hcju/stacksqa huggingface dataset:
Let $P'$ be one of the following properties of homomorphisms of rings: ``finite'', ``surjective'', ``flat'', ``finite type'', ``flat and of finite presentation'', ``\'etale''. Translated into algebra, the statement means the following: If $A \to B$ is an $R$-algebra homomorphism and $A$ and $B$ are glueable for $(R \to R', f)$, then $A_f \to B_f$ and $A \otimes_R R' \to B \otimes_R R'$ have $P'$ if and only if $A \to B$ has $P'$. \medskip\noindent By More on Algebra, Lemmas \ref{more-algebra-lemma-faithful-descent} and \ref{more-algebra-lemma-BL-flat} the algebraic statement is true for $P'$ equal to ``finite'' or ``flat''. \medskip\noindent If $A_f \to B_f$ and $A \otimes_R R' \to B \otimes_R R'$ are surjective, then $N = B/A$ is an $R$-module with $N_f = 0$ and $N \otimes_R R' = 0$ and hence vanishes by More on Algebra, Lemma \ref{more-algebra-lemma-BL-faithful}. Thus $A \to B$ is surjective. \medskip\noindent If $A_f \to B_f$ and $A \otimes_R R' \to B \otimes_R R'$ are finite type, then we can choose an $A$-algebra homomorphism $A[x_1, \ldots, x_n] \to B$ such that $A_f[x_1, \ldots, x_n] \to B_f$ and $(A \otimes_R R')[x_1, \ldots, x_n] \to B \otimes_R R'$ are surjective (small detail omitted). We conclude that $A[x_1, \ldots, x_n] \to B$ is surjective by the previous result. Thus $A \to B$ is of finite type. \medskip\noindent If $A_f \to B_f$ and $A \otimes_R R' \to B \otimes_R R'$ are flat and of finite presentation, then we know that $A \to B$ is flat and of finite type by what we have already shown. Choose a surjection $A[x_1, \ldots, x_n] \to B$ and denote $I$ the kernel. By flatness of $B$ over $A$ we see that $I_f$ is the kernel of $A_f[x_1, \ldots, x_n] \to B_f$ and $I \otimes_R R'$ is the kernel of $A \otimes_R R'[x_1, \ldots, x_n] \to B \otimes_R R'$. Thus $I_f$ is a finite $A_f[x_1, \ldots, x_n]$-module and $I \otimes_R R'$ is a finite $(A \otimes_R R')[x_1, \ldots, x_n]$-module. By More on Algebra, Lemma \ref{more-algebra-lemma-faithful-descent} applied to $I$ viewed as a module over $A[x_1, \ldots, x_n]$ we conclude that $I$ is a finitely generated ideal and we conclude $A \to B$ is flat and of finite presentation. \medskip\noindent If $A_f \to B_f$ and $A \otimes_R R' \to B \otimes_R R'$ are \'etale, then we know that $A \to B$ is flat and of finite presentation by what we have already shown. Since the fibres of $\Spec(B) \to \Spec(A)$ are isomorphic to fibres of $\Spec(B_f) \to \Spec(A_f)$ or $\Spec(B/fB) \to \Spec(A/fA)$, we conclude that $A \to B$ is unramified, see Morphisms, Lemmas \ref{morphisms-lemma-unramified-over-field} and \ref{morphisms-lemma-unramified-etale-fibres}. We conclude that $A \to B$ is \'etale by Morphisms, Lemma \ref{morphisms-lemma-flat-unramified-etale} for example.
