$ is w-local, and \item $p^{-1}(X_0)$ is the set of closed points of $Y$. \end{enumerate} \end{enumerate}


proetale-lemma-construct

Let $A$ be a ring. Let $X = \Spec(A)$. Let $T \subset \pi_0(X)$ be a closed subset. There exists a surjective ind-Zariski ring map $A \to B$ such that $\Spec(B) \to \Spec(A)$ induces a homeomorphism of $\Spec(B)$ with the inverse image of $T$ in $X$.


proetale-lemma-construct-profinite

Let $A$ be a ring and let $X = \Spec(A)$. Let $T$ be a profinite space and let $T \to \pi_0(X)$ be a continuous map. There exists an ind-Zariski ring map $A \to B$ such that with $Y = \Spec(B)$ the diagram $$ \xymatrix{ Y \ar[r] \ar[d] & \pi_0(Y) \ar[d] \\ X \ar[r] & \pi_0(X) } $$ is cartesian in the category of topological spaces and such that $\pi_0(Y) = T$ as spaces over $\pi_0(X)$.


proetale-lemma-w-local-morphism-equal-points-stalks-is-iso

Let $A \to B$ be ring map such that \begin{enumerate} \item $A \to B$ identifies local rings, \item the topological spaces $\Spec(B)$, $\Spec(A)$ are w-local, \item $\Spec(B) \to \Spec(A)$ is w-local, and \item $\pi_0(\Spec(B)) \to \pi_0(\Spec(A))$ is bijective. \end{enumerate} Then $A \to B$ is an isomorphism


proetale-lemma-w-local-morphism-equal-stalks-is-ind-zariski

Let $A \to B$ be ring map such that \begin{enumerate} \item $A \to B$ identifies local rings, \item the topological spaces $\Spec(B)$, $\Spec(A)$ are w-local, and \item $\Spec(B) \to \Spec(A)$ is w-local. \end{enumerate} Then $A \to B$ is ind-Zariski.


proetale-lemma-w-local-extremally-disconnected

Let $A$ be a ring. The following are equivalent \begin{enumerate} \item every faithfully flat ring map $A \to B$ identifying local rings has a section, \item every faithfully flat ind-Zariski ring map $A \to B$ has a section, and \item $A$ satisfies \begin{enumerate} \item $\Spec(A)$ is w-local, and \item $\pi_0(\Spec(A))$ is extremally disconnected. \end{enumerate} \end{enumerate}


proetale-lemma-find-Zariski-w-contractible

Let $A$ be a ring. There exists a faithfully flat, ind-Zariski ring map $A \to B$ such that $B$ satisfies the equivalent conditions of Lemma \ref{lemma-w-local-extremally-disconnected}.


proetale-lemma-base-change-ind-etale

Let $A \to B$ and $A \to A'$ be ring maps. Let $B' = B \otimes_A A'$ be the base change of $B$. If $A \to B$ is ind-\'etale, then $A' \to B'$ is ind-\'etale.


proetale-lemma-composition-ind-etale

Let $A \to B$ and $B \to C$ be ring maps. If $A \to B$ and $B \to C$ are ind-\'etale, then $A \to C$ is ind-\'etale.


proetale-lemma-ind-ind-etale

A filtered colimit of ind-\'etale $A$-algebras is ind-\'etale over $A$.


proetale-lemma-ind-etale-permanence

Let $A$ be a ring. Let $B \to C$ be an $A$-algebra map of ind-\'etale $A$-algebras. Then $C$ is an ind-\'etale $B$-algebra.


proetale-lemma-lift-ind-etale

Let $A$ be a ring and let $I \subset A$ be an ideal. The base change functor $$ \text{ind-\'etale }A\text{-algebras} \longrightarrow \text{ind-\'etale }A/I\text{-algebras},\quad C \longmapsto C/IC $$ has a fully faithful right adjoint $v$. In particular, given an ind-\'etale $A/I$-algebra $\overline{C}$ there exists an ind-\'etale $A$-algebra $C = v(\overline{C})$ such that $\overline{C} = C/IC$.


proetale-lemma-first-construction

Given a ring $A$ there exists a faithfully flat ind-\'etale $A$-algebra $C$ such that every faithfully flat \'etale ring map $C \to B$ has a section.


proetale-lemma-have-sections-quotient

Let $A$ be a ring such that every faithfully flat \'etale ring map $A \to B$ has a section. Then the same is true for every quotient ring $A/I$.


proetale-lemma-have-sections-strictly-henselian

Let $A$ be a ring such that every faithfully flat \'etale ring map $A \to B$ has a section. Then every local ring of $A$ at a maximal ideal is strictly henselian.


proetale-lemma-have-sections-localize

Let $A$ be a ring such that every faithfully flat \'etale ring map $A \to B$ has a section. Let $Z \subset \Spec(A)$ be a closed subscheme of the form $D(f) \cap V(I)$ and let $A \to A_Z^\sim$ be as constructed in Lemma \ref{lemma-localization}. Then every faithfully flat \'etale ring map $A_Z^\sim \to C$ has a section.


proetale-lemma-get-w-local-algebraic-residue-field-extensions

Let $A \to B$ be a ring map inducing algebraic extensions on residue fields. There exists a commutative diagram $$ \xymatrix{ B \ar[r] & D \\ A \ar[r] \ar[u] & C \ar[u] } $$ with the following properties: \begin{enumerate} \item $A \to C$ is faithfully flat and ind-\'etale, \item $B \to D$ is faithfully flat and ind-\'etale, \item $\Spec(C)$ is w-local, \item $\Spec(D)$ is w-local, \item $\Spec(D) \to \Spec(C)$ is w-local, \item the set of closed points of $\Spec(D)$ is the inverse image of the set of closed points of $\Spec(C)$, \item the set of closed points of $\Spec(C)$ surjects onto $\Spec(A)$, \item the set of closed points of $\Spec(D)$ surjects onto $\Spec(B)$, \item for $\mathfrak m \subset C$ maximal the local ring $C_\mathfrak m$ is strictly henselian. \end{enumerate}


proetale-lemma-pro-V-V

Let $Y$ be an affine scheme. Let $X = \lim X_i$ be a directed limit of affine schemes over $Y$. The following are equivalent \begin{enumerate} \item $\{X \to Y\}$ is a standard V covering (Topologies, Definition \ref{topologies-definition-standard-V-covering}), and \item $\{X_i \to Y\}$ is a standard V covering for all $i$. \end{enumerate}


proetale-lemma-pro-h-V

Let $X \to Y$ be a morphism of affine schemes. The following are equivalent \begin{enumerate} \item $\{X \to Y\}$ is a standard V covering (Topologies, Definition \ref{topologies-definition-standard-V-covering}), \item $X = \lim X_i$ is a directed limit of affine schemes over $Y$ such that $\{X_i \to Y\}$ is a ph covering for each $i$, and \item $X = \lim X_i$ is a directed limit of affine schemes over $Y$ such that $\{X_i \to Y\}$ is an h covering for each $i$. \end{enumerate}


proetale-lemma-h-limit-preserving

Let $S$ be a scheme. Let $F$ be a contravariant functor defined on the category of all schemes over $S$. If \begin{enumerate} \item $F$ satisfies the sheaf property for the h topology, and \item $F$ is limit preserving (Limits, Remark \ref{limits-remark-limit-preserving}), \end{enumerate} then $F$ satisfies the sheaf property for the V topology.


proetale-lemma-w-local-strictly-henselian-extremally-disconnected

Let $A$ be a ring. The following are equivalent \begin{enumerate} \item $A$ is w-contractible, \item every faithfully flat, ind-\'etale ring map $A \to B$ has a section, and \item $A$ satisfies \begin{enumerate} \item $\Spec(A)$ is w-local, \item $\pi_0(\Spec(A))$ is extremally disconnected, and \item for every maximal ideal $\mathfrak m \subset A$ the local ring $A_\mathfrak m$ is strictly henselian. \end{enumerate} \end{enumerate}


proetale-lemma-finite-finitely-presented-over-extremally-disconnected

Let $A \to B$ be a quasi-finite and finitely presented ring map. If the residue fields of $A$ are separably algebraically closed and $\Spec(A)$ is extremally disconnected, then $\Spec(B)$ is extremally disconnected.


proetale-lemma-finite-finitely-presented-over-w-contractible

Let $A \to B$ be a finite and finitely presented ring map. If $A$ is w-contractible, so is $B$.


proetale-lemma-localization-w-contractible

Let $A$ be a ring. Let $Z \subset \Spec(A)$ be a closed subset of the form $Z = V(f_1, \ldots, f_r)$. Set $B = A_Z^\sim$, see Lemma \ref{lemma-localization}. If $A$ is w-contractible, so is $B$.


proetale-lemma-recognize-proetale-covering

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent \begin{enumerate} \item $\{f_i : T_i \to T\}_{i \in I}$ is a pro-\'etale covering, \item each $f_i$ is weakly \'etale and $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering, \item each $f_i$ is weakly \'etale and for every affine open $U \subset T$ there exist quasi-compact opens $U_i \subset T_i$ which are almost all empty, such that $U = \bigcup f_i(U_i)$, \item each $f_i$ is weakly \'etale and there exists an affine open covering $T = \bigcup_{\alpha \in A} U_\alpha$ and for each $\alpha \in A$ there exist $i_{\alpha, 1}, \ldots, i_{\alpha, n(\alpha)} \in I$ and quasi-compact opens $U_{\alpha, j} \subset T_{i_{\alpha, j}}$ such that $U_\alpha = \bigcup_{j = 1, \ldots, n(\alpha)} f_{i_{\alpha, j}}(U_{\alpha, j})$. \end{enumerate} If $T$ is quasi-separated, these are also equivalent to \begin{enumerate} \item[(5)] each $f_i$ is weakly \'etale, and for every $t \in T$ there exist $i_1, \ldots, i_n \in I$ and quasi-compact opens $U_j \subset T_{i_j}$ such that $\bigcup_{j = 1, \ldots, n} f_{i_j}(U_j)$ is a (not necessarily open) neighbourhood of $t$ in $T$. \end{enumerate}


proetale-lemma-etale-proetale

Any \'etale covering and any Zariski covering is a pro-\'etale covering.


proetale-lemma-proetale

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is a pro-\'etale covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is a pro-\'etale covering and for each $i$ we have a pro-\'etale covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a pro-\'etale covering. \item If $\{T_i \to T\}_{i\in I}$ is a pro-\'etale covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is a pro-\'etale covering. \end{enumerate}


proetale-lemma-proetale-affine

Let $T$ be an affine scheme. Let $\{T_i \to T\}_{i \in I}$ be a pro-\'etale covering of $T$. Then there exists a pro-\'etale covering $\{U_j \to T\}_{j = 1, \ldots, n}$ which is a refinement of $\{T_i \to T\}_{i \in I}$ such that each $U_j$ is an affine scheme. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$.


proetale-lemma-fibre-products-proetale

Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale site containing $S$. Let $\Sch$ be the category of all schemes. \begin{enumerate} \item The categories $\Sch_\proetale$, $(\Sch/S)_\proetale$, $S_\proetale$, and $(\textit{Aff}/S)_\proetale$ have fibre products agreeing with fibre products in $\Sch$. \item The categories $\Sch_\proetale$, $(\Sch/S)_\proetale$, $S_\proetale$ have equalizers agreeing with equalizers in $\Sch$. \item The categories $(\Sch/S)_\proetale$, and $S_\proetale$ both have a final object, namely $S/S$. \item The category $\Sch_\proetale$ has a final object agreeing with the final object of $\Sch$, namely $\Spec(\mathbf{Z})$. \end{enumerate}


proetale-lemma-affine-big-site-proetale

Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale site containing $S$. The functor $(\textit{Aff}/S)_\proetale \to (\Sch/S)_\proetale$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\Sh((\textit{Aff}/S)_\proetale)$ to $\Sh((\Sch/S)_\proetale)$.


