hcju/stacksqa huggingface dataset:
We first prove (1). Assume $s, t$ locally of finite type. By Lemma \ref{lemma-make-pre-equivalence} we may assume that $R$ is a pre-equivalence relation. Let $k$ be an algebraically closed field over $B$. Suppose $\overline{u}, \overline{u}' \in U(k)$ are $R$-equivalent. Then for some extension field $k \subset \Omega$ there exists a point $\overline{r} \in R(\Omega)$ mapping to $(\overline{u}, \overline{u}') \in (U \times_B U)(\Omega)$, see Lemma \ref{lemma-weak-orbit-pre-equivalence}. Hence $$ Z = R \times_{j, U \times_B U, (\overline{u}, \overline{u}')} \Spec(k) $$ is nonempty. As $s$ is locally of finite type we see that also $j$ is locally of finite type, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}. This implies $Z$ is a nonempty algebraic space locally of finite type over the algebraically closed field $k$ (use Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-type}). Thus $Z$ has a $k$-valued point, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}. Hence we conclude there exists a $\overline{r} \in R(k)$ with $j(\overline{r}) = (\overline{u}, \overline{u}')$, and we conclude that $\overline{u}, \overline{u}'$ are $R$-equivalent as desired. \medskip\noindent The proof of part (2) is the same, except that it uses Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-large-enough} instead of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}. This shows that the assertion holds as soon as $|k| > \lambda(R)$ with $\lambda(R)$ as introduced just above Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}.
Consider the equalizer $$ Z = R \times_{(\phi, \phi) \circ j, X \times_B X, \Delta_{X/B}} X $$ algebraic space. Then $Z \to R$ is an immersion by assumption (3). By assumption (1) $|Z| \to |R|$ is surjective. This implies that $Z \to R$ is a bijective closed immersion (use Schemes, Lemma \ref{schemes-lemma-immersion-when-closed}) and by assumption (2) we conclude that $Z = R$.
The implications (1) $\Rightarrow$ (2) and (2) $\Rightarrow$ (3) follow from Lemma \ref{lemma-set-theoretic-pre-equivalence-geometric} and the definitions. The same lemma shows that (3) implies $j$ is a set-theoretic pre-equivalence relation. But of course condition (3) also implies that $j$ is universally injective, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-injective}, so that $j$ is indeed a set-theoretic equivalence relation. At this point we know that (1), (2), (3) are all equivalent. \medskip\noindent Condition (4) implies (3) without any further hypotheses on $j$. Assume $j$ is decent, or locally separated, or quasi-separated and the equivalent conditions (1), (2), (3) hold. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-when-universally-injective-radicial} we see that $j$ is radicial. Let $k$ be any algebraically closed field over $B$. Let $\overline{u}, \overline{u}' \in U(k)$ with $\overline{u} \sim_R \overline{u}'$. We see that $R \times_{U \times_B U, (\overline{u}, \overline{u}')} \Spec(k)$ is nonempty. Hence, as $j$ is radicial, its reduction is the spectrum of a field purely inseparable over $k$. As $k = \overline{k}$ we see that it is the spectrum of $k$. Whence a point $\overline{r} \in R(k)$ with $t(\overline{r}) = \overline{u}$ and $s(\overline{r}) = \overline{u}'$ as desired.