Let $f, g : Y \to Z$ be two morphisms of algebraic spaces over $X$ with $Y$ and $Z$ glueable for $(R \to R', f)$ such that $f$ and $g$ are mapped to the same morphism in the category $\textit{Spaces}(U \leftarrow U' \to X')$. We have to show the equalizer $E \to Y$ of $f$ and $g$ is an isomorphism. Working \'etale locally on $Y$ we may assume $Y$ is an affine scheme. Then $E$ is a scheme and the morphism $E \to Y$ is a monomorphism and locally quasi-finite, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-properties-diagonal}. Moreover, the base change of $E \to Y$ to $U$ and to $X'$ is an isomorphism. As $Y$ is the disjoint union of the affine open $V = U \times_X Y$ and the affine closed $V(f) \times_X Y$, we conclude $E$ is the disjoint union of their isomorphic inverse images. It follows in particular that $E$ is quasi-compact. By Zariski's main theorem (More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}) we conclude that $E$ is quasi-affine. Set $B = \Gamma(E, \mathcal{O}_E)$ and $A = \Gamma(Y, \mathcal{O}_Y)$ so that we have an $R$-algebra homomorphism $A \to B$. Since $E \to Y$ becomes an isomorphism after base change to $U$ and $X'$ we obtain ring maps $B \to A_f$ and $B \to A \otimes_R R'$ agreeing as maps into $A \otimes_R R'_f$. Since $A$ is glueable for $(R \to R', f)$ we get a ring map $B \to A$ which is left inverse to the map $A \to B$. The corresponding morphism $Y = \Spec(A) \to \Spec(B)$ maps into the open subscheme $E \subset \Spec(B)$ pointwise because this is true after base change to $U$ and $X'$. Hence we get a morphism $Y \to E$ over $Y$. Since $E \to Y$ is a monomorhism we conclude $Y \to E$ is an isomorphism as desired.
Let $Y, Z$ be two algebraic spaces over $X$ which are both glueable for $(R \to R', f)$ and assume the diagonal of $Z$ is affine. Let $a : U \times_X Y \to U \times_X Z$ over $U$ and $b : X' \times_X Y \to X' \times_X Z$ over $X'$ be two morphisms of algebraic spaces which induce the same morphism $c : U' \times_X Y \to U' \times_X Z$ over $U'$. We want to construct a morphism $f : Y \to Z$ over $X$ which produces the morphisms $a$, $b$ on base change to $U$, $X'$. By the faithfulness of Lemma \ref{lemma-glueing-f}, it suffices to construct the morphism $f$ \'etale locally on $Y$ (details omitted). Thus we may and do assume $Y$ is affine. \medskip\noindent Let $y \in |Y|$ be a point. If $y$ maps into the open $U \subset X$, then $U \times_X Y$ is an open of $Y$ on which the morphism $f$ is defined (we can just take $a$). Thus we may assume $y$ maps into the closed subset $V(f)$ of $X$. Since $R/fR = R'/fR'$ there is a unique point $y' \in |X' \times_X Y|$ mapping to $y$. Denote $z' = b(y') \in |X' \times_X Z|$ and $z \in |Z|$ the images of $y'$. Choose an \'etale neighbourhood $(W, w) \to (Z, z)$ with $W$ affine. Observe that $$ (U \times_X W) \times_{U \times_X Z, a} (U \times_X Y),\quad (U' \times_X W) \times_{U' \times_X Z, c} (U' \times_X Y), $$ and $$ (X' \times_X W) \times_{X' \times_X Z, b} (X' \times_X Y) $$ form an object of $\textit{Spaces}(U \leftarrow U' \to X')$ with affine parts (this is where we use that $Z$ has affine diagonal). Hence by Lemma \ref{lemma-glueing-affines} there exists a unique affine scheme $V$ glueable for $(R \to R', f)$ such that $$ (U \times_X V, U' \times_X V, X' \times_X V) $$ is the triple displayed above. By fully faithfulness for the affine case (Lemma \ref{lemma-glueing-affines}) we get a unique morphisms $V \to W$ and $V \to Y$ agreeing with the first and second projection morphisms over $U$ and $X'$ in the construction above. By Lemma \ref{lemma-glueing-affines-etale} the morphism $V \to Y$ is \'etale. To finish the proof, it suffices to show that there is a point $v \in |V|$ mapping to $y$ (because then $f$ is defined on an \'etale neighbourhood of $y$, namely $V$). There is a unique point $w' \in |X' \times_X W|$ mapping to $w$. By uniqueness $w'$ is mapped to $z'$ under the map $|X' \times_X W| \to |X' \times_X Z|$. Then we consider the cartesian diagram $$ \xymatrix{ X' \times_X V \ar[r] \ar[d] & X' \times_X W \ar[d] \\ X' \times_X Y \ar[r] & X' \times_X Z } $$ to see that there is a point $v' \in |X' \times_X V|$ mapping to $y'$ and $w'$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}. Of course the image $v$ of $v'$ in $|V|$ maps to $y$ and the proof is complete.
