hcju/stacksqa huggingface dataset:
If $f$ is quasi-separated, then $g$ is quasi-separated by Lemma \ref{lemma-quasi-separated-gives-quasi-separated}. Conversely, assume $g$ is quasi-separated. We have to show that $f$ is quasi-separated. Exactly as in the proof of Lemma \ref{lemma-output-quasi-compact} we may check this over the members of a \'etale covering of $X$ by affine schemes using Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-separated-local} and Formal Spaces, Lemma \ref{formal-spaces-lemma-separated-local}. Thus we may and do assume $X$ is affine. \medskip\noindent Let $V \to X'$ be a surjective \'etale morphism where $V = \coprod_{j \in J} V_j$ is a disjoint union of affines. To show that $X'$ is quasi-separated, it suffices to show that $V_j \times_{X'} V_{j'}$ is quasi-compact for all $j, j' \in J$. Since $W$ is quasi-separated the fibre products $(V_j \times_Y V_{j'})_{/T} = (V_j)_{/T} \times_{X'_{/T}} (V_{j'})_{/T}$ are quasi-compact for all $j, j' \in J$. Since $X$ is Noetherian affine and $U' \to U$ is an isomorphism, we see that $$ (V_j \times_{X'} V_{j'}) \times_X U = (V_j \times_X V_{j'}) \times_X U $$ is quasi-compact. Hence we conclude by the equality $$ |V_j \times_{X'} V_{j'}| = |(V_j \times_{X'} V_{j'}) \times_X U| \amalg |(V_j \times_{X'} V_{j'})_{/T, red}| $$ and the fact that a formal algebraic space is quasi-compact if and only if its associated reduced algebraic space is so.
If $f$ is separated, then $g$ is separated and $\Delta_g$ is rig-surjective by Lemmas \ref{lemma-quasi-separated-gives-quasi-separated} and \ref{lemma-separated-mono-open-diagonal-rig-surjective}. Assume $g$ is separated and $\Delta_g$ is rig-surjective. Exactly as in the proof of Lemma \ref{lemma-output-quasi-compact} we may check this over the members of a \'etale covering of $X$ by affine schemes using Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-separated} (locality on the base of being separated for morphisms of algebraic spaces), Formal Spaces, Lemma \ref{formal-spaces-lemma-base-change-separated} (being separated for morphisms of formal algebraic spaces is preserved by base change), and Lemma \ref{lemma-base-change-rig-surjective} (being rig-surjective is preserved by base change). Thus we may and do assume $X$ is affine. Furthermore, we already know that $f : X' \to X$ is quasi-separated by Lemma \ref{lemma-output-quasi-separated}. \medskip\noindent By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-check-separated-dvr} and Remark \ref{spaces-cohomology-remark-variant} it suffices to show that given any commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X' \ar[d] \\ \Spec(R) \ar[r]^p \ar@{-->}[ru] & X' \times_X X' } $$ where $R$ is a complete discrete valuation ring with fraction field $K$, there is a dotted arrow making the diagram commute (as this will give the uniqueness part of the valuative criterion). Let $h : \Spec(R) \to X$ be the composition of $p$ with the morphism $Y \times_X Y \to X$. There are three cases: Case I: $h(\Spec(R)) \subset U$. This case is trivial because $U' = X' \times_X U \to U$ is an isomorphism. Case II: $h$ maps $\Spec(R)$ into $T$. This case follows from our assumption that $g : W \to X_{/T}$ is separated. Namely, if $Z$ denotes the reduced induced closed subspace structure on $T$, then $h$ factors through $Z$ and $$ W \times_{X_{/T}} Z = X' \times_X Z \longrightarrow Z $$ is separated by assumption (and for example Formal Spaces, Lemma \ref{formal-spaces-lemma-separated-local}) which implies we get the lifting property by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-check-separated-dvr} applied to the displayed arrow. Case III: $h(\Spec(K))$ is not in $T$ but $h$ maps the closed point of $\Spec(R)$ into $T$. In this case the corresponding morphism $$ p_{/T} : \text{Spf}(R) \longrightarrow (X' \times_X X')_{/T} = W \times_{X_{/T}} W $$ is an adic morphism (by Formal Spaces, Lemma \ref{formal-spaces-lemma-map-completions-representable} and Definition \ref{formal-spaces-definition-adic-morphism}). Hence our assumption that $\Delta_g : W \to W \times_{X_{/T}} W$ is rig-surjective implies we can lift $p_{/T}$ to a morphism $\text{Spf}(R) \to W = X'_{/T}$, see Lemma \ref{lemma-monomorphism-rig-surjective}. Algebraizing the composition $\text{Spf}(R) \to X'$ using Formal Spaces, Lemma \ref{formal-spaces-lemma-map-into-algebraic-space} we find a morphism $\Spec(R) \to X'$ lifting $p$ as desired.