Assume $\phi$ is set-theoretically invariant. This means that for any algebraically closed field $k$ over $B$ and any $\overline{r} \in R(k)$ we have $\phi(s(\overline{r})) = \phi(t(\overline{r}))$. Hence $((\phi(t(\overline{r})), \phi(s(\overline{r}))), \overline{r})$ defines a point in the fibre product mapping to $\overline{r}$ via $p$. This shows that $p$ is surjective. Conversely, assume $p$ is surjective. Pick $\overline{r} \in R(k)$. As $p$ is surjective, we can find a field extension $k \subset K$ and a $K$-valued point $\tilde r$ of the fibre product with $p(\tilde r) = \overline{r}$. Then $q(\tilde r) \in U \times_X U$ maps to $(t(\overline{r}), s(\overline{r}))$ in $U \times_B U$ and we conclude that $\phi(s(\overline{r})) = \phi(t(\overline{r}))$. This proves that $\phi$ is set-theoretically invariant. \medskip\noindent The proofs of (2), (3), and (4) are omitted. Hint: Assume $k$ is an algebraically closed field over $B$ of large cardinality. Consider the associated diagram of sets $$ \xymatrix{ (U(k) \times_{X(k)} U(k)) \times_{U(k) \times U(k)} R(k) \ar[d]^q \ar[r]_-p & R(k) \ar[d]^j \\ U(k) \times_{X(k)} U(k) \ar[r]^c & U(k) \times U(k) } $$ By the lemmas above the equivalences posed in (2), (3), and (4) become set-theoretic questions related to the diagram we just displayed, using that surjectivity translates into surjectivity on $k$-valued points by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-large-enough}.
Note that since $U$, $R$ are locally of finite type over $B$ all of the morphisms $s, t, j, \phi$ are locally of finite type, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}. We will also use without further mention Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points}. Assume $\phi$ is an orbit space. Let $k$ be any algebraically closed field over $B$. Let $\overline{x} \in X(k)$. Consider $U \times_{\phi, X, \overline{x}} \Spec(k)$. This is a nonempty algebraic space which is locally of finite type over $k$. Hence it has a $k$-valued point. This shows the displayed map of the lemma is surjective. Suppose that $\overline{u}, \overline{u}' \in U(k)$ map to the same element of $X(k)$. By Definition \ref{definition-set-theoretically-invariant} this means that $\overline{u}, \overline{u}'$ are in the same $R$-orbit. By Lemma \ref{lemma-geometric-orbits} this means that they are equivalent under the equivalence relation generated by $j(R(k))$. Thus the displayed morphism is injective. \medskip\noindent Conversely, assume the displayed map is bijective for all algebraically closed fields $k$ over $B$. This condition clearly implies that $\phi$ is surjective. We have already assumed that $\phi$ is $R$-invariant. Finally, the injectivity of all the displayed maps implies that $\phi$ separates orbits. Hence $\phi$ is an orbit space.
The implications (3) $\Rightarrow$ (2) $\Rightarrow$ (1) are immediate. \medskip\noindent Let $X \to Y$ be an alteration with $X$ regular. Then $Y^\nu \to Y$ is finite by Lemma \ref{lemma-regular-alteration-implies}. Consider the factorization $f : X \to Y^\nu$ from  Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-normalization-normal}. The morphism $f$ is finite over an open $V \subset Y^\nu$ containing every point of codimension $\leq 1$ in $Y^\nu$ by Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-finite-in-codim-1}. Then $f$ is flat over $V$ by Algebra, Lemma \ref{algebra-lemma-CM-over-regular-flat} and the fact that a normal local ring of dimension $\leq 2$ is Cohen-Macaulay by Serre's criterion (Algebra, Lemma \ref{algebra-lemma-criterion-normal}). Then $V$ is regular by Algebra, Lemma \ref{algebra-lemma-descent-regular}. As $Y^\nu$ is Noetherian we conclude that $Y^\nu \setminus V = \{y_1, \ldots, y_m\}$ is finite. For each $i$ let $\mathcal{O}_{Y^\nu, y_i}^h$ be the henselian local ring. Then $X \times_Y \Spec(\mathcal{O}_{Y^\nu, y_i}^h)$ is a regular alteration of $\Spec(\mathcal{O}_{Y^\nu, y_i}^h)$ (some details omitted). By Lemma \ref{lemma-regular-alteration-implies-local} the completion of $\mathcal{O}_{Y^\nu, y_i}^h$ is normal. In this way we see that (1) $\Rightarrow$ (4). \medskip\noindent Assume (4). We have to prove (3). We may immediately replace $Y$ by its normalization. Let $y_1, \ldots, y_m \in |Y|$ be the singular points. Choose a collection of elementary \'etale neighbourhoods $(V_i, v_i) \to (Y, y_i)$ as in Section \ref{section-strategy}. For each $i$ the henselian local ring $\mathcal{O}_{Y^\nu, y_i}^h$ is the henselization of $\mathcal{O}_{V_i, v_i}$. Hence these rings have isomorphic completions. Thus by the result for schemes (Resolution of Surfaces, Theorem \ref{resolve-theorem-resolve}) we see that there exist finite sequences of normalized blowups $$ X_{i, n_i} \to X_{i, n_i - 1} \to \ldots \to V_i $$ blowing up only in points lying over $v_i$ such that $X_{i, n_i}$ is regular. By Lemma \ref{lemma-equivalence-sequence-normalized-blowups} there is a sequence of normalized blowing ups $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to Y $$ and of course $X_n$ is regular too (look at the local rings). This completes the proof.