proetale-lemma-put-in-T

Let $\Sch_\proetale$ be a big pro-\'etale site. Let $f : T \to S$ be a morphism in $\Sch_\proetale$. The functor $T_\proetale \to (\Sch/S)_\proetale$ is cocontinuous and induces a morphism of topoi $$ i_f : \Sh(T_\proetale) \longrightarrow \Sh((\Sch/S)_\proetale) $$ For a sheaf $\mathcal{G}$ on $(\Sch/S)_\proetale$ we have the formula $(i_f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.


proetale-lemma-at-the-bottom

Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale site containing $S$. The inclusion functor $S_\proetale \to (\Sch/S)_\proetale$ satisfies the hypotheses of Sites, Lemma \ref{sites-lemma-bigger-site} and hence induces a morphism of sites $$ \pi_S : (\Sch/S)_\proetale \longrightarrow S_\proetale $$ and a morphism of topoi $$ i_S : \Sh(S_\proetale) \longrightarrow \Sh((\Sch/S)_\proetale) $$ such that $\pi_S \circ i_S = \text{id}$. Moreover, $i_S = i_{\text{id}_S}$ with $i_{\text{id}_S}$ as in Lemma \ref{lemma-put-in-T}. In particular the functor $i_S^{-1} = \pi_{S, *}$ is described by the rule $i_S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.


proetale-lemma-morphism-big

Let $\Sch_\proetale$ be a big pro-\'etale site. Let $f : T \to S$ be a morphism in $\Sch_\proetale$. The functor $$ u : (\Sch/T)_\proetale \longrightarrow (\Sch/S)_\proetale, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\Sch/S)_\proetale \longrightarrow (\Sch/T)_\proetale, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\Sch/T)_\proetale) \longrightarrow \Sh((\Sch/S)_\proetale) $$ We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


proetale-lemma-morphism-big-small

Let $\Sch_\proetale$ be a big pro-\'etale site. Let $f : T \to S$ be a morphism in $\Sch_\proetale$. \begin{enumerate} \item We have $i_f = f_{big} \circ i_T$ with $i_f$ as in Lemma \ref{lemma-put-in-T} and $i_T$ as in Lemma \ref{lemma-at-the-bottom}. \item The functor $S_\proetale \to T_\proetale$, $(U \to S) \mapsto (U \times_S T \to T)$ is continuous and induces a morphism of topoi $$ f_{small} : \Sh(T_\proetale) \longrightarrow \Sh(S_\proetale). $$ We have $f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. \item We have a commutative diagram of morphisms of sites $$ \xymatrix{ T_\proetale \ar[d]_{f_{small}} & (\Sch/T)_\proetale \ar[d]^{f_{big}} \ar[l]^{\pi_T}\\ S_\proetale & (\Sch/S)_\proetale \ar[l]_{\pi_S} } $$ so that $f_{small} \circ \pi_T = \pi_S \circ f_{big}$ as morphisms of topoi. \item We have $f_{small} = \pi_S \circ f_{big} \circ i_T = \pi_S \circ i_f$. \end{enumerate}


proetale-lemma-morphism-big-small-cartesian-diagram

Let $\Sch_\proetale$ be a big pro-\'etale site. Consider a cartesian diagram $$ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S } $$ in $\Sch_\proetale$. Then $i_g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.


proetale-lemma-alternative

Let $S$ be a scheme. Let $S_{affine, \proetale}$ denote the full subcategory of $S_\proetale$ consisting of affine objects. A covering of $S_{affine, \proetale}$ will be a standard pro-\'etale covering, see Definition \ref{definition-standard-proetale}. Then restriction $$ \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, \etale}} $$ defines an equivalence of topoi $\Sh(S_\proetale) \cong \Sh(S_{affine, \proetale})$.


proetale-lemma-affine-alternative

Let $S$ be an affine scheme. Let $S_{app}$ denote the full subcategory of $S_\proetale$ consisting of affine objects $U$ such that $\mathcal{O}(S) \to \mathcal{O}(U)$ is ind-\'etale. A covering of $S_{app}$ will be a standard pro-\'etale covering, see Definition \ref{definition-standard-proetale}. Then restriction $$ \mathcal{F} \longmapsto \mathcal{F}|_{S_{app}} $$ defines an equivalence of topoi $\Sh(S_\proetale) \cong \Sh(S_{app})$.


proetale-lemma-proetale-subcanonical

Let $S$ be a scheme. The topology on each of the pro-\'etale sites $\Sch_\proetale$, $S_\proetale$, $(\Sch/S)_\proetale$, $S_{affine, \proetale}$, and $(\textit{Aff}/S)_\proetale$ is subcanonical.


proetale-lemma-w-contractible-proetale-cover

Let $T = \Spec(A)$ be an affine scheme. The following are equivalent \begin{enumerate} \item $A$ is w-contractible, and \item every pro-\'etale covering of $T$ can be refined by a Zariski covering of the form $T = \coprod_{i = 1, \ldots, n} U_i$. \end{enumerate}


proetale-lemma-w-contractible-is-weakly-contractible

Let $\Sch_\proetale$ be a big pro-\'etale site as in Definition \ref{definition-big-proetale-site}. Let $T = \Spec(A)$ be an affine object of $\Sch_\proetale$. The following are equivalent \begin{enumerate} \item $A$ is w-contractible, \item $T$ is a weakly contractible (Sites, Definition \ref{sites-definition-w-contractible}) object of $\Sch_\proetale$, and \item every pro-\'etale covering of $T$ can be refined by a Zariski covering of the form $T = \coprod_{i = 1, \ldots, n} U_i$. \end{enumerate}


proetale-lemma-get-many-weakly-contractible

Let $\Sch_\proetale$ be a big pro-\'etale site as in Definition \ref{definition-big-proetale-site}. For every object $T$ of $\Sch_\proetale$ there exists a covering $\{T_i \to T\}$ in $\Sch_\proetale$ with each $T_i$ affine and the spectrum of a w-contractible ring. In particular, $T_i$ is weakly contractible in $\Sch_\proetale$.


proetale-lemma-proetale-enough-w-contractible

Let $S$ be a scheme. The pro-\'etale sites $S_\proetale$, $(\Sch/S)_\proetale$, $S_{affine, \proetale}$, and $(\textit{Aff}/S)_\proetale$ and if $S$ is affine $S_{app}$ have enough (affine) quasi-compact, weakly contractible objects, see Sites, Definition \ref{sites-definition-w-contractible}.


proetale-lemma-weakly-contractible-cover

Let $S$ be a scheme. The pro-\'etale sites $\Sch_\proetale$, $S_\proetale$, $(\Sch/S)_\proetale$ have the following property: for any object $U$ there exists a covering $\{V \to U\}$ with $V$ a weakly contractible object. If $U$ is quasi-compact, then we may choose $V$ affine and weakly contractible.


proetale-lemma-w-contractible-hypercovering

Let $X$ be a scheme. \begin{enumerate} \item For every object $U$ of $X_\proetale$ there exists a hypercovering $K$ of $U$ in $X_\proetale$ such that each term $K_n$ consists of a single weakly contractible object of $X_\proetale$ covering $U$. \item For every quasi-compact and quasi-separated object $U$ of $X_\proetale$ there exists a hypercovering $K$ of $U$ in $X_\proetale$ such that each term $K_n$ consists of a single affine and weakly contractible object of $X_\proetale$ covering $U$. \end{enumerate}


proetale-lemma-compute-cohomology

Let $X$ be a scheme. Let $E \in D^+(X_\proetale)$ be represented by a bounded below complex $\mathcal{E}^\bullet$ of abelian sheaves. Let $K$ be a hypercovering of $U \in \Ob(X_\proetale)$ with $K_n = \{U_n \to U\}$ where $U_n$ is a weakly contractible object of $X_\proetale$. Then $$ R\Gamma(U, E) = \text{Tot}(s(\mathcal{E}^\bullet(K))) $$ in $D(\textit{Ab})$.


proetale-lemma-presheaf-value-weakly-contractible

Let $X$ be a scheme. Let $\mathcal{F}$ be a presheaf of sets on $X_\proetale$ which sends finite disjoint unions to products. Then $\mathcal{F}^\#(W) = \mathcal{F}(W)$ if $W$ is an affine weakly contractible object of $X_\proetale$.


proetale-lemma-small-pullback-weakly-contractible

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a sheaf of sets on $X_\proetale$. If $W$ is an affine weakly contractible object of $X_\proetale$, then $$ f_{small}^{-1}\mathcal{F}(W) = \colim_{W \to V} \mathcal{F}(V) $$ where the colimit is over morphisms $W \to V$ over $Y$ with $V \in Y_\proetale$.


proetale-lemma-compare-injectives

Let $S$ be a scheme. Let $T$ be an object of $(\Sch/S)_\proetale$. \begin{enumerate} \item If $\mathcal{I}$ is injective in $\textit{Ab}((\Sch/S)_\proetale)$, then \begin{enumerate} \item $i_f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(T_\proetale)$, \item $\mathcal{I}|_{S_\proetale}$ is injective in $\textit{Ab}(S_\proetale)$, \end{enumerate} \item If $\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}((\Sch/S)_\proetale)$, then \begin{enumerate} \item $i_f^{-1}\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}(T_\proetale)$, \item $\mathcal{I}^\bullet|_{S_\proetale}$ is a K-injective complex in $\textit{Ab}(S_\proetale)$, \end{enumerate} \end{enumerate}


proetale-lemma-cohomological-descent-proetale

Let $S$ be a scheme. For $K \in D(S_\proetale)$ the map $$ K \longrightarrow R\pi_{S, *}\pi_S^{-1}K $$ is an isomorphism.


proetale-lemma-limit-pullback

Let $X$ be a scheme. Let $Y = \lim Y_i$ be the limit of a directed inverse system of quasi-compact and quasi-separated objects of $X_\proetale$ with affine transition morphisms. For any sheaf $\mathcal{F}$ on $X_\etale$ we have $\epsilon^{-1}\mathcal{F}(Y) = \colim \mathcal{F}(Y_i)$.


proetale-lemma-fully-faithful

Let $X$ be a scheme. For every sheaf $\mathcal{F}$ on $X_\etale$ the adjunction map $\mathcal{F} \to \epsilon_*\epsilon^{-1}\mathcal{F}$ is an isomorphism.


proetale-lemma-affine-vanishing

Let $X$ be an affine scheme. For injective abelian sheaf $\mathcal{I}$ on $X_\etale$ we have $H^p(X_\proetale, \epsilon^{-1}\mathcal{I}) = 0$ for $p > 0$.


proetale-lemma-relative-comparison

Let $X$ be a scheme. \begin{enumerate} \item For an abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $R\epsilon_*(\epsilon^{-1}\mathcal{F}) = \mathcal{F}$. \item For $K \in D^+(X_\etale)$ the map $K \to R\epsilon_*\epsilon^{-1}K$ is an isomorphism. \end{enumerate}


proetale-lemma-compare-cohomology-nonabelian

Let $X$ be a scheme. Let $\mathcal{G}$ be a sheaf of (possibly noncommutative) groups on $X_\etale$. We have $$ H^1(X_\etale, \mathcal{G}) = H^1(X_\proetale, \epsilon^{-1}\mathcal{G}) $$ where $H^1$ is defined as the set of isomorphism classes of torsors (see Cohomology on Sites, Section \ref{sites-cohomology-section-h1-torsors}).