If $f$ is proper, then $g$ is a formal modification by Lemma \ref{lemma-modification-gives-formal-modification}. Assume $g$ is a formal modification. By Lemmas \ref{lemma-output-quasi-compact} and \ref{lemma-output-separated} we see that $f$ is quasi-compact and separated. \medskip\noindent By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-check-proper-dvr} and Remark \ref{spaces-cohomology-remark-variant} it suffices to show that given any commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X' \ar[d]^f \\ \Spec(R) \ar[r]^p \ar@{-->}[ru] & X } $$ where $R$ is a complete discrete valuation ring with fraction field $K$, there is a dotted arrow making the diagram commute. There are three cases: Case I: $p(\Spec(R)) \subset U$. This case is trivial because $U' \to U$ is an isomorphism. Case II: $p$ maps $\Spec(R)$ into $T$. This case follows from our assumption that $g : W \to X_{/T}$ is proper. Namely, if $Z$ denotes the reduced induced closed subspace structure on $T$, then $p$ factors through $Z$ and $$ W \times_{X_{/T}} Z = X' \times_X Z \longrightarrow Z $$ is proper by assumption which implies we get the lifting property by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-check-proper-dvr} applied to the displayed arrow. Case III: $p(\Spec(K))$ is not in $T$ but $p$ maps the closed point of $\Spec(R)$ into $T$. In this case the corresponding morphism $$ p_{/T} : \text{Spf}(R) \longrightarrow X'_{/T} = W $$ is an adic morphism (by Formal Spaces, Lemma \ref{formal-spaces-lemma-map-completions-representable} and Definition \ref{formal-spaces-definition-adic-morphism}). Hence our assumption that $g : W \to X_{/T}$ be rig-surjective implies we can lift $g_{/T}$ to a morphism $\text{Spf}(R') \to W = X'_{/T}$ for some extension of complete discrete valuation rings $R \subset R'$. Algebraizing the composition $\text{Spf}(R') \to X'$ using Formal Spaces, Lemma \ref{formal-spaces-lemma-map-into-algebraic-space} we find a morphism $\Spec(R') \to X'$ lifting $p$ as desired.
If $f$ is \'etale, then $g$ is \'etale by Lemma \ref{lemma-map-completions-etale}. Conversely, assume $g$ is \'etale. Since $f$ is an isomorphism over $U$ we see that $f$ is \'etale over $U$. Thus it suffices to prove that $f$ is \'etale at any point of $X'$ lying over $T$. Denote $Z \subset X$ the reduced closed subspace whose underlying topological space is $|Z| = T \subset |X|$, see Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}. Letting $Z_n \subset X$ be the $n$th infinitesimal neighbourhood we have $X_{/T} = \colim Z_n$. Since $X'_{/T} = W \to X_{/T}$ we conclude that $f^{-1}(Z_n) = X' \times_X Z_n \to Z_n$ is \'etale by the assumed \'etaleness of $g$. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds} we conclude that $f$ is \'etale at points lying over $T$.
Set $X = \Spec(A)$ and $T = V(I)$. Set $X_1 = \Spec(B)$ and $T_1 = V(IB)$. By Theorem \ref{theorem-dilatations-general} (in fact we only need the affine case treated in Lemma \ref{lemma-dilatations-affine}) the category (\ref{equation-modification}) for $X$ and $T$ is equivalent to the the category of rig-\'etale morphisms $W \to X_{/T}$ of locally Noetherian formal algebraic spaces. Similarly, the the category (\ref{equation-modification}) for $X_1$ and $T_1$ is equivalent to the category of rig-\'etale morphisms $W_1 \to X_{1, /T_1}$ of locally Noetherian formal algebraic spaces. Since $X_{/T} = \text{Spf}(A^\wedge)$ and $X_{1, /T_1} = \text{Spf}(B^\wedge)$ (Formal Spaces, Lemma \ref{formal-spaces-lemma-affine-formal-completion-types}) we see that these categories are equivalent by our assumption that $A^\wedge \to B^\wedge$ is an isomorphism. We omit the verification that this equivalence is given by base change.