Cases (1), (2), and (3) are special cases of (4). Choose an affine scheme $U = \Spec(B)$ and an \'etale morphism $U \to X$. The ring map $A \to B$ is of finite type. We have to show that $\dim(U_\kappa) \leq \dim(A) - 1$. Since $X$ is decent, the generic points of irreducible components of $U$ are the points lying over generic points of irreducible components of $|X|$, see Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-generic-points}. Hence the fibre of $\Spec(B) \to \Spec(A)$ over $(0)$ is the (finite) set of minimal primes $\mathfrak q_1, \ldots, \mathfrak q_r$ of $B$. Thus $A_f \to B_f$ is finite for some nonzero $f \in A$ (Algebra, Lemma \ref{algebra-lemma-generically-finite}). We conclude $\kappa(\mathfrak q_i)$ is a finite extension of the fraction field of $A$. Let $\mathfrak q \subset B$ be a prime lying over $\mathfrak m$. Then $$ \dim(B_\mathfrak q) = \max \dim((B/\mathfrak q_i)_{\mathfrak q}) \leq \dim(A) $$ the inequality by the dimension formula for $A \subset B/\mathfrak q_i$, see Algebra, Lemma \ref{algebra-lemma-dimension-formula}. However, the dimension of $B_\mathfrak q/\mathfrak m B_\mathfrak q$ (which is the local ring of $U_\kappa$ at the corresponding point) is at least one less because the minimal primes $\mathfrak q_i$ are not in $V(\mathfrak m)$. We conclude by Properties, Lemma \ref{properties-lemma-dimension}.
For $n = 1$ this is Limits of Spaces, Lemma \ref{spaces-limits-lemma-excision-modifications}. For $n > 1$ the lemma can be proved in exactly the same way or it can be deduced from it. For example, suppose that $g_i : Y_i \to U_i$ are objects of $\mathcal{C}_{U_i, u_i}$. Then by the case $n = 1$ we can find $f'_i : Y'_i \to X$ which are isomorphisms over $X \setminus \{x_i\}$ and whose base change to $U_i$ is $f_i$. Then we can set $$ f : Y = Y'_1 \times_X \ldots \times_X Y'_n \to X $$ This is an object of $\mathcal{C}_{X, \{x_1, \ldots, x_n\}}$ whose base change by $U_i \to X$ recovers $g_i$. Thus the functor is essentially surjective. We omit the proof of fully faithfulness.
