hcju/stacksqa huggingface dataset:
Let $\{u_{j1}, \ldots, u_{jn_j}\}$ be the orbit of $u_j$. Let $W' \subset W \subset U$ be as in Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme}. Since $U = t(s^{-1}(\overline{W}))$ we see that at least one $u_{ji} \in \overline{W}$. Since $u_{ji}$ is a generic point of an irreducible component and $U$ locally Noetherian, this implies that $u_{ji} \in W$. Since $W$ is $R$-invariant, we conclude that $u_j \in W$ and in fact the whole orbit is contained in $W$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-image-affine-finite-morphism-affine-Noetherian} it suffices to find an $R$-invariant affine open subscheme $V'$ of $W'$ containing $u_1, \ldots, u_m$ (because then we can let $V \subset W$ be the corresponding open subscheme which will be affine). Thus we may replace $(U, R, s, t, c)$ by the restriction $(W', R', s', t', c')$ to $W'$. In other words, we may assume we have a groupoid scheme $(U, R, s, t, c)$ whose morphisms $s$ and $t$ are finite locally free. By Properties, Lemma \ref{properties-lemma-maximal-points-affine} we can find an affine open containing $\{u_{ij}\}$ (a locally Noetherian scheme is quasi-separated by Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated}). Finally, we can apply Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine} to conclude.
Let $W' \subset U'$ be an $R'$-invariant affine open. Set $\tilde R = U' \times_{g, U, t} R$ with maps $\text{pr}_0 : \tilde R \to U'$ and $h = s \circ \text{pr}_1 : \tilde R \to U$. Observe that $\text{pr}_0$ and $h$ are integral. It follows that $\tilde W = \text{pr}_0^{-1}(W')$ is affine. Since $W'$ is $R'$-invariant, the image $W = h(\tilde W)$ is set theoretically $R$-invariant and $\tilde W = h^{-1}(W)$ set theoretically (details omitted). Thus, if we can show that $W$ is open, then $W$ is a scheme and the morphism $\tilde W \to W$ is integral surjective which implies that $W$ is affine by Limits, Proposition \ref{limits-proposition-affine}. However, our assumption on orbits meeting $U'$ implies that $h : \tilde R \to U$ is surjective. Since an integral surjective morphism is submersive (Topology, Lemma \ref{topology-lemma-closed-morphism-quotient-topology} and Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed}) it follows that $W$ is open.
Let $u_1, \ldots, u_m \in V' \subset V \subset U$ be as in Lemma \ref{lemma-invariant-affine-open-around-generic-point}. Since $U \setminus V$ is closed in $U$, $j$ an immersion, and $V(I) \cup j(U)$ is closed in $\Spec(A)$, we can find an ideal $J \subset I$ such that $V(J) = V(I) \cup j(U \setminus V)$. For example we can take the ideal of elements of $I$ which vanish on $j(U \setminus V)$. Thus we can replace $(U, R, s, t, c)$, $j : U \to \Spec(A)$, and $I$ by $(V', R', s', t', c')$, $j|_{V'} : V' \to \Spec(A)$, and $J$. In other words, we may assume that $U$ is affine and that $s$ and $t$ are finite locally free. Take any $f \in I$ which does not vanish at all the points in the $R$-orbits of $u_1, \ldots, u_m$ (Algebra, Lemma \ref{algebra-lemma-silly}). Consider $$ g = \text{Norm}_s(t^\sharp(j^\sharp(f))) \in \Gamma(U, \mathcal{O}_U) $$ Since $f \in I$ and since $V(I) \cup j(U)$ is closed we see that $U \cap D(f) \to D(f)$ is a closed immersion. Hence $f^ng$ is the image of an element $h \in I$ for some $n > 0$. We claim that $h$ works. Namely, we have seen in Groupoids, Lemma \ref{groupoids-lemma-determinant-trick} that $g$ is an $R$-invariant function, hence $D(g) \subset U$ is $R$-invariant. Since $f$ does not vanish on the orbit of $u_j$, the function $g$ does not vanish at $u_j$. Moreover, we have $V(g) \supset V(j^\sharp(f))$ and hence $j^{-1}D(h) = D(g)$.