Choose $n'$, $T' \to Y'$ and $n_1$, $T_1 \to V$ as in Properties, Lemma \ref{properties-lemma-quasi-affine-presentation}. Picture $$ \xymatrix{ & & T_1 \times_V V' \times_Y T' \ar[ld] \ar[rd] \\ T_1 \ar[d] & T_1 \times_V V' \ar[l] \ar[dr] & & V' \times_{Y'} T' \ar[r] \ar[dl] & T' \ar[d] \\ V & & V' \ar[rr] \ar[ll] & & Y' } $$ Observe that $T_1 \times_V V'$ and $V' \times_{Y'} T'$ are affine (namely the morphisms $V' \to V$ and $V' \to Y'$ are affine as base changes of the affine morphisms $U' \to U$ and $U' \to X'$). By construction we see that $$ \mathbf{A}^{n'}_{T_1 \times_V V'} \cong T_1 \times_V V' \times_{Y'} T' \cong \mathbf{A}^{n_1}_{V' \times_{Y'} T'} $$ In other words, the affine schemes $\mathbf{A}^{n'}_{T_1}$ and $\mathbf{A}^{n_1}_{T'}$ are part of a triple making an affine object of $\textit{Spaces}(U \leftarrow U' \to X')$. By Lemma \ref{lemma-glueing-affines} there exists a morphism of affine schemes $T \to X$ and isomorphisms $U \times_X T \cong \mathbf{A}^{n'}_{T_1}$ and $X' \times_X T \cong \mathbf{A}^{n_1}_{T'}$ compatible with the isomorphisms displayed above. These isomorphisms produce morphisms $$ U \times_X T \longrightarrow V \quad\text{and}\quad X' \times_X T \longrightarrow Y' $$ satisfying the property of Properties, Lemma \ref{properties-lemma-quasi-affine-presentation} with $n = n' + n_1$ and moreover define a morphism from the triple $(U \times_X T, U' \times_X T, X' \times_X T)$ to our triple $(V, V', Y')$ in the category $\textit{Spaces}(U \leftarrow U' \to X')$. \medskip\noindent By Lemma \ref{lemma-glueing-affines} there is an affine scheme $W$ whose image in $\textit{Spaces}(U \leftarrow U' \to X')$ is isomorphic to the triple $$ ((U \times_X T) \times_V (U \times_X T), (U' \times_X T) \times_{V'} (U' \times_X T), (X' \times_X T) \times_{Y'} (X' \times_X T)) $$ By fully faithfulness of this construction, we obtain two maps $p_0, p_1 : W \to T$ whose base changes to $U, U', X'$ are the projection morphisms. By Lemma \ref{lemma-glueing-affines-etale} the morphisms $p_0, p_1$ are flat and of finite presentation and the morphism $(p_0, p_1) : W \to T \times_X T$ is a closed immersion. In fact, $W \to T \times_X T$ is an equivalence relation: by the lemmas used above we may check symmetry, reflexivity, and transitivity after base change to $U$ and $X'$, where these are obvious (details omitted). Thus the quotient sheaf $$ Y = T/W $$ is an algebraic space for example by Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}. Since it is clear that $Y/X$ is sent to the triple $(V, V', Y')$. The base change of the diagonal $\Delta : Y \to Y \times_X Y$ by the quasi-compact surjective flat morphism $T \times_X T \to Y \times_X Y$ is the closed immersion $W \to T \times_X T$. Thus $\Delta$ is a closed immersion by Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-closed-immersion}. Thus the algebraic space $Y$ is separated and the proof is complete.