A blowing up is a representable morphism. Hence in either case we inductively see that $Z_j \to X$ or $Z_{i, j} \to U_i$ is representable. Whence each $Z_j$ or $Z_{i, j}$ is a decent algebraic space by Decent Spaces, Lemma \ref{decent-spaces-lemma-representable-named-properties}. This shows that the assertions make sense (since blowing up is only defined for decent spaces). To prove the equivalence, let's start with a sequence of blowups $Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X$. The first morphism $Z_1 \to X$ is given by blowing up one of the $x_i$, say $x_1$. Applying $F$ to $Z_1 \to X$ we find a blowup $Z_{1, 1} \to U_1$ at $u_1$ is the blowing up at $u_1$ and otherwise $Z_{i, 0} = U_i$ for $i > 1$. In the next step, we either blow up one of the $x_i$, $i \geq 2$ on $Z_1$ or we pick a closed point $z_1$ of the fibre of $Z_1 \to X$ over $x_1$. In the first case it is clear what to do and in the second case we use that $(Z_1)_{x_1} \cong (Z_{1, 1})_{u_1}$ (Lemma \ref{lemma-equivalence-fibre}) to get a closed point $z_{1, 1} \in Z_{1, 1}$ corresponding to $z_1$. Then we set $Z_{1, 2} \to Z_{1, 1}$ equal to the blowing up in $z_{1, 1}$. Continuing in this manner we construct the factorizations of each $g_i$. \medskip\noindent Conversely, given sequences of blowups $Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_i$ we construct the sequence of blowing ups of $X$ in exactly the same manner.
Say $T = \{x_1, \ldots, x_r\}$. Pick elementary \'etale neighbourhoods $(U_i, u_i) \to (X, x_i)$ as in Section \ref{section-strategy}. For each $i$ the restriction $\mathcal{I}_i = \mathcal{I}|_{U_i} \subset \mathcal{O}_{U_i}$ is a quasi-coherent sheaf of ideals supported at $u_i$. The local ring of $U_i$ at $u_i$ is regular and has dimension $2$. Thus we may apply Resolution of Surfaces, Lemma \ref{resolve-lemma-make-ideal-principal} to find a sequence $$ X_{i, m_i} \to X_{i, m_i - 1} \to \ldots \to X_1 \to X_{i, 0} = U_i $$ of blowing ups in closed points lying over $u_i$ such that $\mathcal{I}_i \mathcal{O}_{X_{i, m_i}}$ is invertible. By Lemma \ref{lemma-equivalence-sequence-blowups} we find a sequence of blowing ups $$ X_m \to X_{m - 1} \to \ldots \to X_1 \to X_0 = X $$ as in the statement of the lemma whose base change to our $U_i$ produces the given sequences. It follows that $\mathcal{I}\mathcal{O}_{X_n}$ is an invertible ideal sheaf. Namely, we know this is true over $X \setminus \{x_1, \ldots, x_n\}$ and in an \'etale neighbourhood of the fibre of each $x_i$ it is true by construction.
By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-dominate-modification-by-blowup}  there exists a $U$-admissible blowup $X' \to X$ which dominates $Y \to X$. Hence we may assume there exists an ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$ such that $\mathcal{O}_X/\mathcal{I}$ is supported on $T$ and such that $Y$ is the blowing up of $X$ in $\mathcal{I}$. By Lemma \ref{lemma-make-ideal-principal}  there exists a sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X $$ where $X_{i + 1} \to X_i$ is the blowing up of $X_i$ at a closed point $x_i$ lying above a point of $T$ such that $\mathcal{I}\mathcal{O}_{X_n}$ is an invertible ideal sheaf. By the universal property of blowing up (Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-universal-property-blowing-up}) we find the desired factorization.
The blowup morphism $X' \to X$ is proper (as $X$ is locally Noetherian we may apply Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-blowing-up-projective}). Thus $X'$ is Nagata (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-type-nagata}). Therefore the normalization $X'' \to X'$ is finite (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-nagata-normalization}) and we conclude that $X'' \to X$ is proper as well (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-finite-proper} and \ref{spaces-morphisms-lemma-composition-proper}). It follows that the normalized blowing up is a normal (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-normalization-normal}) Nagata algebraic space.