Let $r \in R$ with $s(r) = u$ and $t(r) = u'$. If $u \leadsto u'$ then we can find a nontrivial specialization $r \leadsto r'$ with $s(r') = u'$, see  Schemes, Lemma \ref{schemes-lemma-quasi-compact-closed}. Set $u'' = t(r')$. Note that $u'' \not = u'$ as there are no specializations in the fibres of a finite morphism. Hence we can continue and find a nontrivial specialization $r' \leadsto r''$ with $s(r'') = u''$, etc. This shows that the orbit of $u$ contains an infinite sequence $u \leadsto u' \leadsto u'' \leadsto \ldots$ of specializations which is nonsense as the orbit $t(s^{-1}(\{u\}))$ is finite.
This follows from Morphisms, Lemma \ref{morphisms-lemma-get-affine} but we will also give a direct proof. Let $A' = \Gamma(V, \mathcal{O}_V)$. Then $j' : V \to \Spec(A')$ is a quasi-compact open immersion, see Properties, Lemma \ref{properties-lemma-quasi-affine}. Let $f' \in A'$ be the image of $f$. Then $(j')^{-1}D(f') = j^{-1}D(f)$ is affine. On the other hand, $j'(V) \cap V(f')$ is a subscheme of $\Spec(A')$ which maps isomorphically to the closed subscheme $j(V) \cap V(f)$ of $\Spec(A)$. Hence it is closed in $\Spec(A')$ for example by Schemes, Lemma \ref{schemes-lemma-section-immersion}. Thus we may replace $A$ by $A'$ and assume that $j$ is an open immersion and $A = \Gamma(V, \mathcal{O}_V)$. \medskip\noindent In this case we claim that $j(V) = \Spec(A)$ which finishes the proof. If not, then we can find a principal affine open $D(g) \subset \Spec(A)$ which meets the complement and avoids the closed subset $j(V) \cap V(f)$. Note that $j$ maps $j^{-1}D(f)$ isomorphically onto $D(f)$, see Properties, Lemma \ref{properties-lemma-invert-f-affine}. Hence $D(g)$ meets $V(f)$. On the other hand, $j^{-1}D(g)$ is a principal open of the affine open $j^{-1}D(f)$ hence affine. Hence by Properties, Lemma \ref{properties-lemma-invert-f-affine} again we see that $D(g)$ is isomorphic to $j^{-1}D(g) \subset j^{-1}D(f)$ which implies that $D(g) \subset D(f)$. This contradiction finishes the proof.
Note that $(E, \mathcal{O}_X|_E)$ is a locally ringed space so that Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} applies to $A \to \Gamma(E, \mathcal{O}_X|_E)$. Write $E = \bigcap_{i \in I} U_i$ with $I \not = \emptyset$ and $U_i \subset X$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $U_k \subset U_i \cap U_j$. Set $A_i = \Gamma(U_i, \mathcal{O}_{U_i})$. We obtain commutative diagrams $$ \xymatrix{ (E, \mathcal{O}_X|_E) \ar[r] \ar[d] & (\Spec(A), \mathcal{O}_{\Spec(A)}) \ar[d] \\ (U_i, \mathcal{O}_{U_i}) \ar[r] & (\Spec(A_i), \mathcal{O}_{\Spec(A_i)}) } $$ Since $U_i$ is quasi-affine, we see that $U_i \to \Spec(A_i)$ is a quasi-compact open immersion. On the other hand $A = \colim A_i$. Hence $\Spec(A) = \lim \Spec(A_i)$ as topological spaces (Limits, Lemma \ref{limits-lemma-topology-limit}). Since $E = \lim U_i$ (by Topology, Lemma \ref{topology-lemma-make-spectral-space}) we see that $E \to \Spec(A)$ is a homeomorphism onto its image $E'$ and that $E'$ is the intersection of the inverse images of the opens $U_i \subset \Spec(A_i)$ in $\Spec(A)$. For any $e \in E$ the local ring $\mathcal{O}_{X, e}$ is the value of $\mathcal{O}_{U_i, e}$ which is the same as the value on $\Spec(A)$. \medskip\noindent To prove the final assertion of the lemma we argue as follows. Pick $i, j \in I$ with $U_i \subset U_j$. Consider the following commutative diagrams $$ \xymatrix{ U_i \ar[r] \ar[d] & \Spec(A_i) \ar[d] \\ U_i \ar[r] & \Spec(A_j) } \quad\quad \xymatrix{ W \ar[r] \ar[d] & \Spec(A_i) \ar[d] \\ W \ar[r] & \Spec(A_j) } \quad\quad \xymatrix{ W \ar[r] \ar[d] & \Spec(A) \ar[d] \\ W \ar[r] & \Spec(A_j) } $$ By Properties, Lemma \ref{properties-lemma-cartesian-diagram-quasi-affine} the first diagram is cartesian. Hence the second is cartesian as well. Passing to the limit we find that the third diagram is cartesian, so the top horizontal arrow of this diagram is an open immersion.