You can deduce this for the statements quasi-compact, quasi-separated, separated, and proper by using Lemmas \ref{lemma-output-quasi-compact} \ref{lemma-output-quasi-separated}, \ref{lemma-output-separated}, \ref{lemma-output-quasi-separated}, and \ref{lemma-output-proper} to translate the corresponding property into a property of the formal completion and using the argument of the proof of Lemma \ref{lemma-Noetherian-local-ring}. However, there is a direct argument using fpqc descent as follows. First, you can reduce to proving the lemma for $A \to A^\wedge$ and $B \to B^\wedge$ since $A^\wedge  \to B^\wedge$ is an isomorphism. Then note that $\{U \to \Spec(A), \Spec(A^\wedge) \to \Spec(A)\}$ is an fpqc covering with $U = \Spec(A) \setminus V(I)$ as before. The base change of $f$ by $U \to \Spec(A)$ is $\text{id}_U$ by definition of our category (\ref{equation-modification}). Let $P$ be a property of morphisms of algebraic spaces which is fpqc local on the base (Descent on Spaces, Definition \ref{spaces-descent-definition-property-morphisms-local}) such that $P$ holds for identity morphisms. Then we see that $P$ holds for $f$ if and only if $P$ holds for $g$. This applies to $P$ equal to quasi-compact, quasi-separated, separated, proper, and finite by Descent on Spaces, Lemmas \ref{spaces-descent-lemma-descending-property-quasi-compact}, \ref{spaces-descent-lemma-descending-property-quasi-separated}, \ref{spaces-descent-lemma-descending-property-separated}, \ref{spaces-descent-lemma-descending-property-proper}, and \ref{spaces-descent-lemma-descending-property-finite}.
Choose a presentation $B = A[x_1, \ldots, x_r]^\wedge/J$. Write $P = A[x_1, \ldots, x_r]^\wedge$. Choose generators $g_1, \ldots, g_m \in J$. Choose generators $k_1, \ldots, k_t$ of the module of relations between $g_1, \ldots, g_m$, i.e., such that $$ P^{\oplus t} \xrightarrow{k_1, \ldots, k_t} P^{\oplus m} \xrightarrow{g_1, \ldots, g_m} P \to B \to 0 $$ is a resolution. Write $k_i = (k_{i1}, \ldots, k_{im})$ so that we have \begin{equation} \label{equation-relations-straight-up} \sum\nolimits_j k_{ij}g_j = 0 \end{equation} for $i = 1, \ldots, t$. Denote $K = (k_{ij})$ the $m \times t$-matrix with entries $k_{ij}$. \medskip\noindent Let $A[x_1, \ldots, x_r]^h$ be the henselization of the pair $(A[x_1, \ldots, x_r], IA[x_1, \ldots, x_r])$, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization}. We may and do think of $A[x_1, \ldots, x_r]^h$ as a subring of $P = A[x_1, \ldots, x_r]^\wedge$, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization-Noetherian-pair}. Since $A$ is a Noetherian G-ring, so is $A[x_1, \ldots, x_r]$, see More on Algebra, Proposition \ref{more-algebra-proposition-finite-type-over-G-ring}. Hence we have approximation for the map $A[x_1, \ldots, x_r]^h \to A[x_1, \ldots, x_r]^\wedge = P$ with respect to the ideal generated by $I$, see Smoothing Ring Maps, Lemma \ref{smoothing-lemma-henselian-pair}. Choose a large enough integer $n$ as in Lemma \ref{lemma-close-enough}. By the approximation property we may choose $g'_1, \ldots, g'_m \in A[x_1, \ldots, x_r]^h$ and a matrix $K' = (k'_{ij}) \in \text{Mat}(m \times t, A[x_1, \ldots, x_r]^h)$ such that $\sum\nolimits_j k'_{ij}g'_j = 0$ in $A[x_1, \ldots, x_r]^h$ and such that $g_i - g'_i \in I^nP$ and $K - K' \in I^n\text{Mat}(m \times t, P)$. By our choice of $n$ we conclude that there is an isomorphism $$ B \to P/(g'_1, \ldots, g'_m) = \left(A[x_1, \ldots, x_r]^h/(g'_1, \ldots, g'_m)\right)^\wedge $$ This finishes the proof by Lemma \ref{lemma-algebraize-easy}.