Although one can prove this lemma directly for algebraic spaces, we will continue the approach used above to reduce it to the case of schemes. \medskip\noindent We will use that Noetherian algebraic spaces are quasi-separated and hence points have well defined residue fields (for example by Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood}). We will use the results of Morphisms of Spaces, Sections \ref{spaces-morphisms-section-nagata}, \ref{spaces-morphisms-section-dimension-formula}, and \ref{spaces-morphisms-section-normalization} without further mention. We may replace $Y$ by its normalization. Let $X_0 \to X$ be the normalization. The morphism $Y \to X$ factors through $X_0$. Thus we may assume that both $X$ and $Y$ are normal. \medskip\noindent Assume $X$ and $Y$ are normal. The morphism $f : Y \to X$ is an isomorphism over an open which contains every point of codimension $0$ and $1$ in $Y$ and every point of $Y$ over which the fibre is finite, see Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-modification-normal-iso-over-codimension-1}. Hence we see that there is a finite set of closed points $T \subset |X|$ such that $f$ is an isomorphism over $X \setminus T$. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-dominate-modification-by-blowup} there exists an $X \setminus T$-admissible blowup $Y' \to X$ which dominates $Y$. After replacing $Y$ by the normalization of $Y'$ we see that we may assume that $Y \to X$ is representable. \medskip\noindent Say $T = \{x_1, \ldots, x_r\}$. Pick elementary \'etale neighbourhoods $(U_i, u_i) \to (X, x_i)$ as in Section \ref{section-strategy}. For each $i$ the morphism $Y_i = Y \times_X U_i \to U_i$ is a proper birational morphism which is an isomorphism over $U_i \setminus \{u_i\}$. Thus we may apply Resolution of Surfaces, Lemma \ref{resolve-lemma-dominate-by-normalized-blowing-up} to find a sequence $$ X_{i, m_i} \to X_{i, m_i - 1} \to \ldots \to X_1 \to X_{i, 0} = U_i $$ of normalized blowing ups in closed points lying over $u_i$ such that $X_{i, m_i}$ dominates $Y_i$. By Lemma \ref{lemma-equivalence-sequence-normalized-blowups} we find a sequence of normalized blowing ups $$ X_m \to X_{m - 1} \to \ldots \to X_1 \to X_0 = X $$ as in the statement of the lemma whose base change to our $U_i$ produces the given sequences. It follows that $X_m$ dominates $Y$ by the equivalence of categories of Lemma \ref{lemma-equivalence}.
This follows immediately from the case of schemes by choosing an elementary \'etale neighbourhood $(U, u) \to (X, p)$ as in Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood}, setting $V = U \times_X Y = U \times_{\Spec(A)} \Spec(A^\wedge)$ and $v = (u, q)$. The case of schemes is Resolution of Surfaces, Lemma \ref{resolve-lemma-iso-completions}.
By \'etale localization the first two statements follow immediately from the counter part to this lemma for schemes, see Resolution of Surfaces, Lemma \ref{resolve-lemma-port-regularity-to-completion}. For part (3), since $Y \to X$ is surjective (as $A \to A^\wedge$ is faithfully flat) we see that $Y$ regular implies $X$ regular by part (1). Conversely, if $X$ is regular, then the henselian local rings of $Y$ are regular for all points of the special fibre. Let $y \in |Y|$ be a general point. Since $|Y| \to |\Spec(A^\wedge)|$ is closed in the proper case, we can find a specialization $y \leadsto y_0$ with $y_0$ in the closed fibre. Choose an elementary \'etale neighbourhood $(V, v_0) \to (Y, y_0)$ as in Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-space-elementary-etale-neighbourhood}. Since $Y$ is decent we can lift $y \leadsto y_0$ to a specialization $v \leadsto v_0$ in $V$ (Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-specialization}). Then we conclude that $\mathcal{O}_{V, v}$ is a localization of $\mathcal{O}_{V, v_0}$ hence regular and the proof is complete.