Write $E = \bigcap_{i \in I} V_i$ with $V_i \subset Y$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $V_k \subset V_i \cap V_j$. Then we have similarly that $f^{-1}(E) = \bigcap_{i \in I} f^{-1}(V_i)$ in $X$. Thus the result follows from equation (\ref{equation-sections-of-intersection}) and the corresponding result for $V_i$ and $f^{-1}(V_i)$ which is Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
If $M$ is a finite module, then the map is surjective by Algebra, Proposition \ref{algebra-proposition-fg-tensor}. Conversely, assume the map is surjective. Let $m_1, m_2, m_3, \ldots$ be an enumeration of the elements of $M$. Let $\sum_{j = 1, \ldots, m} x_j \otimes a_j$ be an element of the tensor product mapping to the element $(m_n) \in M^\mathbf{N}$. Then we see that $x_1, \ldots, x_m$ generate $M$ over $R$ as in the proof of Algebra, Proposition \ref{algebra-proposition-fg-tensor}.
If $M$ is a finitely presented module, then the map is an isomorphism by Algebra, Proposition \ref{algebra-proposition-fp-tensor}. Conversely, assume the map is an isomorphism. By Lemma \ref{lemma-countable-fg-tensor} the module $M$ is finite. Choose a surjection $R^{\oplus m} \to M$ with kernel $K$. Then $K$ is countable as a submodule of $R^{\oplus m}$. Arguing as in the proof of Algebra, Proposition \ref{algebra-proposition-fp-tensor} we see that $K \otimes_R R^\mathbf{N} \to K^\mathbf{N}$ is surjective. Hence we conclude that $K$ is a finite $R$-module by Lemma \ref{lemma-countable-fg-tensor}. Thus $M$ is finitely presented.
If $R$ is coherent, then $R^\mathbf{N}$ is a flat module by Algebra, Proposition \ref{algebra-proposition-characterize-coherent}. Assume $R^\mathbf{N}$ is flat. Let $I \subset R$ be a finitely generated ideal. To prove the lemma we show that $I$ is finitely presented as an $R$-module. Namely, the map $I \otimes_R R^\mathbf{N} \to R^\mathbf{N}$ is injective as $R^\mathbf{N}$ is flat and its image is $I^\mathbf{N}$ by Lemma \ref{lemma-countable-fg-tensor}. Thus we conclude by Lemma \ref{lemma-countable-fp-tensor}.
Suppose we have $\alpha \in K$ and a nonzero $r \in R$ such that $r \alpha^n \in R$ for all $n \geq 1$. Then we consider $f = \sum r \alpha^{n - 1} x^n$ in $R[[x]]$. Write $\alpha = a/b$ for $a, b \in R$ with $b$ nonzero. Then we see that $(a x - b)f = -rb$. It follows that $rb$ is in the ideal $(ax - b)R[[x]]$. Let $S = \{h \in R[x] : h(0) = 1\}$. This is a multiplicative subset and flatness of $R[x] \to R[[x]]$ implies that $S^{-1}R[x] \to R[[x]]$ is faithfully flat (details omitted; hint: use Algebra, Lemma \ref{algebra-lemma-ff-rings}). Hence $$ S^{-1}R/(ax - b)S^{-1}R \to R[[x]]/(ax - b)R[[x]] $$ is injective. We conclude that $h rb = (ax - b) g$ for some $h \in S$ and $g \in R[x]$. Writing $h = 1 + h_1 x + \ldots + h_d x^d$ shows that we obtain $$ 1 + h_1 x + \ldots + h_d x^d = (1/r)(\alpha x - 1)g $$ This factorization in $K[x]$ gives a corresponding factorization in $K[x^{-1}]$ which shows that $\alpha$ is the root of a monic polynomial with coefficients in $R$ as desired.