proetale-lemma-compare-derived

Let $X$ be a scheme. Let $\Lambda$ be a ring. \begin{enumerate} \item The essential image of the fully faithful functor $\epsilon^{-1} : \textit{Mod}(X_\etale, \Lambda) \to \textit{Mod}(X_\proetale, \Lambda)$ is a weak Serre subcategory $\mathcal{C}$. \item The functor $\epsilon^{-1}$ defines an equivalence of categories of $D^+(X_\etale, \Lambda)$ with $D^+_\mathcal{C}(X_\proetale, \Lambda)$ with question inverse given by $R\epsilon_*$. \end{enumerate}


proetale-lemma-compare-locally-constant

Let $X$ be a scheme. Let $\Lambda$ be a ring. The functor $\epsilon^{-1}$ defines an equivalence of categories $$ \left\{ \begin{matrix} \text{locally constant sheaves}\\ \text{of }\Lambda\text{-modules on }X_\etale\\ \text{of finite presentation} \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{locally constant sheaves}\\ \text{of }\Lambda\text{-modules on }X_\proetale\\ \text{of finite presentation} \end{matrix} \right\} $$


proetale-lemma-naive-completion

Let $\mathcal{C}$ be a site. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. The left adjoint to the inclusion functor $D_{comp}(\mathcal{C}, \Lambda) \to D(\mathcal{C}, \Lambda)$ of Algebraic and Formal Geometry, Proposition \ref{algebraization-proposition-derived-completion} sends $K$ to $$ K^\wedge = R\lim(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}) $$ In particular, $K$ is derived complete if and only if $K = R\lim(K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n})$.


proetale-lemma-pushforward-Noetherian-case

Let $\Lambda$ be a Noetherian ring. Let $I \subset \Lambda$ be an ideal. Let $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a morphism of topoi. Then \begin{enumerate} \item $Rf_*$ sends $D_{comp}(\mathcal{D}, \Lambda)$ into $D_{comp}(\mathcal{C}, \Lambda)$, \item the map $Rf_* : D_{comp}(\mathcal{D}, \Lambda) \to D_{comp}(\mathcal{C}, \Lambda)$ has a left adjoint $Lf_{comp}^* : D_{comp}(\mathcal{C}, \Lambda) \to D_{comp}(\mathcal{D}, \Lambda)$ which is $Lf^*$ followed by derived completion, \item $Rf_*$ commutes with derived completion, \item for $K$ in $D_{comp}(\mathcal{D}, \Lambda)$ we have $Rf_*K = R\lim Rf_*(K \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n})$. \item for $M$ in $D_{comp}(\mathcal{C}, \Lambda)$ we have $Lf^*_{comp}M = R\lim Lf^*(M \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n})$. \end{enumerate}


proetale-lemma-morphism-comparison

Let $f : X \to Y$ be a morphism of schemes. \begin{enumerate} \item Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$. Then we have $f_{\proetale, *}\epsilon^{-1}\mathcal{F} = \epsilon^{-1}f_{\etale, *}\mathcal{F}$. \item Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Then we have $Rf_{\proetale, *}\epsilon^{-1}\mathcal{F} = \epsilon^{-1}Rf_{\etale, *}\mathcal{F}$. \end{enumerate}


proetale-lemma-finite

Let $f : Z \to X$ be a finite morphism of schemes which is locally of finite presentation. Then $f_{\proetale, *} : \textit{Ab}(Z_\proetale) \to \textit{Ab}(X_\proetale)$ is exact.


proetale-lemma-closed-immersion-affines

Let $i : Z \to X$ be a closed immersion morphism of affine schemes. Denote $X_{app}$ and $Z_{app}$ the sites introduced in Lemma \ref{lemma-affine-alternative}. The base change functor $$ u : X_{app} \to Z_{app},\quad U \longmapsto u(U) = U \times_X Z $$ is continuous and has a fully faithful left adjoint $v$. For $V$ in $Z_{app}$ the morphism $V \to v(V)$ is a closed immersion identifying $V$ with $u(v(V)) = v(V) \times_X Z$ and every point of $v(V)$ specializes to a point of $V$. The functor $v$ is cocontinuous and sends coverings to coverings.


proetale-lemma-closed-immersion-affines-apply

Let $Z \to X$ be a closed immersion morphism of affine schemes. The corresponding morphism of topoi $i = i_\proetale$ is equal to the morphism of topoi associated to the fully faithful cocontinuous functor $v : Z_{app} \to X_{app}$ of Lemma \ref{lemma-closed-immersion-affines}. It follows that \begin{enumerate} \item $i^{-1}\mathcal{F}$ is the sheaf associated to the presheaf $V \mapsto \mathcal{F}(v(V))$, \item for a weakly contractible object $V$ of $Z_{app}$ we have $i^{-1}\mathcal{F}(V) = \mathcal{F}(v(V))$, \item $i^{-1} : \Sh(X_\proetale) \to \Sh(Z_\proetale)$ has a left adjoint $i^{Sh}_!$, \item $i^{-1} : \textit{Ab}(X_\proetale) \to \textit{Ab}(Z_\proetale)$ has a left adjoint $i_!$, \item $\text{id} \to i^{-1}i^{Sh}_!$, $\text{id} \to i^{-1}i_!$, and $i^{-1}i_* \to \text{id}$ are isomorphisms, and \item $i_*$, $i^{Sh}_!$ and $i_!$ are fully faithful. \end{enumerate}


proetale-lemma-closed-immersion

Let $i : Z \to X$ be a closed immersion of schemes. Then \begin{enumerate} \item $i_\proetale^{-1}$ commutes with limits, \item $i_{\proetale, *}$ is fully faithful, and \item $i_\proetale^{-1}i_{\proetale, *} \cong \text{id}_{\Sh(Z_\proetale)}$. \end{enumerate}


proetale-lemma-thickening

Let $i : Z \to X$ be an integral universally injective and surjective morphism of schemes. Then $i_{\proetale, *}$ and $i_\proetale^{-1}$ are quasi-inverse equivalences of categories of pro-\'etale topoi.


proetale-lemma-compute-i-star

Let $i : Z \to X$ be a closed immersion of schemes. Let $U \to X$ be an object of $X_\proetale$ such that \begin{enumerate} \item $U$ is affine and weakly contractible, and \item every point of $U$ specializes to a point of $U \times_X Z$. \end{enumerate} Then $i_\proetale^{-1}\mathcal{F}(U \times_X Z) = \mathcal{F}(U)$ for all abelian sheaves on $X_\proetale$.


proetale-lemma-jshriek-comparison

Let $j : U \to X$ be an \'etale morphism of schemes. Let $\mathcal{G}$ be an abelian sheaf on $U_\etale$. Then $\epsilon^{-1} j_!\mathcal{G} = j_!\epsilon^{-1}\mathcal{G}$ as sheaves on $X_\proetale$.


proetale-lemma-jshriek-zero

Let $j : U \to X$ be a weakly \'etale morphism of schemes. Let $i : Z \to X$ be a closed immersion such that $U \times_X Z = \emptyset$. Let $V \to X$ be an affine object of $X_\proetale$ such that every point of $V$ specializes to a point of $V_Z = Z \times_X V$. Then $j_!\mathcal{F}(V) = 0$ for all abelian sheaves on $U_\proetale$.


proetale-lemma-ses-associated-to-open

Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme and let $U \subset X$ be the complement. Denote $i : Z \to X$ and $j : U \to X$ the inclusion morphisms. Assume that $j$ is a quasi-compact morphism. For every abelian sheaf on $X_\proetale$ there is a canonical short exact sequence $$ 0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0 $$ on $X_\proetale$ where all the functors are for the pro-\'etale topology.


proetale-lemma-compare-constructible

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. The functor $\epsilon^{-1}$ defines an equivalence of categories $$ \left\{ \begin{matrix} \text{constructible sheaves of}\\ \Lambda\text{-modules on }X_\etale\\ \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} \text{constructible sheaves of}\\ \Lambda\text{-modules on }X_\proetale\\ \end{matrix} \right\} $$ between constructible sheaves of $\Lambda$-modules on $X_\etale$ and constructible sheaves of $\Lambda$-modules on $X_\proetale$.


proetale-lemma-constructible-serre

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\Lambda$-modules on $X_\proetale$ is a weak Serre subcategory of $\textit{Mod}(X_\proetale, \Lambda)$.


proetale-lemma-compare-constructible-derived

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. Let $D_c(X_\etale, \Lambda)$, resp.\ $D_c(X_\proetale, \Lambda)$ be the full subcategory of $D(X_\etale, \Lambda)$, resp.\ $D(X_\proetale, \Lambda)$ consisting of those complexes whose cohomology sheaves are constructible sheaves of $\Lambda$-modules. Then $$ \epsilon^{-1} : D_c^+(X_\etale, \Lambda) \longrightarrow D_c^+(X_\proetale, \Lambda) $$ is an equivalence of categories.


proetale-lemma-tensor-c

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. Let $K, L \in D_c^-(X_\proetale, \Lambda)$. Then $K \otimes_\Lambda^\mathbf{L} L$ is in $D_c^-(X_\proetale, \Lambda)$.


proetale-lemma-compare-truncations-constructible

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. Let $K$ be an object of $D(X_\proetale, \Lambda)$. Set $K_n = K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}$. If $K_1$ is in $D^-_c(X_\proetale, \Lambda/I)$, then $K_n$ is in $D^-_c(X_\proetale, \Lambda/I^n)$ for all $n$.


proetale-lemma-Noetherian-constructible

Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $\mathcal{F}$ be a constructible $\Lambda$-sheaf on $X_\proetale$. Then there exists a finite partition $X = \coprod X_i$ by locally closed subschemes such that the restriction $\mathcal{F}|_{X_i}$ is lisse.


proetale-lemma-weird

Let $X$ be a weakly contractible affine scheme. Let $\Lambda$ be a Noetherian ring and $I \subset \Lambda$ be an ideal. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_\proetale$ such that \begin{enumerate} \item $\mathcal{F} = \lim \mathcal{F}/I^n\mathcal{F}$, \item $\mathcal{F}/I^n\mathcal{F}$ is a constant sheaf of $\Lambda/I^n$-modules, \item $\mathcal{F}/I\mathcal{F}$ is of finite type. \end{enumerate} Then $\mathcal{F} \cong \underline{M}^\wedge$ where $M$ is a finite $\Lambda^\wedge$-module.


proetale-lemma-connected-lisse

Let $X$ be a connected scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. If $\mathcal{F}$ is a lisse constructible $\Lambda$-sheaf on $X_\proetale$, then $\mathcal{F}$ is adic lisse.


proetale-lemma-derived-complete-zero

Let $X$ be a scheme. Let $\Lambda$ be a ring and let $I \subset \Lambda$ be a finitely generated ideal. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_\proetale$. If $\mathcal{F}$ is derived complete and $\mathcal{F}/I\mathcal{F} = 0$, then $\mathcal{F} = 0$.


proetale-lemma-derived-complete-limit

Let $X$ be a weakly contractible affine scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $\mathcal{F}$ be a derived complete sheaf of $\Lambda$-modules on $X_\proetale$ with $\mathcal{F}/I\mathcal{F}$ a locally constant sheaf of $\Lambda/I$-modules of finite type. Then there exists an integer $t$ and a surjective map $$ (\underline{\Lambda}^\wedge)^{\oplus t} \to \mathcal{F} $$


proetale-lemma-describe-constructible-complexes

In the situation above suppose $K$ is in $D_{cons}(X, \Lambda)$ and $X$ is quasi-compact. Set $K_n = K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}$. There exist $a, b$ such that \begin{enumerate} \item $K = R\lim K_n$ and $H^i(K) = 0$ for $i \not \in [a, b]$, \item each $K_n$ has tor amplitude in $[a, b]$, \item each $K_n$ has constructible cohomology sheaves, \item each $K_n = \epsilon^{-1}L_n$ for some $L_n \in D_{ctf}(X_\etale, \Lambda/I^n)$ (\'Etale Cohomology, Definition \ref{etale-cohomology-definition-ctf}). \end{enumerate}


proetale-lemma-weakly-contractible-locally-constant-ML

Let $X$ be a weakly contractible affine scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $K$ be an object of $D_{cons}(X, \Lambda)$ such that $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}$ is isomorphic in $D(X_\proetale, \Lambda/I^n)$ to a complex of constant sheaves of $\Lambda/I^n$-modules. Then $$ H^0(X, K \otimes_\Lambda^\mathbf{L} \Lambda/I^n) $$ has the Mittag-Leffler condition.