Let $w' : X' \to W$ and $\hat w : X_{/T} \to W$ be morphisms which determine the same morphism $X'_{/T} \to W$ by composition with $X'_{/T} \to X$ and $X'_{/T} \to X_{/T}$. We have to prove there exists a unique morphism $w : X \to W$ whose composition with $X' \to X$ and $X_{/T} \to X$ recovers $w'$ and $\hat w$. The uniqueness is immediate from Lemma \ref{lemma-faithful-general}. \medskip\noindent The assumptions on $T$ and $f$ are preserved by base change by any \'etale morphism $X_1 \to X$ of algebraic spaces. Since formal algebraic spaces are sheaves for the \'etale topology and since we aready have the uniqueness, it suffices to prove existence after replacing $X$ by the members of an \'etale covering. Thus we may assume $X$ is an affine Noetherian scheme. \medskip\noindent Assume $X$ is an affine Noetherian scheme. We will construct the morphism $w : X \to W$ using the material in Pushouts of Spaces, Section \ref{spaces-pushouts-section-coequalizer-glue}. It makes sense to read a little bit of the material in that section before continuing the read the proof. \medskip\noindent Set $X'' = X' \times_X X'$ and consider the two morphisms $a = w' \circ \text{pr}_1 : X'' \to W$ and  $b = w' \circ \text{pr}_2 : X'' \to W$. Then we see that $a$ and $b$ agree over the open $U$ and that $a_{/T}$ and $b_{a/T}$ agree (as these are both equal to the composition $X''_{/T} \to X_{/T} \to W$ where the second arrow is $\hat w$). Thus by Lemma \ref{lemma-faithful-general} we see $a = b$. \medskip\noindent Denote $Z \subset X$ the reduced induced closed subscheme structure on $T$. For $n \geq 1$ denote $Z_n \subset X$ the $n$th infinitesimal neighbourhood of $Z$. Denote $w_n = \hat w|_{Z_n} : Z_n \to W$ so that we have $\hat w = \colim w_n$ on $X_{/T} = \colim Z_n$. Set $Y_n = X' \amalg Z_n$. Consider the two projections $$ s_n, t_n : R_n = Y_n \times_X Y_n \longrightarrow Y_n $$ Let $Y_n \to X_n \to X$ be the coequalizer of $s_n$ and $t_n$ as in Pushouts of Spaces, Section \ref{spaces-pushouts-section-coequalizer-glue} (in particular this coequalizer exists, has good properties, etc, see Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-coequalizer}). By the result $a = b$ of the previous parapgraph and the agreement of $w'$ and $\hat w$ over $X'_{/T}$ we see that the morphism $$ w' \amalg w_n : Y_n \longrightarrow W $$ equalizes the morphisms $s_n$ and $t_n$. Hence we see that for all $n \geq 1$ there is a morphism $w^n : X_n \to W$ compatible with $w'$ and $w_n$. Moreover, for $m \geq 1$ the composition $$ X_n \to X_{n + m} \xrightarrow{w^{n + m}} W $$ is equal to $w^n$ by construction (as the corresponding statement holds for $w' \amalg w_{n + m}$ and $w' \amalg w_n$). By Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-essentially-constant} and Remark \ref{spaces-pushouts-remark-essentially-constant} the system of algebraic spaces $X_n$ is essentially constant with value $X$ and we conclude.