The arguments above show that this is true if we can show that $R$ is not completely normal (valuation rings are normal, see Algebra, Lemma \ref{algebra-lemma-valuation-ring-normal}). Let $\mathfrak p \subset \mathfrak m \subset R$ be a chain of primes. Pick nonzero $x \in \mathfrak p$ and $y \in \mathfrak m \setminus \mathfrak p$. Then $x y^{-n} \in R$ for all $n \geq 1$ (if not then $y^n/x \in R$ which is absurd because $y \not \in \mathfrak p$). Hence $1/y$ is almost integral over $R$ but not in $R$.
Set $A = k[z]_{(z)}$ and let $B = (k[x, y, z] \oplus E)_{(x, y, z, E)}$. Since $x, y, z$ is a regular sequence in $B$, see proof of Lemma \ref{lemma-strange-regular-sequence}, we see that $x, y$ is a regular sequence in $B$ and that $B/(x, y)$ is a torsion free $A$-module, hence flat. On the other hand, there exists a nonzero element $\delta \in B/\mathfrak m_AB = B/zB$ which is annihilated by $\overline{x}, \overline{y}$. Hence $H_2(K_\bullet(B/\mathfrak m_AB, \overline{x}, \overline{y})) \not = 0$. Thus $\overline{x}, \overline{y}$ is not Koszul-regular, in particular it is not a regular sequence, see More on Algebra, Lemma \ref{more-algebra-lemma-regular-koszul-regular}.
Say $I = (e_1, e_2, e_3, \ldots)$ with $e_n$ an idempotent of $R$. After inductively replacing $e_{n + 1}$ by $e_n + (1 - e_n)e_{n + 1}$ we may assume that $(e_1) \subset (e_2) \subset (e_3) \subset \ldots$ and hence $I = \bigcup_{n \geq 1} (e_n) = \colim_n e_nR$. In this case $$ \Hom_R(I, M) = \Hom_R(\colim_n e_nR, M) = \lim_n \Hom_R(e_nR, M) = \lim_n e_nM $$ Note that the transition maps $e_{n + 1}M \to e_nM$ are given by multiplication by $e_n$ and are surjective. Hence by Algebra, Lemma \ref{algebra-lemma-ML-exact-sequence} the functor $\Hom_R(I, M)$ is exact, i.e., $I$ is a projective $R$-module.
Note that under the Frobenius map $F_A : A \to A$, the target copy of $A$ is a free-module over the domain with basis $\{1, T, \dots, T^{p - 1}\}$. Thus, $F_A$ is faithfully flat, and consequently,  so is $A \to A_{perf}$ since it is a colimit of faithfully flat maps. Since $A_{perf}$ is a perfect ring, the relative Frobenius $F_{A_{perf}/A}$ is a surjection. In  other words, $A_{perf} = A[A_{perf}^p]$, which readily implies $\Omega_{A_{perf}/A} = 0$. Then $A \rightarrow A_{perf}$ is formally unramified by More on Morphisms, Lemma \ref{more-morphisms-lemma-formally-unramified-differentials} \medskip\noindent It suffices to show that $A \rightarrow A_{perf}$ is not formally smooth. Note that since $A$ is a smooth $\mathbb{F}_p$-algebra, the cotangent complex $L_{A/\mathbb{F}_P} \simeq \Omega_{A/\mathbb{F}_p}[0]$ is concentrated in degree $0$, see Cotangent, Lemma \ref{cotangent-lemma-when-projective}. Moreover, $L_{A_{perf}/\mathbb{F}_p} = 0$ in $D(A_{perf})$ by Cotangent, Lemma \ref{cotangent-lemma-perfect-zero}. Consider the distinguished triangle of cotangent complexes $$ L_{A/\mathbb{F}_p} \otimes_A A_{perf} \to L_{A_{perf}/\mathbb{F}_p} \to L_{A_{perf}/A} \to (L_{A/\mathbb{F}_p} \otimes_A A_{perf})[1] $$ in $D(A_{perf})$, see Cotangent, Section \ref{cotangent-section-triangle}. We find $L_{A_{perf}/A} = \Omega_{A/\mathbb{F}_p} \otimes_A A_{perf}[1]$, that is, $L_{A_{perf}/A}$ is equal to a free rank $1$ $A_{perf}$ module placed in degree $-1$. Thus $A \rightarrow A_{perf}$ is not formally smooth by More on Morphisms, Lemma \ref{more-morphisms-lemma-NL-formally-smooth} and Cotangent, Lemma \ref{cotangent-lemma-relation-with-naive-cotangent-complex}.