proetale-lemma-connected-adic-lisse

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 \ref{definition-adic-constructible}).


proetale-lemma-proetale-induced

Let $\Sch_\proetale$ be a big pro-\'etale site as in Definition \ref{definition-big-proetale-site}. Let $T \in \Ob(\Sch_\proetale)$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary pro-\'etale covering of $T$. There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_\proetale$ which refines $\{T_i \to T\}_{i \in I}$.


proetale-lemma-change-alpha

Suppose given big sites $\Sch_\proetale$ and $\Sch'_\proetale$ as in Definition \ref{definition-big-proetale-site}. Assume that $\Sch_\proetale$ is contained in $\Sch'_\proetale$. The inclusion functor $\Sch_\proetale \to \Sch'_\proetale$ satisfies the assumptions of Sites, Lemma \ref{sites-lemma-bigger-site}. There are morphisms of topoi \begin{eqnarray*} g : \Sh(\Sch_\proetale) & \longrightarrow & \Sh(\Sch'_\proetale) \\ f : \Sh(\Sch'_\proetale) & \longrightarrow & \Sh(\Sch_\proetale) \end{eqnarray*} such that $f \circ g \cong \text{id}$. For any object $S$ of $\Sch_\proetale$ the inclusion functor $(\Sch/S)_\proetale \to (\Sch'/S)_\proetale$ satisfies the assumptions of Sites, Lemma \ref{sites-lemma-bigger-site} also. Hence similarly we obtain morphisms \begin{eqnarray*} g : \Sh((\Sch/S)_\proetale) & \longrightarrow & \Sh((\Sch'/S)_\proetale) \\ f : \Sh((\Sch'/S)_\proetale) & \longrightarrow & \Sh((\Sch/S)_\proetale) \end{eqnarray*} with $f \circ g \cong \text{id}$.


proetale-proposition-maps-wich-identify-local-rings

Let $A \to B$ be a ring map which identifies local rings. Then there exists a faithfully flat, ind-Zariski ring map $B \to B'$ such that $A \to B'$ is ind-Zariski.


proetale-proposition-weakly-etale

Let $A \to B$ be a weakly \'etale ring map. Then there exists a faithfully flat, ind-\'etale ring map $B \to B'$ such that $A \to B'$ is ind-\'etale.


proetale-proposition-find-w-contractible

For every ring $A$ there exists a faithfully flat, ind-\'etale ring map $A \to D$ such that $D$ is w-contractible.


proetale-proposition-enough-weakly-contractibles

Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has enough weakly contractible objects. Let $\Lambda$ be a Noetherian ring. Let $I \subset \Lambda$ be an ideal. \begin{enumerate} \item The category of derived complete sheaves $\Lambda$-modules is a weak Serre subcategory of $\textit{Mod}(\mathcal{C}, \Lambda)$. \item A sheaf $\mathcal{F}$ of $\Lambda$-modules satisfies $\mathcal{F} = \lim \mathcal{F}/I^n\mathcal{F}$ if and only if $\mathcal{F}$ is derived complete and $\bigcap I^n\mathcal{F} = 0$. \item The sheaf $\underline{\Lambda}^\wedge$ is derived complete. \item If $\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$ is an inverse system of derived complete sheaves of $\Lambda$-modules, then $\lim \mathcal{F}_n$ is derived complete. \item An object $K \in D(\mathcal{C}, \Lambda)$ is derived complete if and only if each cohomology sheaf $H^p(K)$ is derived complete. \item An object $K \in D_{comp}(\mathcal{C}, \Lambda)$ is bounded above if and only if $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}$ is bounded above. \item An object $K \in D_{comp}(\mathcal{C}, \Lambda)$ is bounded if $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}$ has finite tor dimension. \end{enumerate}


formal-spaces-lemma-fully-faithful

Choose a category of schemes $\Sch_\alpha$ as in Sets, Lemma \ref{sets-lemma-construct-category}. Given a formal scheme $\mathfrak X$ let $$ h_\mathfrak X : (\Sch_\alpha)^{opp} \longrightarrow \textit{Sets},\quad h_\mathfrak X(S) = \Mor_{\textit{Formal Schemes}}(S, \mathfrak X) $$ be its functor of points. Then we have $$ \Mor_{\textit{Formal Schemes}}(\mathfrak X, \mathfrak Y) = \Mor_{\textit{PSh}(\Sch_\alpha)}(h_\mathfrak X, h_\mathfrak Y) $$ provided the size of $\mathfrak X$ is not too large.


formal-spaces-lemma-formal-scheme-sheaf-fppf

\begin{slogan} Formal schemes are fpqc sheaves \end{slogan} Let $\mathfrak X$ be a formal scheme. The functor of points $h_\mathfrak X$ (see Lemma \ref{lemma-fully-faithful}) satisfies the sheaf condition for fpqc coverings.


formal-spaces-lemma-closed

Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module and let $M_\lambda$, $\lambda \in \Lambda$ be a fundamental system of open submodules. Let $N \subset M$ be a submodule. The closure of $N$ is $\bigcap_{\lambda \in \Lambda} (N + M_\lambda)$.


formal-spaces-lemma-closure

Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Then \begin{enumerate} \item $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge$ is exact, and \item $N^\wedge$ is the closure of the image of $N \to M^\wedge$. \end{enumerate}


formal-spaces-lemma-quotient-by-closed

Let $R$ be a topological ring. Let $M$ be a complete, linearly topologized $R$-module. Let $N \subset M$ be a closed submodule. If $M$ has a countable fundamental system of neighbourhoods of $0$, then $M/N$ is complete and the map $M \to M/N$ is open.


formal-spaces-lemma-ses

\begin{reference} \cite[Theorem 8.1]{Ma} \end{reference} Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Assume $M$ has a countable fundamental system of neighbourhoods of $0$. Then \begin{enumerate} \item $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge \to 0$ is exact, \item $N^\wedge$ is the closure of the image of $N \to M^\wedge$, \item $M^\wedge \to (M/N)^\wedge$ is open. \end{enumerate}


formal-spaces-lemma-topologically-nilpotent

Let $B$ be a linearly topologized ring. The set of topologically nilpotent elements of $B$ is a closed, radical ideal of $B$. Let $\varphi : A \to B$ be a continuous map of linearly topologized rings. \begin{enumerate} \item If $f \in A$ is topologically nilpotent, then $\varphi(f)$ is topologically nilpotent. \item If $I \subset A$ consists of topologically nilpotent elements, then the closure of $\varphi(I)B$ consists of topologically nilpotent elements. \end{enumerate}


formal-spaces-lemma-closure-image-ideal

Let $A \to B$ be a continuous map of linearly topologized rings. Let $I \subset A$ be an ideal. The closure of $IB$ is the kernel of $B \to B \widehat{\otimes}_A A/I$.


formal-spaces-lemma-dense-image-surjective

Let $\varphi : A \to B$ be a continuous homomorphism of linearly topologized rings. If \begin{enumerate} \item $\varphi$ is taut, \item $\varphi$ has dense image, \item $A$ is complete, \item $B$ is separated, and \item $A$ has a countable fundamental system of neighbourhoods of $0$. \end{enumerate} Then $\varphi$ is surjective and open, $B$ is complete, and $B = A/K$ for some closed ideal $K \subset A$.


formal-spaces-lemma-taut-is-adic

Let $\varphi : A \to B$ be a continuous map of linearly topologized rings. Let $I \subset A$ be an ideal. Assume \begin{enumerate} \item $I$ is finitely generated, \item $A$ has the $I$-adic topology, \item $B$ is complete, and \item $\varphi$ is taut. \end{enumerate} Then the topology on $B$ is the $I$-adic topology.


formal-spaces-lemma-completed-tensor-product

Let $B \to A$ and $B \to C$ be continuous homomorphisms of linearly topologized rings. \begin{enumerate} \item If $A$ and $C$ are weakly pre-admissible, then $A \widehat{\otimes}_B C$ is weakly admissible. \item If $A$ and $C$ are pre-admissible, then $A \widehat{\otimes}_B C$ is admissible. \item If $A$ and $C$ have a countable fundamental system of open ideals, then $A \widehat{\otimes}_B C$ has a countable fundamental system of open ideals. \item If $A$ and $C$ are pre-adic and have finitely generated ideals of definition, then $A \widehat{\otimes}_B C$ is adic and has a finitely generated ideal of definition. \item If $A$ and $C$ are pre-adic Noetherian rings and $B/\mathfrak b \to A/\mathfrak a$ is of finite type where $\mathfrak a \subset A$ and $\mathfrak b \subset B$ are the ideals of topologically nilpotent elements, then $A \widehat{\otimes}_B C$ is adic Noetherian. \end{enumerate}


formal-spaces-lemma-diagonal-affine-formal-algebraic-space

Let $S$ be a scheme. If $X$ is an affine formal algebraic space over $S$, then the diagonal morphism $\Delta : X \to X \times_S X$ is representable and a closed immersion.


formal-spaces-lemma-covering-by-thickenings

Let $X_\lambda, \lambda \in \Lambda$ and $X = \colim X_\lambda$ be as in Definition \ref{definition-affine-formal-algebraic-space}. Then $X_\lambda \to X$ is representable and a thickening.


formal-spaces-lemma-factor-through-thickening

Let $X_\lambda, \lambda \in \Lambda$ and $X = \colim X_\lambda$ be as in Definition \ref{definition-affine-formal-algebraic-space}. If $Y$ is a quasi-compact algebraic space over $S$, then any morphism $Y \to X$ factors through an $X_\lambda$.


formal-spaces-lemma-mcquillan-affine-formal-algebraic-space

Let $S$ be a scheme. Let $X$ be an fppf sheaf on $(\Sch/S)_{fppf}$ which satisfies the set theoretic condition of Remark \ref{remark-set-theoretic}. The following are equivalent: \begin{enumerate} \item there exists a weakly admissible topological ring $A$ over $S$ (see Remark \ref{remark-mcquillan}) such that $X = \colim_{I \subset A\text{ weak ideal of definition}} \Spec(A/I)$, \item $X$ is an affine formal algebraic space and there exists an $S$-algebra $A$ and a map $X \to \Spec(A)$ such that for a closed immersion $T \to X$ with $T$ an affine scheme the composition $T \to \Spec(A)$ is a closed immersion, \item $X$ is an affine formal algebraic space and there exists an $S$-algebra $A$ and a map $X \to \Spec(A)$ such that for a closed immersion $T \to X$ with $T$ a scheme the composition $T \to \Spec(A)$ is a closed immersion, \item $X$ is an affine formal algebraic space and for some choice of $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space} the projections $\lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda}) \to \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$ are surjective, \item $X$ is an affine formal algebraic space and for any choice of $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space} the projections $\lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda}) \to \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$ are surjective. \end{enumerate} Moreover, the weakly admissible topological ring is $A = \lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$ endowed with its limit topology and the weak ideals of definition classify exactly the morphisms $T \to X$ which are representable and thickenings.