Note that $e : U \to R$ is an immersion as it is a section of the morphism $s$, see Schemes, Lemma \ref{schemes-lemma-section-immersion}. Consider the following diagram $$ \xymatrix{ R \ar[r]_-{(1, i)} \ar[d]_t & R \times_{s, U, t} R \ar[d]^c \ar[rr]_{(\text{pr}_0, i \circ \text{pr}_1)} & & R \times_{t, U, t} R \\ U \ar[r]^e & R } $$ The square on the left is cartesian, because if $a \circ b = e$, then $b = i(a)$. The composition of the horizontal maps is the diagonal morphism of $t : R \to U$. The right top horizontal arrow is an isomorphism. Hence since $\Omega_{R/U}$ is the conormal sheaf of the composition it is isomorphic to the conormal sheaf of $(1, i)$. By Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial-flat} we get the surjection $t^*\mathcal{C}_{U/R} \to \Omega_{R/U}$ and if $c$ is flat, then this is an isomorphism. Since $c$ is a base change of $s$ by the properties of Diagram (\ref{equation-pull}) we conclude that if $s$ is flat, then $c$ is flat, see Morphisms, Lemma \ref{morphisms-lemma-base-change-flat}.
The existence and properties of the open $W \subset U$ are described in Descent, Lemma \ref{descent-lemma-largest-open-of-the-base}. In Diagram (\ref{equation-diagram}) let $W_1 \subset R$ be the maximal open subscheme over which the morphism $\text{pr}_1 : R \times_{s, U, t} R \to R$ has property $\mathcal{P}$. It follows from the aforementioned Descent, Lemma \ref{descent-lemma-largest-open-of-the-base} and the assumption that $\{s : R \to U\}$ and $\{t : R \to U\}$ are coverings for the $\tau$-topology that $t^{-1}(W) = W_1 = s^{-1}(W)$ as desired.
The existence and properties of the open $W \subset U$ are described in Descent, Lemma \ref{descent-lemma-largest-open-of-the-base}. The morphism $$ G \times_{U, t} R \longrightarrow R \times_{s, U} G, \quad (g, r) \longmapsto (r, r^{-1} \circ g \circ r) $$ is an isomorphism over $R$ (where $\circ$ denotes composition in the groupoid). Hence $s^{-1}(W) = t^{-1}(W)$ by the properties of $W$ proved in the aforementioned Descent, Lemma \ref{descent-lemma-largest-open-of-the-base}.
We repeatedly use the properties and the existence of diagram (\ref{equation-diagram}). By the properties of the diagram (and Schemes, Lemma \ref{schemes-lemma-points-fibre-product}) there exists a point $\xi$ of $R \times_{s, U, t} R$ with $\text{pr}_0(\xi) = r$ and $c(\xi) = r'$. Let $\tilde r = \text{pr}_1(\xi) \in R$. \medskip\noindent Proof of (1). Set $k = \kappa(\tilde r)$. Since $t(\tilde r) = u$ and $s(\tilde r) = u'$ we see that $k$ is a common extension of both $\kappa(u)$ and $\kappa(u')$ and in fact that both $(F_u)_k$ and $(F_{u'})_k$ are isomorphic to the fibre of $\text{pr}_1 : R \times_{s, U, t} R \to R$ over $\tilde r$. Hence (1) is proved. \medskip\noindent Part (2) follows since the point $\xi$ maps to $r$, resp.\ $r'$. \medskip\noindent Part (3) is clear from the above (using the point $\xi$ for $\tilde u$ and $\tilde u'$) and the definitions. \medskip\noindent If $s$ and $t$ are flat and of finite presentation, then they are open morphisms (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}). Hence the image of some affine open neighbourhood $V''$ of $\tilde r$ will cover an open neighbourhood $V$ of $u$, resp.\ $V'$ of $u'$. These can be used to show that properties (1) and (2) of the definition of ``locally on the base isomorphic in the $\tau$-topology''.
Apply Lemma \ref{lemma-local-source} with $g = \text{id}$ and $\mathcal{Q} =$``locally of finite presentation'', $\mathcal{R} =$``flat and locally of finite presentation'', and $\mathcal{P}=$``Cohen-Macaulay'', see Remark \ref{remark-local-source-apply}. This gives us an open $U' \subset U$ such that Let $t^{-1}(U') \subset R$ is the largest open subscheme of $R$ on which the morphism $s$ is Cohen-Macaulay. This proves (1). Let $i : R \to R$ be the inverse of the groupoid. Since $i$ is an isomorphism, and $s \circ i = t$ and $t \circ i = s$ we see that $s^{-1}(U')$ is also the largest open of $R$ on which $t$ is Cohen-Macaulay. This proves (2). By More on Morphisms, Lemma \ref{more-morphisms-lemma-flat-finite-presentation-CM-open} the open subset $t^{-1}(U')$ is dense in every fibre of $s : R \to U$. This proves (3). Same argument for (4). Part (5) is a formal consequence of (1) and (2) and the discussion of restrictions in Groupoids, Section \ref{groupoids-section-restrict-groupoid}.