The backward implication follows from Lemma \ref{lemma-completion-etale}. For the forward implication, note that we already know from Lemma \ref{lemma-completion-etale} that $A \to A^\wedge$ is formally unramified or equivalently that $\Omega_{A^\wedge/A}$ is zero. Thus, it suffices to show that the completion map is formally smooth when $A$ is excellent. By N\'eron-Popescu desingularization $A \to A^\wedge$ can be  written as a filtered colimit of smooth $A$-algebras (Smoothing Ring Maps, Theorem \ref{smoothing-theorem-popescu}). Hence $\NL_{A^\wedge/A}$ has vanishing cohomology in degree $-1$. Thus $A \to A^\wedge$ is formally smooth by Algebra, Proposition \ref{algebra-proposition-characterize-formally-smooth}.
Recall that we work on the ``big fppf site'' $\mathcal{X}_{fppf}$ of $\mathcal{X}$ (by our conventions for sheaves of $\mathcal{O}_\mathcal{X}$-modules in the chapters Sheaves on Stacks and Cohomology on Stacks). Let $\mathcal{B}$ be the set of objects $x$ of $\mathcal{X}_{fppf}$ which lie over an affine scheme $U$.  Combining Sheaves on Stacks, Lemmas \ref{stacks-sheaves-lemma-compare-fppf-etale}, \ref{stacks-sheaves-lemma-cohomology-restriction}, Descent, Lemma \ref{descent-lemma-quasi-coherent-and-flat-base-change}, and Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} we see that $H^p(x, \mathcal{F}) = 0$ if $\mathcal{F}$ is locally quasi-coherent and $x \in \mathcal{B}$. Now the claim follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-is-limit-dimension} with $d = 0$.
Consider the example above. The map $\gamma : C^\bullet \to A^\bullet[1]$ is given by multiplication by $\epsilon$ in degree $0$, see Derived Categories, Definition \ref{derived-definition-distinguished-triangle}. Hence it is also true that $$ \xymatrix{ C^\bullet \ar[d] \ar[r]_\gamma & A^\bullet[1] \ar[d] \\ C^\bullet \ar[r]^\gamma & A^\bullet[1] } $$ commutes in $K(R)$ as $\epsilon(1 + \epsilon) = \epsilon$. Thus we indeed have a morphism of distinguished triangles.