Existence of $X_1$ and the fact that $X_1 \to X$ is a finite universal homeomorphism is a special case of Lemma \ref{lemma-coequalizer}. The formation of $X_1$ commutes with \'etale localization on $X$ (see proof of Lemma \ref{lemma-coequalizer}). Thus the morphism $X_1 \to X$ is an isomorphism over $U$. It is immediate from the construction that $Z \to X$ lifts to $X_1$.
Proof of (1). The assumption that $Z_i \to X$ is of finite presentation signifies that the quasi-coherent ideal sheaf $\mathcal{I}_i$ of $Z_i$ is of finite type, see  Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}. Denote $Z \subset X$ the closed subspace cut out by the product $\mathcal{I}_1 \mathcal{I}_2$. Observe that $Z \cap U$ is the disjoint union of $Z_1 \cap U$ and $Z_2 \cap U$. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-separate-disjoint-opens-by-blowing-up} there is a $U \cap Z$-admissible blowup $Z' \to Z$ such that the strict transforms of $Z_1$ and $Z_2$ are disjoint. Denote $Y \subset Z$ the center of this blowing up. Then $Y \to X$ is a closed immersion of finite presentation as the composition of $Y \to Z$ and $Z \to X$ (Divisors on Spaces, Definition \ref{spaces-divisors-definition-admissible-blowup} and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-finite-presentation}). Thus the blowing up $X' \to X$ of $Y$ is a $U$-admissible blowing up. By general properties of strict transforms, the strict transform of $Z_1, Z_2$ with respect to $X' \to X$ is the same as the strict transform of $Z_1, Z_2$ with respect to $Z' \to Z$, see Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-strict-transform}. Thus (1) is proved. \medskip\noindent Proof of (2). By Limits of Spaces, Lemma \ref{spaces-limits-lemma-quasi-coherent-finite-type-ideals} there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J}_i \subset \mathcal{O}_U$ such that $T_i = V(\mathcal{J}_i)$ (set theoretically). By Limits of Spaces, Lemma \ref{spaces-limits-lemma-extend} there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I}_i \subset \mathcal{O}_X$ whose restriction to $U$ is $\mathcal{J}_i$. Apply the result of part (1) to the closed subspaces $Z_i = V(\mathcal{I}_i)$ to conclude.
Let $T' \subset |V|$ be the complement of the maximal open over which $f|_U$ is an isomorphism. Then $T', T$ are closed in $|V|$ and $T \cap T' = \emptyset$. Since $|V|$ is a spectral topological space (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-quasi-separated-spectral}) we can find constructible closed subsets $T_c, T'_c$ of $|V|$ with $T \subset T_c$, $T' \subset T'_c$ such that $T_c \cap T'_c = \emptyset$ (choose a quasi-compact open $W$ of $|V|$ containing $T'$ not meeting $T$ and set $T_c = |V| \setminus W$, then choose a quasi-compact open $W'$ of $|V|$ containing $T_c$ not meeting $T'$ and set $T'_c = |V| \setminus W'$). By Lemma \ref{lemma-separate-disjoint-locally-closed-by-blowing-up} we may, after replacing $Y$ by a $V$-admissible blowing up, assume that $T_c$ and $T'_c$ have disjoint closures in $|Y|$. Let $Y_0$ be the open subspace of $Y$ corresponding to the open $|Y| \setminus \overline{T}'_c$ and set $V_0 = V \cap Y_0$, $U_0 = U \times_V V_0$, and $X_0 = X \times_Y Y_0$. Since $U_0 \to V_0$ is an isomorphism, we can find a $V_0$-admissible blowing up $Y'_0 \to Y_0$ such that the strict transform $X'_0$ of $X_0$ maps isomorphically to $Y'_0$, see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-zariski-after-blowup}. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-extend-admissible-blowups} there exists a $V$-admissible blow up $Y' \to Y$ whose restriction to $Y_0$ is $Y'_0 \to Y_0$. If $f' : X' \to Y'$ denotes the strict transform of $f$, then we see what we want is true because $f'$ restricts to an isomorphism over $Y'_0$.