formal-spaces-lemma-morphism-between-formal-spectra

Let $S$ be a scheme. Let $A$, $B$ be weakly admissible topological rings over $S$. Any morphism $f : \text{Spf}(B) \to \text{Spf}(A)$ of affine formal algebraic spaces over $S$ is equal to $\text{Spf}(f^\sharp)$ for a unique continuous $S$-algebra map $f^\sharp : A \to B$.


formal-spaces-lemma-presentation-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a map of presheaves on $(\Sch/S)_{fppf}$. If $X$ is an affine formal algebraic space and $f$ is representable by algebraic spaces and locally quasi-finite, then $f$ is representable (by schemes).


formal-spaces-lemma-countable-affine-formal-algebraic-space

Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item there exists a system $X_1 \to X_2 \to X_3 \to \ldots$ of thickenings of affine schemes over $S$ such that $X = \colim X_n$, \item there exists a choice $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space} such that $\Lambda$ is countable. \end{enumerate}


formal-spaces-lemma-implications-between-types

Let $X$ be an affine formal algebraic space over a scheme $S$. \begin{enumerate} \item If $X$ is Noetherian, then $X$ is adic*. \item If $X$ is adic*, then $X$ is adic. \item If $X$ is adic, then $X$ is countably indexed. \item If $X$ is countably indexed, then $X$ is McQuillan. \end{enumerate}


formal-spaces-lemma-countably-indexed

Let $S$ be a scheme. Let $X$ be a presheaf on $(\Sch/S)_{fppf}$. The following are equivalent \begin{enumerate} \item $X$ is a countably indexed affine formal algebraic space, \item $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a countable fundamental system of neighbourhoods of $0$, \item $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a fundamental system $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots$ of weak ideals of definition, \item $X = \text{Spf}(A)$ where $A$ is a complete topological $S$-algebra with a fundamental system of open neighbourhoods of $0$ given by a countable sequence $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots$ of ideals such that $I_n/I_{n + 1}$ is locally nilpotent, and \item $X = \text{Spf}(A)$ where $A = \lim B/J_n$ with the limit topology where $B \supset J_1 \supset J_2 \supset J_3 \supset \ldots$ is a sequence of ideals in an $S$-algebra $B$ with $J_n/J_{n + 1}$ locally nilpotent. \end{enumerate}


formal-spaces-lemma-characterize-noetherian-affine

Let $S$ be a scheme. Let $X$ be an affine formal algebraic space. The following are equivalent \begin{enumerate} \item $X$ is Noetherian, \item $X$ is adic* and for some choice of $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space} the schemes $X_\lambda$ are Noetherian, \item $X$ is adic* and for any closed immersion $T \to X$ with $T$ a scheme, $T$ is Noetherian. \end{enumerate}


formal-spaces-lemma-diagonal-formal-algebraic-space

Let $S$ be a scheme. If $X$ is a formal algebraic space over $S$, then the diagonal morphism $\Delta : X \to X \times_S X$ is representable, a monomorphism, locally quasi-finite, locally of finite type, and separated.


formal-spaces-lemma-space-to-formal-space

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism from an algebraic space over $S$ to a formal algebraic space over $S$. Then $f$ is representable by algebraic spaces.


formal-spaces-lemma-reduction-formal-algebraic-space

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. There exists a reduced algebraic space $X_{red}$ and a representable morphism $X_{red} \to X$ which is a thickening. A morphism $U \to X$ with $U$ a reduced algebraic space factors uniquely through $X_{red}$.


formal-spaces-lemma-reduction-smooth

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces and smooth (for example \'etale). Then $X_{red} = X \times_Y Y_{red}$.


formal-spaces-lemma-reduction-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces. Then $f$ is surjective in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation} if and only if $f_{red} : X_{red} \to Y_{red}$ is a surjective morphism of algebraic spaces.


formal-spaces-lemma-colimit-is-formal

Let $S$ be a scheme. Suppose given a directed set $\Lambda$ and a system of algebraic spaces $(X_\lambda, f_{\lambda \mu})$ over $\Lambda$ where each $f_{\lambda \mu} : X_\lambda \to X_\mu$ is a thickening. Then $X = \colim_{\lambda \in \Lambda} X_\lambda$ is a formal algebraic space over $S$.


formal-spaces-lemma-completion-affine-is-affine-formal-algebraic-space

Let $S$ be a scheme. Let $X$ be an affine scheme over $S$. Let $T \subset |X|$ be a closed subset. Then the functor $$ (\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad U \longmapsto \{f : U \to X \mid f(|U|) \subset T\} $$ is a McQuillan affine formal algebraic space.


formal-spaces-lemma-completion-is-formal-algebraic-space

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Then the functor $$ (\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad U \longmapsto \{f : U \to X \mid f(|U|) \subset T\} $$ is a formal algebraic space.


formal-spaces-lemma-map-completions-representable

Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $T \subset |X|$ be a closed subset and let $T' = |f|^{-1}(T) \subset |X'|$. Then $$ \xymatrix{ X'_{/T'} \ar[r] \ar[d] & X' \ar[d]^f \\ X_{/T} \ar[r] & X } $$ is a cartesian diagram of sheaves. In particular, the morphism $X'_{/T'} \to X_{/T}$ is representable by algebraic spaces.


formal-spaces-lemma-reduction-completion

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. The reduction $(X_{/T})_{red}$ of the completion $X_{/T}$ of $X$ along $T$ is the reduced induced closed subspace $Z$ of $X$ corresponding to $T$.


formal-spaces-lemma-affine-formal-completion-types

Let $S$ be a scheme. Let $X = \Spec(A)$ be an affine scheme over $S$. Let $T \subset X$ be a closed subset. Let $X_{/T}$ be the formal completion of $X$ along $T$. \begin{enumerate} \item If $X \setminus T$ is quasi-compact, i.e., $T$ is constructible, then $X_{/T}$ is adic*. \item If $T = V(I)$ for some finitely generated ideal $I \subset A$, then $X_{/T} = \text{Spf}(A^\wedge)$ where $A^\wedge$ is the $I$-adic completion of $A$. \item If $X$ is Noetherian, then $X_{/T}$ is Noetherian. \end{enumerate}


formal-spaces-lemma-etale-covering-by-formal-algebraic-spaces

Let $S$ be a scheme. Let $\{X_i \to X\}_{i \in I}$ be a family of maps of sheaves on $(\Sch/S)_{fppf}$. Assume (a) $X_i$ is a formal algebraic space over $S$, (b) $X_i \to X$ is representable by algebraic spaces and \'etale, and (c) $\coprod X_i \to X$ is a surjection of sheaves. Then $X$ is a formal algebraic space over $S$.


formal-spaces-lemma-fibre-products-general

Let $S$ be a scheme. Let $X, Y$ be formal algebraic spaces over $S$ and let $Z$ be a sheaf whose diagonal is representable by algebraic spaces. Let $X \to Z$ and $Y \to Z$ be maps of sheaves. Then $X \times_Z Y$ is a formal algebraic space.


formal-spaces-lemma-fibre-products

Let $S$ be a scheme. The category of formal algebraic spaces over $S$ has fibre products.


formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The diagonal morphism $\Delta : X \to X \times_Y X$ is representable (by schemes), a monomorphism, locally quasi-finite, locally of finite type, and separated.


formal-spaces-lemma-characterize-quasi-separated

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item the reduction of $X$ (Lemma \ref{lemma-reduction-formal-algebraic-space}) is a quasi-separated algebraic space, \item for $U \to X$, $V \to X$ with $U$, $V$ quasi-compact schemes the fibre product $U \times_X V$ is quasi-compact, \item for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times_X V$ is quasi-compact. \end{enumerate}


formal-spaces-lemma-characterize-separated

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item the reduction of $X$ (Lemma \ref{lemma-reduction-formal-algebraic-space}) is a separated algebraic space, \item for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times_X V$ is affine and $$ \mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}(U \times_X V) $$ is surjective. \end{enumerate}


formal-spaces-lemma-fibre-product-affines-over-separated

Let $S$ be a scheme. Let $X \to Z$ and $Y \to Z$ be morphisms of formal algebraic spaces over $S$. Assume $Z$ separated. \begin{enumerate} \item If $X$ and $Y$ are affine formal algebraic spaces, then so is $X \times_Z Y$. \item If $X$ and $Y$ are McQuillan affine formal algebraic spaces, then so is $X \times_Z Y$. \item If $X$, $Y$, and $Z$ are McQuillan affine formal algebraic spaces corresponding to the weakly admissible topological $S$-algebras $A$, $B$, and $C$, then $X \times_Z Y$ corresponds to $A \widehat{\otimes}_C B$. \end{enumerate}


formal-spaces-lemma-separated-from-separated

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Let $U \to X$ be a morphism where $U$ is a separated algebraic space over $S$. Then $U \to X$ is separated.


formal-spaces-lemma-characterize-quasi-compact

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item the reduction of $X$ (Lemma \ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact algebraic space, \item we can find $\{X_i \to X\}_{i \in I}$ as in Definition \ref{definition-formal-algebraic-space} with $I$ finite, \item there exists a morphism $Y \to X$ representable by algebraic spaces which is \'etale and surjective and where $Y$ is an affine formal algebraic space. \end{enumerate}


formal-spaces-lemma-characterize-quasi-compact-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item the induced map $f_{red} : X_{red} \to Y_{red}$ between reductions (Lemma \ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact morphism of algebraic spaces, \item for every quasi-compact scheme $T$ and morphism $T \to Y$ the fibre product $X \times_Y T$ is a quasi-compact formal algebraic space, \item for every affine scheme $T$ and morphism $T \to Y$ the fibre product $X \times_Y T$ is a quasi-compact formal algebraic space, and \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} such that each $X \times_Y Y_j$ is a quasi-compact formal algebraic space. \end{enumerate}


formal-spaces-lemma-quasi-compact-representable

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces. Then $f$ is quasi-compact in the sense of Definition \ref{definition-quasi-compact-morphism} if and only if $f$ is quasi-compact in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation}.


formal-spaces-lemma-structure-quasi-compact-quasi-separated

\begin{reference} \cite[Proposition 3.32]{Yasuda} \end{reference} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated formal algebraic space over $S$. Then $X = \colim X_\lambda$ for a system of algebraic spaces $(X_\lambda, f_{\lambda \mu})$ over a directed set $\Lambda$ where each $f_{\lambda \mu} : X_\lambda \to X_\mu$ is a thickening.


formal-spaces-lemma-characterize-affine

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X$ is an affine formal algebraic space if and only if its reduction $X_{red}$ (Lemma \ref{lemma-reduction-formal-algebraic-space}) is affine.


formal-spaces-lemma-permanence-representable

Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of formal algebraic spaces over $S$. If $g \circ f : X \to Z$ is representable by algebraic spaces, then $f : X \to Y$ is representable by algebraic spaces.


formal-spaces-lemma-representable-by-algebraic-spaces-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item the morphism $f$ is representable by algebraic spaces, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are formal algebraic spaces, the vertical arrows are representable by algebraic spaces, $U \to X$ is surjective \'etale, and $U \to V$ is representable by algebraic spaces, \item for any commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are formal algebraic spaces and the vertical arrows are representable by algebraic spaces, the morphism $U \to V$ is representable by algebraic spaces, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $j$ a covering $\{X_{ji} \to Y_j \times_Y X\}$ as in Definition \ref{definition-formal-algebraic-space} such that $X_{ji} \to Y_j$ is representable by algebraic spaces for each $j$ and $i$, \item there exist a covering $\{X_i \to X\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \to Y_i \to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \to Y$ is representable by algebraic spaces, such that $X_i \to Y_i$ is representable by algebraic spaces, and \item add more here. \end{enumerate}


formal-spaces-lemma-algebraic-space-over-affine-formal

Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space over $S$. Let $f : X \to Y$ be a map of sheaves on $(\Sch/S)_{fppf}$ which is representable by algebraic spaces. Then $X$ is a formal algebraic space.