Following Hironaka \cite[Example B.3.4.1]{H}, we define a proper morphism of smooth complex 3-folds $f:V_Y\to Y$ which is Zariski-locally projective, but not projective. Since $f$ is proper and not projective, it is also not quasi-projective. \medskip\noindent Let $Y$ be projective 3-space over the complex numbers ${\mathbf C}$. Let $C$ and $D$ be smooth conics in $Y$ such that the closed subscheme $C\cap D$ is reduced and consists of two complex points $P$ and $Q$. (For example, let $C=\{ [x,y,z,w]: xy=z^2, w=0\}$, $D=\{ [x,y,z,w]: xy=w^2, z=0\}$, $P=[1,0,0,0]$, and $Q=[0,1,0,0]$.)  On $Y-Q$, first blow up the curve $C$, and then blow up the strict transform of the curve $D$ (Divisors, Definition  \ref{divisors-definition-strict-transform}). On $Y-P$, first blow up the curve $D$, and then blow up the strict transform of the curve $C$. Over $Y-P-Q$, the two varieties we have constructed are canonically isomorphic, and so we can glue them over $Y-P-Q$. The result is a smooth proper 3-fold $V_Y$ over ${\mathbf C}$. The morphism $f:V_Y\to Y$ is proper and Zariski-locally projective (since it is a blowup over $Y-P$ and over $Y-Q$), by Divisors, Lemma \ref{divisors-lemma-blowing-up-projective}. We will show that $V_Y$ is not projective over ${\mathbf C}$. That will imply that $f$ is not projective. \medskip\noindent To do this, let $L$ be the inverse image in $V_Y$ of a complex point of $C-P-Q$, and $M$ the inverse image of a complex point of $D-P-Q$. Then $L$ and $M$ are isomorphic to the projective line ${\mathbf P}^1_{{\mathbf C}}$.  Next, let $E$ be the inverse image in $V_Y$ of $C\cup D\subset Y$ in $V_Y$; thus $E\rightarrow C\cup D$ is a proper morphism, with fibers isomorphic to ${\mathbf P}^1$ over $(C\cup D)-\{P,Q\}$. The inverse image of $P$ in $E$ is a union of two lines $L_0$ and $M_0$, and we have rational equivalences of cycles $L\sim L_0+M_0$ and $M\sim M_0$ on $E$ (using that $C$ and $D$ are isomorphic to ${\mathbf P}^1$). Note the asymmetry resulting from the order in which we blew up the two curves. Near $Q$, the opposite happens. So the inverse image of $Q$ is the union of two lines $L_0'$ and $M_0'$, and we have rational equivalences $L\sim L_0'$ and $M\sim L_0'+M_0'$ on $E$. Combining these equivalences, we find that $L_0+M_0'\sim 0$ on $E$ and hence on $V_Y$. If $V_Y$ were projective over ${\mathbf C}$, it would have an ample line bundle $H$, which would have degree $> 0$ on all curves in $V_Y$. In particular $H$ would have positive degree on $L_0+M_0'$, contradicting that the degree of a line bundle is well-defined on 1-cycles modulo rational equivalence on a proper scheme over a field (Chow Homology, Lemma \ref{chow-lemma-proper-pushforward-rational-equivalence} and Lemma \ref{chow-lemma-factors}). So $V_Y$ is not projective over ${\mathbf C}$.
[Sketch of proof] Let $P \to W$ be an fppf $G$-torsor which is trivial in the generic point. Then we have a morphism $\Spec(K) \to P$ over $W$ and we can take its scheme theoretic image $Z \subset P$. Since $P \to W$ is proper (as a torsor for a proper group algebraic space over $W$) we see that $Z \to W$ is a proper birational morphism. By Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-finite-in-codim-1} the morphism $Z \to W$ is finite away from finitely many closed points of $W$. By (insert future reference on resolving indeterminacies of morphisms by blowing quadratic transformations for surfaces) the irreducible components of the geometric fibres of $Z \to W$ are rational curves. By More on Groupoids in Spaces, Lemma \ref{spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group} there are no nonconstant morphisms from rational curves to group schemes or torsors over such. Hence $Z \to W$ is finite, whence $Z$ is a scheme and $Z \to W$ is an isomorphism by Morphisms, Lemma \ref{morphisms-lemma-finite-birational-over-normal}. In other words, the torsor $P$ is trivial.