Let $T' \subset |U|$ be the complement of the maximal open on which $f|_U$ is \'etale. Then $T'$ is closed in $|U|$ and disjoint from $|A|$. Since $|U|$ is a spectral topological space (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-quasi-separated-spectral}) we can find constructible closed subsets $T_c, T'_c$ of $|U|$ with $|A| \subset T_c$, $T' \subset T'_c$ such that $T_c \cap T'_c = \emptyset$ (see proof of Lemma \ref{lemma-blowup-iso-along}). By Lemma \ref{lemma-separate-disjoint-locally-closed-by-blowing-up} there is a $U$-admissible blowing up $X_1 \to X$ such that $T_c$ and $T'_c$ have disjoint closures in $|X_1|$. Let $X_{1, 0}$ be the open subspace of $X_1$ corresponding to the open $|X_1| \setminus \overline{T}'_c$ and set $U_0 = U \cap X_{1, 0}$. Observe that the scheme theoretic image $\overline{A}_1 \subset X_1$ of $A$ is contained in $X_{1, 0}$ by construction. \medskip\noindent After replacing $Y$ by a $V$-admissible blowing up and taking strict transforms, we may assume $X_{1, 0} \to Y$ is flat, quasi-finite, and of finite presentation, see More on Morphisms of Spaces, Lemmas \ref{spaces-more-morphisms-lemma-flat-after-blowing-up} and \ref{spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open}. Consider the commutative diagram $$ \vcenter{ \xymatrix{ X_1 \ar[rr] \ar[rd] & & X \ar[ld] \\ & Y } } \quad\text{and the diagram}\quad \vcenter{ \xymatrix{ \overline{A}_1 \ar[rr] \ar[rd] & & \overline{A} \ar[ld] \\ & \overline{B} } } $$ of scheme theoretic images. The morphism $\overline{A}_1 \to \overline{A}$ is surjective because it is proper and hence the scheme theoretic image of $\overline{A}_1 \to \overline{A}$ must be equal to $\overline{A}$ and then we can use Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-scheme-theoretic-image-is-proper}. The statement on \'etale local rings follows by choosing a lift of the geometric point $\overline{a}$ to a geometric point $\overline{a}_1$ of $\overline{A}_1$ and setting $\mathcal{O} = \mathcal{O}_{X_1, \overline{a}_1}$. Namely, since $X_1 \to Y$ is flat and quasi-finite on $X_{1, 0} \supset \overline{A}_1$, the map $\mathcal{O}_{Y', f'(\overline{a})} \to \mathcal{O}_{X_1, \overline{a}_1}$ is finite flat, see Algebra, Lemmas \ref{algebra-lemma-quasi-finite-strict-henselization} and \ref{algebra-lemma-characterize-henselian}.
Denote $Z' \to Z$ the strict transform. The morphism $Z' \to X'$ induces a morphism $Z' \to X' \times_B Y$ which is a closed immersion (as $Z'$ is a closed subspace of $X' \times_X Z$ by definition). Thus to finish the proof it suffices to show that the scheme theoretic image $Z''$ of $V \to Z'$ is $Z'$. Observe that $Z'' \subset Z'$ is a closed subspace such that $V \to Z'$ factors through $Z''$. Since both $V \to X \times_B Y$ and $V \to X' \times_B Y$ are quasi-compact (for the latter this follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-permanence} and the fact that $X' \times_B Y \to X \times_B Y$ is separated as a base change of a proper morphism), by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image} we see that $Z \cap (U \times_B Y) = Z'' \cap (U \times_B Y)$. Thus the inclusion morphism $Z'' \to Z'$ is an isomorphism away from the exceptional divisor $E$ of $Z' \to Z$. However, the structure sheaf of $Z'$ does not have any nonzero sections supported on $E$ (by definition of strict transforms) and we conclude that the surjection $\mathcal{O}_{Z'} \to \mathcal{O}_{Z''}$ must be an isomorphism.