By \'etale localization, it suffices to prove this when $Y = \Spec(A)$ where $A$ is a Noetherian domain. Let $B$ be the integral closure of $A$ in its fraction field. Set $C = \Gamma(X, \mathcal{O}_X)$. By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-pushforward-coherent} we see that $C$ is a finite $A$-module. As $X$ is normal (Properties of Spaces, Lemma \ref{spaces-properties-lemma-regular-normal}) we see that $C$ is normal domain (Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-normal-integral-sections}). Thus $B \subset C$ and we conclude that $B$ is finite over $A$ as $A$ is Noetherian. \medskip\noindent There exists a nonempty open $V \subset Y$ such that $f^{-1}V \to V$ is finite, see Spaces over Fields, Definition \ref{spaces-over-fields-definition-alteration}. After shrinking $V$ we may assume that $f^{-1}V \to V$ is flat (Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-generic-flatness-reduced}). Thus $f^{-1}V \to V$ is faithfully flat. Then $V$ is regular by Algebra, Lemma \ref{algebra-lemma-descent-regular}.
Parts (1) and (2) follow from Lemma \ref{lemma-regular-alteration-implies}. We have to redo part of the proof of that lemma in order to set up notation for the proof of (3). Set $C = \Gamma(X, \mathcal{O}_X)$. By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-pushforward-coherent} we see that $C$ is a finite $A$-module. As $X$ is normal (Properties of Spaces, Lemma \ref{spaces-properties-lemma-regular-normal}) we see that $C$ is normal domain (Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-normal-integral-sections}). Thus $B \subset C$ and we conclude that $B$ is finite over $A$ as $A$ is Noetherian. By Resolution of Surfaces, Lemma \ref{resolve-lemma-algebra-helper} in order to prove (3) it suffices to show that the $\mathfrak m$-adic completion $C^\wedge$ is normal. \medskip\noindent By Algebra, Lemma \ref{algebra-lemma-completion-finite-extension} the completion $C^\wedge$ is the product of the completions of $C$ at the prime ideals of $C$ lying over $\mathfrak m$. There are finitely many of these and these are the maximal ideals $\mathfrak m_1, \ldots, \mathfrak m_r$ of $C$. (The corresponding result for $B$ explains the final statement of the lemma.) Thus replacing $A$ by $C_{\mathfrak m_i}$ and $X$ by $X_i = X \times_{\Spec(C)} \Spec(C_{\mathfrak m_i})$ we reduce to the case discussed in the next paragraph. (Note that $\Gamma(X_i, \mathcal{O}) = C_{\mathfrak m_i}$ by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-flat-base-change-cohomology}.) \medskip\noindent Here $A$ is a Noetherian local normal domain and $f : X \to \Spec(A)$ is a regular alteration with $\Gamma(X, \mathcal{O}_X) = A$. We have to show that the completion $A^\wedge$ of $A$ is a normal domain. By Lemma \ref{lemma-port-regularity-to-completion} $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$ is regular. Since $\Gamma(Y, \mathcal{O}_Y) = A^\wedge$ by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-flat-base-change-cohomology}. We conclude that $A^\wedge$ is normal as before. Namely, $Y$ is normal by Properties of Spaces, Lemma \ref{spaces-properties-lemma-regular-normal}. It is connected because $\Gamma(Y, \mathcal{O}_Y) = A^\wedge$ is local. Hence $Y$ is normal and integral (as connected and normal implies integral for separated algebraic spaces). Thus $\Gamma(Y, \mathcal{O}_Y) = A^\wedge$ is a normal domain by Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-normal-integral-sections}. This proves (3). \medskip\noindent Proof of (4). Let $\eta \in \Spec(A)$ denote the generic point and denote by a subscript $\eta$ the base change to $\eta$. Since $f$ is an alteration, the scheme $X_\eta$ is finite and faithfully flat over $\eta$. Since $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$ is regular by Lemma \ref{lemma-port-regularity-to-completion} we see that $Y_\eta$ is regular (as a limit of opens in $Y$). Then $Y_\eta \to \Spec(A^\wedge \otimes_A \kappa(\eta))$ is finite faithfully flat onto the generic formal fibre. We conclude by Algebra, Lemma \ref{algebra-lemma-descent-regular}.