formal-spaces-lemma-affine-representable-by-algebraic-spaces

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then $f$ is representable (by schemes) and affine.


formal-spaces-lemma-property-goes-up-affine-morphism

Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space. Let $f : X \to Y$ be a map of sheaves on $(\Sch/S)_{fppf}$ which is representable and affine. Then \begin{enumerate} \item $X$ is an affine formal algebraic space. \item if $Y$ is countably indexed, then $X$ is countably indexed. \item if $Y$ is adic*, then $X$ is adic*, \item if $Y$ is Noetherian and $f$ is (locally) of finite type, then $X$ is Noetherian. \end{enumerate}


formal-spaces-lemma-property-goes-up-affine

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then \begin{enumerate} \item if $Y$ is countably indexed, then $X$ is countably indexed. \item if $Y$ is adic*, then $X$ is adic*, \item if $Y$ is Noetherian and $f$ is (locally) of finite type, then $X$ is Noetherian. \end{enumerate}


formal-spaces-lemma-representable-affine

Let $S$ be a scheme. Let $\varphi : A \to B$ be a continuous map of weakly admissible topological rings over $S$. The following are equivalent \begin{enumerate} \item $\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)$ is representable by algebraic spaces, \item $\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)$ is representable (by schemes), \item $\varphi$ is taut, see Definition \ref{definition-taut}. \end{enumerate}


formal-spaces-lemma-etale

Let $S$ be a scheme. Let $Y$ be a McQuillan affine formal algebraic space over $S$, i.e., $Y = \text{Spf}(B)$ for some weakly admissible topological $S$-algebra $B$. Then there is an equivalence of categories between \begin{enumerate} \item the category of morphisms $f : X \to Y$ of affine formal algebraic spaces which are representable by algebraic spaces and \'etale, and \item the category of topological $B$-algebras of the form $A^\wedge$ where $A$ is an \'etale $B$-algebra and $A^\wedge = \lim A/JA$ with $J \subset B$ running over the weak ideals of definition of $B$. \end{enumerate} The equivalence is given by sending $A^\wedge$ to  $X = \text{Spf}(A^\wedge)$. In particular, any $X$ as in (1) is McQuillan.


formal-spaces-lemma-etale-surjective

With notation and assumptions as in Lemma \ref{lemma-etale} let $f : X \to Y$ correspond to $B \to A^\wedge$. The following are equivalent \begin{enumerate} \item $f : X \to Y$ is surjective, \item $B \to A$ is faithfully flat, \item for every weak ideal of definition $J \subset B$ the ring map $B/J \to A/JA$ is faithfully flat, and \item for some weak ideal of definition $J \subset B$ the ring map $B/J \to A/JA$ is faithfully flat. \end{enumerate}


formal-spaces-lemma-iff-countably-indexed

Let $S$ be a scheme. Let $X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then $X$ is countably indexed if and only if $Y$ is countably indexed.


formal-spaces-lemma-iff-adic-star

Let $S$ be a scheme. Let $X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then $X$ is adic* if and only if $Y$ is adic*.


formal-spaces-lemma-iff-noetherian

Let $S$ be a scheme. Let $X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and (locally) of finite type. Then $X$ is Noetherian if and only if $Y$ is Noetherian.


formal-spaces-lemma-type-local

Let $S$ be a scheme. Let $P \in \{countably\ indexed, adic*, Noetherian\}$. Let $X$ be a formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item if $Y$ is an affine formal algebraic space and $f : Y \to X$ is representable by algebraic spaces and \'etale, then $Y$ has property $P$, \item for some $\{X_i \to X\}_{i \in I}$ as in Definition \ref{definition-formal-algebraic-space} each $X_i$ has property $P$. \end{enumerate}


formal-spaces-lemma-formal-completion-types

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Let $X_{/T}$ be the formal completion of $X$ along $T$. \begin{enumerate} \item If $X \setminus T \to X$ is quasi-compact, then $X_{/T}$ is locally adic*. \item If $X$ is locally Noetherian, then $X_{/T}$ is locally Noetherian. \end{enumerate}


formal-spaces-lemma-types-fibre-products

Let $S$ be a scheme. Let $X \to Y$ and $Z \to Y$ be morphisms of formal algebraic space over $S$. Then \begin{enumerate} \item If $X$ and $Z$ are locally countably indexed, then $X \times_Y Z$ is locally countably indexed. \item If $X$ and $Z$ are locally adic*, then $X \times_Y Z$ is locally adic*. \item If $X$ and $Z$ are locally Noetherian and $X_{red} \to Y_{red}$ is locally of finite type, then $X \times_Y Z$ is locally Noetherian. \end{enumerate}


formal-spaces-lemma-structure-locally-noetherian

Let $S$ be a scheme. Let $X$ be a locally Noetherian formal algebraic space over $S$. Then $X = \colim X_n$ for a system $X_1 \to X_2 \to X_3 \to \ldots$ of finite order thickenings of locally Noetherian algebraic spaces over $S$ where $X_1 = X_{red}$ and $X_n$ is the $n$th infinitesimal neighbourhood of $X_1$ in $X_m$ for all $m \geq n$.


formal-spaces-lemma-completion-in-sub

Let $A \in \Ob(\textit{WAdm})$. Let $A \to A'$ be a ring map (no topology). Let $(A')^\wedge = \lim_{I \subset A\text{ w.i.d}} A'/IA'$ be the object of $\textit{WAdm}$ constructed in Example \ref{example-representable-morphism-from-completion}. \begin{enumerate} \item If $A$ is in $\textit{WAdm}^{count}$, so is $(A')^\wedge$. \item If $A$ is in $\textit{WAdm}^{adic*}$, so is $(A')^\wedge$. \item If $A$ is in $\textit{WAdm}^{Noeth}$ and $A'$ is Noetherian, then $(A')^\wedge$ is in $\textit{WAdm}^{Noeth}$. \end{enumerate}


formal-spaces-lemma-property-defines-property-morphisms

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$. The following are equivalent \begin{enumerate} \item for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and \'etale, the morphism $U \to V$ corresponds to a morphism of $\textit{WAdm}^{count}$ with property $P$, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $j$ a covering $\{X_{ji} \to Y_j \times_Y X\}$ as in Definition \ref{definition-formal-algebraic-space} such that each $X_{ji} \to Y_j$  corresponds to a morphism of $\textit{WAdm}^{count}$ with property $P$, and \item there exist a covering $\{X_i \to X\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \to Y_i \to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \to Y$ is representable by algebraic spaces and \'etale, and $X_i \to Y_i$ corresponds to a morphism of $\textit{WAdm}^{count}$ with property $P$. \end{enumerate}


formal-spaces-lemma-base-change-property-morphisms

Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which is stable under base change. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $f$ satisfies the equivalent conditions of Lemma \ref{lemma-property-defines-property-morphisms} then so does $\text{pr}_2 : X \times_Y Z \to Z$.


formal-spaces-lemma-composition-property-morphisms

Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which is stable under composition. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $f$ and $g$ satisfies the equivalent conditions of Lemma \ref{lemma-property-defines-property-morphisms} then so does $g \circ f : X \to Z$.


formal-spaces-lemma-representable-property-rings

Let $B \to A$ be an arrow of $\textit{WAdm}^{count}$. The following are equivalent \begin{enumerate} \item[(a)] $B \to A$ is taut (Definition \ref{definition-taut}), \item[(b)] for $B \supset J_1 \supset J_2 \supset J_3 \supset \ldots$ a fundamental system of weak ideals of definitions there exist a commutative diagram $$ \xymatrix{ A \ar[r] & \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 \\ B \ar[r] \ar[u] & \ldots \ar[r] & B/J_3 \ar[r] \ar[u] & B/J_2 \ar[r] \ar[u] & B/J_1 \ar[u] } $$ such that $A_{n + 1}/J_nA_{n + 1} = A_n$ and $A = \lim A_n$ as topological ring. \end{enumerate} Moreover, these equivalent conditions define a local property, i.e., they satisfy axioms (\ref{item-axiom-1}), (\ref{item-axiom-2}), (\ref{item-axiom-3}).


formal-spaces-lemma-representable-local-property

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and \'etale, the morphism $U \to V$ corresponds to a taut map $B \to A$ of $\textit{WAdm}^{count}$, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $j$ a covering $\{X_{ji} \to Y_j \times_Y X\}$ as in Definition \ref{definition-formal-algebraic-space} such that each $X_{ji} \to Y_j$  corresponds to a taut ring map in $\textit{WAdm}^{count}$, \item there exist a covering $\{X_i \to X\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \to Y_i \to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \to Y$ is representable by algebraic spaces and \'etale, and $X_i \to Y_i$ corresponds to a taut ring map in $\textit{WAdm}^{count}$, and \item $f$ is representable by algebraic spaces. \end{enumerate}


formal-spaces-lemma-adic-homomorphism

Let $A$ and $B$ be pre-adic topological rings. Let $\varphi : A \to B$ be a continuous ring homomorphism. \begin{enumerate} \item If $\varphi$ is adic, then $\varphi$ is taut. \item If $B$ is complete, $A$ has a finitely generated ideal of definition, and $\varphi$ is taut, then $\varphi$ is adic. \end{enumerate} In particular the conditions ``$\varphi$ is adic'' and ``$\varphi$ is taut'' are equivalent on the category $\textit{WAdm}^{adic*}$.


formal-spaces-lemma-composition-finite-type

The composition of finite type morphisms is of finite type. The same holds for locally of finite type.


formal-spaces-lemma-base-change-finite-type

A base change of a finite type morphism is finite type. The same holds for locally of finite type.


formal-spaces-lemma-finite-type-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item the morphism $f$ is locally of finite type, \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are formal algebraic spaces, the vertical arrows are representable by algebraic spaces and \'etale, $U \to X$ is surjective, and $U \to V$ is locally of finite type, \item for any commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are formal algebraic spaces and vertical arrows representable by algebraic spaces and \'etale, the morphism $U \to V$ is locally of finite type, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $j$ a covering $\{X_{ji} \to Y_j \times_Y X\}$ as in Definition \ref{definition-formal-algebraic-space} such that $X_{ji} \to Y_j$ is locally of finite type for each $j$ and $i$, \item there exist a covering $\{X_i \to X\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \to Y_i \to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \to Y$ is representable by algebraic spaces and \'etale, such that $X_i \to Y_i$ is locally of finite type, and \item add more here. \end{enumerate}


formal-spaces-lemma-locally-finite-type-locally-noetherian

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. If $Y$ is locally Noetherian and $f$ locally of finite type, then $X$ is locally Noetherian.


formal-spaces-lemma-fibre-product-Noetherian

Let $S$ be a scheme. Let $f : X \to Y$ and $Z \to Y$ be morphisms of formal algebraic spaces over $S$. If $Z$ is locally Noetherian and $f$ locally of finite type, then $Z \times_Y X$ is locally Noetherian.


formal-spaces-lemma-composition-surjective

The composition of two surjective morphisms is a surjective morphism.


formal-spaces-lemma-base-change-surjective

A base change of a surjective morphism is a surjective morphism.


formal-spaces-lemma-characterize-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is surjective, \item for every scheme $T$ and morphism $T \to Y$ the projection $X \times_Y T \to T$ is a surjective morphism of formal algebraic spaces, \item for every affine scheme $T$ and morphism $T \to Y$ the projection $X \times_Y T \to T$ is a surjective morphism of formal algebraic spaces, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} such that each $X \times_Y Y_j \to Y_j$ is a surjective morphism of formal algebraic spaces, \item there exists a surjective morphism $Z \to Y$ of formal algebraic spaces such that $X \times_Y Z \to Z$ is surjective, and \item add more here. \end{enumerate}


formal-spaces-lemma-composition-monomorphism

The composition of two monomorphisms is a monomorphism.