Consider the scheme theoretic image $Z \subset Y \times_B U$ of the ``diagonal'' morphism $V \to Y \times_B U$. If we replace $Y$ by a $V$-admissible blowing up, then $Z$ is replaced by the strict transform with respect to this blowing up, see Lemma \ref{lemma-replaced-by-strict-transform}. Hence by More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-zariski-after-blowup} we may assume $Z \to Y$ is an open immersion. If $V' \subset Y$ denotes the image, then we see that the induced morphism $V' \to U$ is proper because the projection $Y \times_B U \to U$ is proper and $V' \cong Z$ is a closed subspace of $Y \times_B U$.
Choose a stratification $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ and morphisms $f_p : V_p \to U_p$ as in Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}. Let $p$ be the smallest integer such that $U_p \not \subset U$ (this is possible as $U \not = X$). Choose an affine open $V \subset V_p$ such that the \'etale morphism $f_p|_V : V \to X$ does not factor through $U$. Consider the open $W = U \cup \Im(V \to X)$ and the reduced closed subspace $Z \subset W$ with $|Z| = |W| \setminus |U|$. Then $f^{-1}Z \to Z$ is an isomorphism because we have the corresponding property for the morphism $f_p$, see the lemma cited above. Thus $(U \subset W, f : V \to W)$ is a distinguished square. It may not be true that the open $I = \Im(U \times_W V \to U)$ is dense in $U$. The algebraic space $U' \subset U$ whose underlying set is $|U| \setminus \overline{|I|}$ is Noetherian and hence we can find a dense open subscheme $U'' \subset U'$, see for example Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}. Then we can find a dense open affine $U''' \subset U''$, see Properties, Lemmas \ref{properties-lemma-Noetherian-irreducible-components} and \ref{properties-lemma-maximal-points-affine}. After we replace $f$ by $V \amalg U''' \to X$ everything is clear.
Let $X$ and $Y$ be varieties with generic points $x \in X$ and $y \in Y$. Recall that dominant rational maps from $X$ to $Y$ are exactly those rational maps which map $x$ to $y$ (Morphisms, Definition \ref{morphisms-definition-dominant-rational} and discussion following). Thus given a dominant rational map $X \supset U \to Y$ we obtain a map of function fields $$ k(Y) = \kappa(y) = \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, x} = \kappa(x) = k(X) $$ Conversely, such a $k$-algebra map (which is automatically local as the source and target are fields) determines (uniquely) a dominant rational map by Morphisms, Lemma \ref{morphisms-lemma-rational-map-finite-presentation}. In this way we obtain a fully faithful functor. To finish the proof it suffices to show that every finitely generated field extension $K/k$ is in the essential image. Since $K/k$ is finitely generated, there exists a finite type $k$-algebra $A \subset K$ such that $K$ is the fraction field of $A$. Then $X = \Spec(A)$ is a variety whose function field is $K$.