The property of being locally of finite type is stable under composition and arbitrary base change, see Morphisms, Lemmas \ref{morphisms-lemma-composition-finite-type} and \ref{morphisms-lemma-base-change-finite-type}. Hence (1) is clear from Diagram (\ref{equation-restriction}). For the other cases, see Morphisms, Lemmas \ref{morphisms-lemma-composition-finite-presentation}, \ref{morphisms-lemma-base-change-finite-presentation}, \ref{morphisms-lemma-composition-flat}, and \ref{morphisms-lemma-base-change-flat}.
This follows formally from the properties of the diagram of Lemma \ref{lemma-double-restrict}. In the first case, note that the image of the morphism $h''$ is contained in the image of $h$, as $g'' = g \circ g'$. Hence we may replace the $U$ in the lower right corner of the diagram by $h(U' \times_U R)$. This explains the significance of conditions (1) and (2) in the lemma. In the second case, note that $\{\text{pr}_0 : U' \times_{g, U, t} R \to U'\}$ is a $\tau$-covering as a base change of $\tau$ and condition (a).
By Groupoids, Lemma \ref{groupoids-lemma-group-scheme-over-field-separated} the stabilizer group scheme $G \to U$ is separated. By Groupoids, Lemma \ref{groupoids-lemma-diagonal} the morphism $j = (t, s) : R \to U \times_S U$ is separated. As $U$ is the spectrum of a field the scheme $U \times_S U$ is affine (by the construction of fibre products in Schemes, Section \ref{schemes-section-fibre-products}). Hence $R$ is a separated scheme, see Schemes, Lemma \ref{schemes-lemma-separated-permanence}.
The morphism $U' \to U$ equals $\Spec(k') \to \Spec(k)$, hence is affine, surjective and flat. The morphisms $s, t : R \to U$ and the morphism $U' \to U$ are universally open by Morphisms, Lemma \ref{morphisms-lemma-scheme-over-field-universally-open}. Since $R$ is not empty and $U$ is the spectrum of a field the morphisms $s, t : R \to U$ are surjective and flat. Then you conclude by using Morphisms, Lemmas \ref{morphisms-lemma-base-change-surjective}, \ref{morphisms-lemma-composition-surjective}, \ref{morphisms-lemma-composition-open}, \ref{morphisms-lemma-base-change-affine}, \ref{morphisms-lemma-composition-affine}, \ref{morphisms-lemma-base-change-flat}, and \ref{morphisms-lemma-composition-flat}.
Write $U = \Spec(k)$. Consider a field extension $k \subset k'$. Denote $U' = \Spec(k')$. Let $R'$ be the restriction of $R$ via $U' \to U$. Set $Z' = Z \times_R R'$ and $W' = R' \times_R W$. Consider a point $\xi = (z, w)$ of $Z \times_{s, U, t} W$. Let $r \in R$ be the image of $z$ under $Z \to R$. Pick $k' \supset k$ and $r' \in R'$ as in Lemma \ref{lemma-groupoid-on-field-explain-points}. We can choose $z' \in Z'$ mapping to $z$ and $r'$. Then we can find $\xi' \in Z' \times_{s', U', t'} W'$ mapping to $z'$ and $\xi$. The open $c(r', W')$ (Lemma \ref{lemma-groupoid-on-field-move-point}) is contained in the image of $Z' \times_{s', U', t'} W' \to R'$. Observe that $Z' \times_{s', U', t'} W' = (Z \times_{s, U, t} W) \times_{R \times_{s, U, t} R} (R' \times_{s', U', t'} R')$. Hence the image of $Z' \times_{s', U', t'} W' \to R' \to R$ is contained in the image of $Z \times_{s, U, t} W \to R$. As $R' \to R$ is open (Lemma \ref{lemma-restrict-groupoid-on-field}) we conclude the image contains an open neighbourhood of the image of $\xi$ as desired.