formal-spaces-lemma-base-change-monomorphism

A base change of a monomorphism is a monomorphism.


formal-spaces-lemma-characterize-monomorphisms

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is a monomorphism, \item for every scheme $T$ and morphism $T \to Y$ the projection $X \times_Y T \to T$ is a monomorphism of formal algebraic spaces, \item for every affine scheme $T$ and morphism $T \to Y$ the projection $X \times_Y T \to T$ is a monomorphism of formal algebraic spaces, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} such that each $X \times_Y Y_j \to Y_j$ is a monomorphism of formal algebraic spaces, and \item there exists a family of morphisms $\{Y_j \to Y\}$ such that $\coprod Y_j \to Y$ is a surjection of sheaves on $(\Sch/S)_{fppf}$ such that each $X \times_Y Y_j \to Y_j$ is a monomorphism for all $j$, \item there exists a morphism $Z \to Y$ of formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and locally of finite presentation such that $X \times_Y Z \to X$ is a monomorphism, and \item add more here. \end{enumerate}


formal-spaces-lemma-composition-closed-immersion

The composition of two closed immersions is a closed immersion.


formal-spaces-lemma-base-change-closed-immersion

A base change of a closed immersion is a closed immersion.


formal-spaces-lemma-characterize-closed-immersion

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is a closed immersion, \item for every scheme $T$ and morphism $T \to Y$ the projection $X \times_Y T \to T$ is a closed immersion, \item for every affine scheme $T$ and morphism $T \to Y$ the projection $X \times_Y T \to T$ is a closed immersion, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} such that each $X \times_Y Y_j \to Y_j$ is a closed immersion, and \item there exists a morphism $Z \to Y$ of formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and locally of finite presentation such that $X \times_Y Z \to X$ is a closed immersion, and \item add more here. \end{enumerate}


formal-spaces-lemma-closed-immersion-into-McQuillan

Let $S$ be a scheme. Let $X$ be a McQuillan affine formal algebraic space over $S$. Let $f : Y \to X$ be a closed immersion of formal algebraic spaces over $S$. Then $Y$ is a McQuillan affine formal algebraic space and $f$ corresponds to a continuous homomorphism $A \to B$ of weakly admissible topological $S$-algebras which is taut, has closed kernel, and has dense image.


formal-spaces-lemma-monomorphism-iso-over-red

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces. Assume \begin{enumerate} \item $f$ is representable by algebraic spaces, \item $f$ is a monomorphism, \item the inclusion $Y_{red} \to Y$ factors through $f$, and \item $f$ is locally of finite type or $Y$ is locally Noetherian. \end{enumerate} Then $f$ is a closed immersion.


formal-spaces-lemma-topologically-finite-type-finite-type

Let $S$ be a scheme. Let $\varphi : A \to B$ be a continuous map of weakly admissible topological rings over $S$. The following are equivalent \begin{enumerate} \item $\text{Spf}(\varphi) : Y = \text{Spf}(B) \to \text{Spf}(A) = X$ is of finite type, \item $\varphi$ is taut and $B$ is topologically of finite type over $A$. \end{enumerate}


formal-spaces-lemma-category-affine-over

Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Assume $X$ is McQuillan and let $A$ be the weakly admissible topological ring associated to $X$. Then there is an anti-equivalence of categories between \begin{enumerate} \item the category $\mathcal{C}$ introduced above, and \item the category of maps $Y \to X$ of finite type of affine formal algebraic spaces. \end{enumerate}


formal-spaces-lemma-closed-immersion-into-countably-indexed

Let $S$ be a scheme. Let $X$ be a countably indexed affine formal algebraic space over $S$. Let $f : Y \to X$ be a closed immersion of formal algebraic spaces over $S$. Then $Y$ is a countably indexed affine formal algebraic space and $f$ corresponds to $A \to A/K$ where $A$ is an object of $\textit{WAdm}^{count}$ (Section \ref{section-morphisms-rings}) and $K \subset A$ is a closed ideal.


formal-spaces-lemma-quotient-restricted-power-series

Let $B \to A$ be an arrow of $\textit{WAdm}^{count}$, see Section \ref{section-morphisms-rings}. The following are equivalent \begin{enumerate} \item[(a)] $B \to A$ is taut and $B/J \to A/I$ is of finite type for every weak ideal of definition $J \subset B$ where $I \subset A$ is the closure of $JA$, \item[(b)] $B \to A$ is taut and $B/J_\lambda \to A/I_\lambda$ is of finite type for a cofinal system $(J_\lambda)$ of weak ideals of definition of $B$ where $I_\lambda \subset A$ is the closure of $J_\lambda A$, \item[(c)] $B \to A$ is taut and $A$ is topologically of finite type over $B$, \item[(d)] $A$ is isomorphic as a topological $B$-algebra to a quotient of $B\{x_1, \ldots, x_n\}$ by a closed ideal. \end{enumerate} Moreover, these equivalent conditions define a local property, i.e., they satisfy Axioms (\ref{item-axiom-1}), (\ref{item-axiom-2}), (\ref{item-axiom-3}).


formal-spaces-lemma-quotient-restricted-power-series-admissible

In Lemma \ref{lemma-quotient-restricted-power-series} if $B$ is admissible (for example adic), then the equivalent conditions (a) -- (d) are also equivalent to \begin{enumerate} \item[(e)] $B \to A$ is taut and $B/J \to A/I$ is of finite type for some ideal of definition $J \subset B$ where $I \subset A$ is the closure of $JA$. \end{enumerate}


formal-spaces-lemma-representable-affine-finite-type

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of affine formal algebraic spaces. Assume $Y$ countably indexed. The following are equivalent \begin{enumerate} \item $f$ is locally of finite type, \item $f$ is of finite type, \item $f$ corresponds to a morphism $B \to A$ of $\textit{WAdm}^{count}$ (Section \ref{section-morphisms-rings}) satisfying the equivalent conditions of Lemma \ref{lemma-quotient-restricted-power-series}. \end{enumerate}


formal-spaces-lemma-finite-type-local-property

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and \'etale, the morphism $U \to V$ corresponds to a morphism of $\textit{WAdm}^{count}$ which is taut and topologically of finite type, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $j$ a covering $\{X_{ji} \to Y_j \times_Y X\}$ as in Definition \ref{definition-formal-algebraic-space} such that each $X_{ji} \to Y_j$  corresponds to a morphism of $\textit{WAdm}^{count}$ which is taut and topologically of finite type, \item there exist a covering $\{X_i \to X\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \to Y_i \to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \to Y$ is representable by algebraic spaces and \'etale, and $X_i \to Y_i$ corresponds to a morphism of $\textit{WAdm}^{count}$ which is, taut and topologically of finite type, and \item $f$ is locally of finite type. \end{enumerate}


formal-spaces-lemma-base-change-separated

All of the separation axioms listed in Definition \ref{definition-separated-morphism} are stable under base change.


formal-spaces-lemma-fibre-product-after-map

Let $S$ be a scheme. Let $f : X \to Z$, $g : Y \to Z$ and $Z \to T$ be morphisms of formal algebraic spaces over $S$. Consider the induced morphism $i : X \times_Z Y \to X \times_T Y$. Then \begin{enumerate} \item $i$ is representable (by schemes), locally of finite type, locally quasi-finite, separated, and a monomorphism, \item if $Z \to T$ is separated, then $i$ is a closed immersion, and \item if $Z \to T$ is quasi-separated, then $i$ is quasi-compact. \end{enumerate}


formal-spaces-lemma-separated-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. Let $\mathcal{P}$ be any of the separation axioms of Definition \ref{definition-separated-morphism}. The following are equivalent \begin{enumerate} \item $f$ is $\mathcal{P}$, \item for every scheme $Z$ and morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is $\mathcal{P}$, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is $\mathcal{P}$, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the formal algebraic space $Z \times_Y X$ is $\mathcal{P}$ (see Definition \ref{definition-separated}), \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} such that the base change $Y_j \times_Y X \to Y_j$ has $\mathcal{P}$ for all $j$. \end{enumerate}


formal-spaces-lemma-proper-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is proper, \item for every scheme $Z$ and morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is proper, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is proper, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the formal algebraic space $Z \times_Y X$ is an algebraic space proper over $Z$, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} such that the base change $Y_j \times_Y X \to Y_j$ is proper for all $j$. \end{enumerate}


formal-spaces-lemma-base-change-proper

Proper morphisms of formal algebraic spaces are preserved by base change.


formal-spaces-lemma-sheaf-fpqc

\begin{slogan} Formal algebraic spaces are fpqc sheaves \end{slogan} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X$ satisfies the sheaf property for the fpqc topology.


formal-spaces-lemma-map-into-affine

Let $S$ be a scheme. Let $A$ be a weakly admissible topological $S$-algebra. Let $X$ be an affine scheme over $S$. Then the natural map $$ \Mor_S(\Spec(A), X) \longrightarrow \Mor_S(\text{Spf}(A), X) $$ is bijective.


formal-spaces-lemma-map-into-scheme

Let $S$ be a scheme. Let $A$ be a weakly admissible topological $S$-algebra such that $A/I$ is a local ring for some weak ideal of definition $I \subset A$. Let $X$ be a scheme over $S$. Then the natural map $$ \Mor_S(\Spec(A), X) \longrightarrow \Mor_S(\text{Spf}(A), X) $$ is bijective.


formal-spaces-lemma-map-into-algebraic-space

Let $S$ be a scheme. Let $R$ be a complete local Noetherian $S$-algebra. Let $X$ be an algebraic space over $S$. Then the natural map $$ \Mor_S(\Spec(R), X) \longrightarrow \Mor_S(\text{Spf}(R), X) $$ is bijective.


formal-spaces-lemma-adic-into-completion

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that $X \setminus T \to X$ is quasi-compact. Let $R$ be a complete local Noetherian $S$-algebra. Then an adic morphism $p : \text{Spf}(R) \to X_{/T}$ corresponds to a unique morphism $g : \Spec(R) \to X$ such that $g^{-1}(T) = \{\mathfrak m_R\}$.


formal-spaces-lemma-etale-morphism-topoi

Let $S$ be a scheme, and let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. Assume $f$ is representable by algebraic spaces and \'etale. In this case there is a cocontinuous functor $j : X_\etale \to Y_\etale$. The morphism of topoi $f_{small}$ is the morphism of topoi associated to $j$, see Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi}. Moreover, $j$ is continuous as well, hence Sites, Lemma \ref{sites-lemma-when-shriek} applies.


formal-spaces-lemma-affine-identify-affine-etale

Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Then $X_{affine, \etale}$ is equivalent to the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that \begin{enumerate} \item $U$ is an affine formal algebraic space, \item $\varphi$ is representable by algebraic spaces and \'etale. \end{enumerate}


formal-spaces-lemma-affine-etale-mcquillan

Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Assume $X$ is McQuillan, i.e., equal to $\text{Spf}(A)$ for some weakly admissible topological $S$-algebra $A$. Then $(X_{affine, \etale})^{opp}$ is equivalent to the category whose \begin{enumerate} \item objects are $A$-algebras of the form $B^\wedge = \lim B/JB$ where $A \to B$ is an \'etale ring map and $J$ runs over the weak ideals of definition of $A$, and \item morphisms are continuous $A$-algebra homomorphisms. \end{enumerate}


formal-spaces-lemma-identify-spaces-etale

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X_{spaces, \etale}$ is equivalent to the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that $\varphi$ is representable by algebraic spaces and \'etale.