The morphism $X \times_{\Spec(k)} Y \to \Spec(k)$ is of finite type and separated because it is the composition of the morphisms $X \times_{\Spec(k)} Y \to Y \to \Spec(k)$ which are separated and of finite type, see Morphisms, Lemmas \ref{morphisms-lemma-base-change-finite-type} and \ref{morphisms-lemma-composition-finite-type} and Schemes, Lemma \ref{schemes-lemma-separated-permanence}. To finish the proof it suffices to show that $X \times_{\Spec(k)} Y$ is integral. Let $X = \bigcup_{i = 1, \ldots, n} U_i$, $Y = \bigcup_{j = 1, \ldots, m} V_j$ be finite affine open coverings. If we can show that each $U_i \times_{\Spec(k)} V_j$ is integral, then we are done by Properties, Lemmas \ref{properties-lemma-characterize-reduced}, \ref{properties-lemma-characterize-irreducible}, and \ref{properties-lemma-characterize-integral}. This reduces us to the affine case. \medskip\noindent The affine case translates into the following algebra statement: Suppose that $A$, $B$ are integral domains and finitely generated $k$-algebras. Then $A \otimes_k B$ is an integral domain. To get a contradiction suppose that $$ (\sum\nolimits_{i = 1, \ldots, n} a_i \otimes b_i) (\sum\nolimits_{j = 1, \ldots, m} c_j \otimes d_j) = 0 $$ in $A \otimes_k B$ with both factors nonzero in $A \otimes_k B$. We may assume that $b_1, \ldots, b_n$ are $k$-linearly independent in $B$, and that $d_1, \ldots, d_m$ are $k$-linearly independent in $B$. Of course we may also assume that $a_1$ and $c_1$ are nonzero in $A$. Hence $D(a_1c_1) \subset \Spec(A)$ is nonempty. By the Hilbert Nullstellensatz (Algebra, Theorem \ref{algebra-theorem-nullstellensatz}) we can find a maximal ideal $\mathfrak m \subset A$ contained in $D(a_1c_1)$ and $A/\mathfrak m = k$ as $k$ is algebraically closed. Denote $\overline{a}_i, \overline{c}_j$ the residue classes of $a_i, c_j$ in $A/\mathfrak m = k$. The equation above becomes $$ (\sum\nolimits_{i = 1, \ldots, n} \overline{a}_i b_i) (\sum\nolimits_{j = 1, \ldots, m} \overline{c}_j d_j) = 0 $$ which is a contradiction with $\mathfrak m \in D(a_1c_1)$, the linear independence of $b_1, \ldots, b_n$ and $d_1, \ldots, d_m$, and the fact that $B$ is a domain.
We may assume $X = \Spec(A)$ is affine. Then $Y = \Spec(A \otimes_k K)$. Since $K$ is flat over $k$, we see that $A \to A \otimes_k K$ is flat. Hence $Y \to X$ is flat and we get the first statement if we also use Algebra, Lemma \ref{algebra-lemma-local-flat-ff}. The second statement follows from Schemes, Lemma \ref{schemes-lemma-points-fibre-product}. Now $y$ corresponds to a prime ideal $\mathfrak q \subset A \otimes_k K$ and $x$ to $\mathfrak r = A \cap \mathfrak q$. Then $\mathfrak p_0$ is the kernel of the induced map $\kappa(\mathfrak r) \otimes_k K \to \kappa(\mathfrak q)$. The map on local rings is $$ A_\mathfrak r \longrightarrow (A \otimes_k K)_\mathfrak q $$ We can factor this map through $A_\mathfrak r \otimes_k K = (A \otimes_k K)_{\mathfrak r}$ to get $$ A_\mathfrak r \longrightarrow A_\mathfrak r \otimes_k K \longrightarrow (A \otimes_k K)_\mathfrak q $$ and then the second arrow is a localization at some prime. This prime ideal is the inverse image of $\mathfrak p_0$ (details omitted) and this proves (3). To see (4) use (3) and that localization and $- \otimes_k K$ are exact functors.
(The parenthetical statement follows from Algebra, Lemma \ref{algebra-lemma-separable-extension-preserves-reducedness}.) Combining Lemmas \ref{lemma-change-fields-flat} and \ref{lemma-change-fields-algebraic-dim} we see that $\mathcal{O}_{Y, y}/\mathfrak m_x \mathcal{O}_{Y, y}$ has dimension $0$ and is reduced. Hence it is a field.