[Sketch of proof] Unwinding definitions, it is enough to check that $H^1(X, A_0) \to \lim H^1(X_n, A_0)$ is not surjective. As $X$ is regular and projective, by Lemmas \ref{lemma-torsors-over-field-torsion} and \ref{lemma-torsors-over-two-dimensional-regular} each $A_0$-torsor over $X$ is torsion. In particular, the group $H^1(X, A_0)$ is torsion. It is thus enough to show: (a) the group $H^1(X_0, A_0)$ is non-torsion, and (b) the maps $H^1(X_{n + 1}, A_0) \to H^1(X_n, A_0)$ are surjective for all $n$. \medskip\noindent Ad (a). One constructs a nontorsion $A_0$-torsor $P_0$ on the nodal curve $X_0$ by glueing trivial $A_0$-torsors on each component of $X_0$ using non-torsion points on $A_0$ as the isomorphisms over the nodes. More precisely, let $x \in X_0$ be a node which occurs in a loop consisting of rational curves. Let $X'_0 \to X_0$ be the normalization of $X_0$ in $X_0 \setminus \{x\}$. Let $x', x'' \in X'_0$ be the two points mapping to $x_0$. Then we take $A_0 \times_{\Spec(k)} X'_0$ and we identify $A_0 \times {x'}$ with $A_0 \times \{x''\}$ using translation $A_0 \to A_0$ by a nontorsion point $a_0 \in A_0(k)$ (there is such a nontorsion point as $k$ is algebraically closed and not the algebraic closure of a finite field -- this is actually not trivial to prove). One can show that the glueing is an algebraic space (in fact one can show it is a scheme) and that it is an nontorsion $A_0$-torsor over $X_0$. The reason that it is nontorsion is that if $[n](P_0)$ has a section, then that section produces a morphism $s : X'_0 \to A_0$ such that $[n](a_0) = s(x') - s(x'')$ in the group law on $A_0(k)$. However, since the irreducible components of the loop are rational to section $s$ is constant on them ( More on Groupoids in Spaces, Lemma \ref{spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group}). Hence $s(x') = s(x'')$ and we obtain a contradiction. \medskip\noindent Ad (b). Deformation theory shows that the obstruction to deforming an $A_0$-torsor $P_n \to X_n$ to an $A_0$-torsor $P_{n + 1} \to X_{n + 1}$ lies in $H^2(X_0, \omega)$ for a suitable vector bundle $\omega$ on $X_0$. The latter vanishes as $X_0$ is a curve, proving the claim.
Part (1) we saw above. This shows that $\textit{Coh}_{X/A}$ fails axiom [4] of Artin's Axioms, Section \ref{artin-section-axioms}. Hence it cannot be an algebraic stack by Artin's Axioms, Lemma \ref{artin-lemma-effective}. In this way we see that (2) is true. To see (3), note that there are compatible surjections $\mathcal{O}_{X_n} \to \mathcal{F}_n$ for all $n$. Thus we see that $\Quotfunctor_{\mathcal{O}_X/X/A}$ fails axiom [4] and we see that (3) is true as before.
With notation as above consider the family of morphisms $$ \mathcal{W} = \{\Spec(\mathbf{Z}_A) \to \Spec(\mathbf{Z})\}_{A \in U} $$ By Descent, Lemma \ref{descent-lemma-universal-effective-epimorphism} and the two claims proved above this is a universally effective epimorphism. In any category with fibre products, the universal effective epimorphisms give $\mathcal{C}$ the structure of a site (modulo some set theoretical issues which are easy to fix) defining the canonical topology. Thus $\mathcal{W}$ is a covering for the canonical topology. On the other hand, we have seen above that any finite subfamily $$ \{\Spec(\mathbf{Z}_{A_i}) \to \Spec(\mathbf{Z})\}_{i = 1, \ldots, n},\quad n \in \mathbf{N}, A_1, \ldots, A_n \in U $$ factors through $\Spec(\mathbf{Z}[1/p])$ for some $p$. Hence this finite family cannot be a universally effective epimorphism and more generally no universally effective epimorphism $\{g_j : T_j \to \Spec(\mathbf{Z})\}$ can refine $\{\Spec(\mathbf{Z}_{A_i}) \to \Spec(\mathbf{Z})\}_{i = 1, \ldots, n}$. By Sites, Definition \ref{sites-definition-quasi-compact} this means that $\Spec(\mathbf{Z})$ is not quasi-compact in the canonical topology. To see that our notion of quasi-compactness agrees with the usual topos theoretic definition, see Sites, Lemma \ref{sites-lemma-quasi-compact}.