formal-spaces-lemma-identify-affine-etale

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X_{affine, \etale}$ is equivalent to the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that \begin{enumerate} \item $U$ is an affine formal algebraic space, \item $\varphi$ is representable by algebraic spaces and \'etale. \end{enumerate}


pione-theorem-fundamental-group

Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point of $X$. \begin{enumerate} \item The fibre functor $F_{\overline{x}}$ defines an equivalence of categories $$ \textit{F\'Et}_X \longrightarrow \textit{Finite-}\pi_1(X, \overline{x})\textit{-Sets} $$ \item Given a second geometric point $\overline{x}'$ of $X$ there exists an isomorphism $t : F_{\overline{x}} \to F_{\overline{x}'}$. This gives an isomorphism $\pi_1(X, \overline{x}) \to \pi_1(X, \overline{x}')$ compatible with the equivalences in (1). This isomorphism is independent of $t$ up to inner conjugation. \item Given a morphism $f : X \to Y$ of connected schemes denote $\overline{y} = f \circ \overline{x}$. There is a canonical continuous homomorphism $$ f_* : \pi_1(X, \overline{x}) \to \pi_1(Y, \overline{y}) $$ such that the diagram $$ \xymatrix{ \textit{F\'Et}_Y \ar[r]_{\text{base change}} \ar[d]_{F_{\overline{y}}} & \textit{F\'Et}_X \ar[d]^{F_{\overline{x}}} \\ \textit{Finite-}\pi_1(Y, \overline{y})\textit{-Sets} \ar[r]^{f_*} & \textit{Finite-}\pi_1(X, \overline{x})\textit{-Sets} } $$ is commutative. \end{enumerate}


pione-lemma-sheaves-point

Let $K$ be a field. Let $K^{sep}$ be a separable closure of $K$. Consider the profinite group $G = \text{Gal}(K^{sep}/K)$. The functor $$ \begin{matrix} \text{schemes \'etale over }K & \longrightarrow & G\textit{-Sets} \\ X/K & \longmapsto & \Mor_{\Spec(K)}(\Spec(K^{sep}), X) \end{matrix} $$ is an equivalence of categories.


pione-lemma-aut-inverse-limit

Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$ be a functor. The map (\ref{equation-embedding-product}) identifies $\text{Aut}(F)$ with a closed subgroup of $\prod_{X \in \Ob(\mathcal{C})} \text{Aut}(F(X))$. In particular, if $F(X)$ is finite for all $X$, then $\text{Aut}(F)$ is a profinite group.


pione-lemma-single-out-profinite

Let $G$ be a topological group. The automorphism group of the functor (\ref{equation-forgetful}) endowed with its profinite topology from Lemma \ref{lemma-aut-inverse-limit} is the profinite completion of $G$.


pione-lemma-second-fundamental-functor

Let $G$ be a topological group. Let $F : \textit{Finite-}G\textit{-Sets} \to \textit{Sets}$ be an exact functor with $F(X)$ finite for all $X$. Then $F$ is isomorphic to the functor (\ref{equation-forgetful}).


pione-lemma-epi-mono

Let $(\mathcal{C}, F)$ be a Galois category. Let $X \to Y \in \text{Arrows}(\mathcal{C})$. Then \begin{enumerate} \item $F$ is faithful, \item $X \to Y$ is a monomorphism $\Leftrightarrow F(X) \to F(Y)$ is injective, \item $X \to Y$ is an epimorphism $\Leftrightarrow F(X) \to F(Y)$ is surjective, \item an object $A$ of $\mathcal{C}$ is initial if and only if $F(A) = \emptyset$, \item an object $Z$ of $\mathcal{C}$ is final if and only if $F(Z)$ is a singleton, \item if $X$ and $Y$ are connected, then $X \to Y$ is an epimorphism, \item \label{item-one-element} if $X$ is connected and $a, b : X \to Y$ are two morphisms then $a = b$ as soon as $F(a)$ and $F(b)$ agree on one element of $F(X)$, \item if $X = \coprod_{i = 1, \ldots, n} X_i$ and $Y = \coprod_{j = 1, \ldots, m} Y_j$ where $X_i$, $Y_j$ are connected, then there is map $\alpha : \{1, \ldots, n\} \to \{1, \ldots, m\}$ such that $X \to Y$ comes from a collection of morphisms $X_i \to Y_{\alpha(i)}$. \end{enumerate}


pione-lemma-galois

Let $(\mathcal{C}, F)$ be a Galois category. For any connected object $X$ of $\mathcal{C}$ there exists a Galois object $Y$ and a morphism $Y \to X$.


pione-lemma-tame

\begin{reference} Compare with \cite[Definition 7.2.4]{BS}. \end{reference} Let $(\mathcal{C}, F)$ be a Galois category. Let $G = \text{Aut}(F)$ be as in Example \ref{example-from-C-F-to-G-sets}. For any connected $X$ in $\mathcal{C}$ the action of $G$ on $F(X)$ is transitive.


pione-lemma-functoriality-galois

Let $(\mathcal{C}, F)$ and $(\mathcal{C}', F')$ be Galois categories. Let $H : \mathcal{C} \to \mathcal{C}'$ be an exact functor. There exists an isomorphism $t : F' \circ H \to F$. The choice of $t$ determines a continuous homomorphism $h : G' = \text{Aut}(F') \to \text{Aut}(F) = G$ and a $2$-commutative diagram $$ \xymatrix{ \mathcal{C} \ar[r]_H \ar[d] & \mathcal{C}' \ar[d] \\ \textit{Finite-}G\textit{-Sets} \ar[r]^h & \textit{Finite-}G'\textit{-Sets} } $$ The map $h$ is independent of $t$ up to an inner automorphism of $G$. Conversely, given a continuous homomorphism $h : G' \to G$ there is an exact functor $H : \mathcal{C} \to \mathcal{C}'$ and an isomorphism $t$ recovering $h$ as above.


pione-lemma-functoriality-galois-surjective

In diagram (\ref{equation-translation}) the following are equivalent \begin{enumerate} \item $h : G' \to G$ is surjective, \item $H : \mathcal{C} \to \mathcal{C}'$ is fully faithful, \item if $X \in \Ob(\mathcal{C})$ is connected, then $H(X)$ is connected, \item if $X \in \Ob(\mathcal{C})$ is connected and there is a morphism $*' \to H(X)$ in $\mathcal{C}'$, then there is a morphism $* \to X$, and \item for any object $X$ of $\mathcal{C}$ the map $\Mor_\mathcal{C}(*, X) \to \Mor_{\mathcal{C}'}(*', H(X))$ is bijective. \end{enumerate} Here $*$ and $*'$ are final objects of $\mathcal{C}$ and $\mathcal{C}'$.


pione-lemma-composition-trivial

In diagram (\ref{equation-translation}) the following are equivalent \begin{enumerate} \item $h \circ h'$ is trivial, and \item the image of $H' \circ H$ consists of objects isomorphic to finite coproducts of final objects. \end{enumerate}


pione-lemma-functoriality-galois-ses

In diagram (\ref{equation-translation}) the following are equivalent \begin{enumerate} \item the sequence $G'' \xrightarrow{h'} G' \xrightarrow{h} G \to 1$ is exact in the following sense: $h$ is surjective, $h \circ h'$ is trivial, and $\Ker(h)$ is the smallest closed normal subgroup containing $\Im(h')$, \item $H$ is fully faithful and an object $X'$ of $\mathcal{C}'$ is in the essential image of $H$ if and only if $H'(X')$ is isomorphic to a finite coproduct of final objects, and \item $H$ is fully faithful, $H \circ H'$ sends every object to a finite coproduct of final objects, and for an object $X'$ of $\mathcal{C}'$ such that $H'(X')$ is a finite coproduct of final objects there exists an object $X$ of $\mathcal{C}$ and an epimorphism $H(X) \to X'$. \end{enumerate}


pione-lemma-functoriality-galois-injective

In diagram (\ref{equation-translation}) the following are equivalent \begin{enumerate} \item $h'$ is injective, and \item for every connected object $X''$ of $\mathcal{C}''$ there exists an object $X'$ of $\mathcal{C}'$ and a diagram $$ X'' \leftarrow Y'' \rightarrow H(X') $$ in $\mathcal{C}''$ where $Y'' \to X''$ is an epimorphism and $Y'' \to H(X')$ is a monomorphism. \end{enumerate}


pione-lemma-functoriality-galois-normal

In diagram (\ref{equation-translation}) the following are equivalent \begin{enumerate} \item the image of $h'$ is normal, and \item for every connected object $X'$ of $\mathcal{C}'$ such that there is a morphism from the final object of $\mathcal{C}''$ to $H'(X')$ we have that $H'(X')$ is isomorphic to a finite coproduct of final objects. \end{enumerate}


pione-lemma-finite-etale-covers-limits-colimits

Let $X$ be a scheme. The category $\textit{F\'Et}_X$ has finite limits and finite colimits and for any morphism $X' \to X$ the base change functor $\textit{F\'Et}_X \to \textit{F\'Et}_{X'}$ is exact.


pione-lemma-internal-hom-finite-etale

Let $X$ be a scheme. Given $U, V$ finite \'etale over $X$ there exists a scheme $W$ finite \'etale over $X$ such that $$ \Mor_X(X, W) = \Mor_X(U, V) $$ and such that the same remains true after any base change.


pione-lemma-finite-etale-connected-galois-category

Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point. The functor $$ F_{\overline{x}} : \textit{F\'Et}_X \longrightarrow \textit{Sets},\quad Y \longmapsto |Y_{\overline{x}}| $$ defines a Galois category (Definition \ref{definition-galois-category}).


pione-lemma-fundamental-group-Galois-group

Let $K$ be a field and set $X = \Spec(K)$. Let $\overline{K}$ be an algebraic closure and denote $\overline{x} : \Spec(\overline{K}) \to X$ the corresponding geometric point. Let $K^{sep} \subset \overline{K}$ be the separable algebraic closure. \begin{enumerate} \item The functor of Lemma \ref{lemma-sheaves-point} induces an equivalence $$ \textit{F\'Et}_X \longrightarrow \textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-Sets}. $$ compatible with $F_{\overline{x}}$ and the functor $\textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-Sets} \to \textit{Sets}$. \item This induces a canonical isomorphism $$ \text{Gal}(K^{sep}/K) \longrightarrow \pi_1(X, \overline{x}) $$ of profinite topological groups. \end{enumerate}


pione-lemma-what-equivalence-gives

Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes such that the base change functor $\textit{F\'Et}_Y \to \textit{F\'Et}_X$ is an equivalence of categories. In this case \begin{enumerate} \item $f$ induces a homeomorphism $\pi_0(X) \to \pi_0(Y)$, \item if $X$ or equivalently $Y$ is connected, then $\pi_1(X, \overline{x}) = \pi_1(Y, \overline{y})$. \end{enumerate}


pione-lemma-gabber

Let $(A, I)$ be a henselian pair. Set $X = \Spec(A)$ and $Z = \Spec(A/I)$. The functor $$ \textit{F\'Et}_X \longrightarrow \textit{F\'Et}_Z,\quad U \longmapsto U \times_X Z $$ is an equivalence of categories.


pione-lemma-thickening

Let $X \subset X'$ be a thickening of schemes. The functor $$ \textit{F\'Et}_{X'} \longrightarrow \textit{F\'Et}_X,\quad U' \longmapsto U' \times_{X'} X $$ is an equivalence of categories.


pione-lemma-finite-etale-on-proper-over-henselian

Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$ with closed fibre $X_0$. Then the functor $$ \textit{F\'Et}_X \to \textit{F\'Et}_{X_0},\quad U \longmapsto U_0 = U \times_X X_0 $$ is an equivalence of categories.