hcju/stacksqa huggingface dataset:
To see this we will check the hypotheses (1) -- (4) of Lemma \ref{lemma-cech-to-cohomology-relative} and Lemma \ref{lemma-cech-to-cohomology-relative-modules}. The $1$-morphism $f$ is faithful by Algebraic Stacks, Lemma \ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}. This proves (4). Hypothesis (3) follows from the fact that $\mathcal{U}$ is an algebraic stack, see Lemma \ref{lemma-fibre-products}. To see (2) apply Lemma \ref{lemma-surjective-flat-locally-finite-presentation}. Condition (1) is satisfied by fiat in all four cases.
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, Lemma \ref{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 Varieties, Lemma \ref{varieties-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. By Lemma \ref{lemma-regular-alteration-implies-local} the completion of $\mathcal{O}_{Y^\nu, y_i}$ 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. Applying Lemmas \ref{lemma-resolve-complete} and \ref{lemma-existence-implies-existence-by-normalized-blowing-ups} we find  there exists a finite sequence of normalized blowups $$ Y_{i, n_i} \to Y_{i, n_i - 1} \to \ldots \to \Spec(\mathcal{O}^\wedge_{Y, y_i}) $$ such that $Y_{i, n_i}$ is regular. By Lemma \ref{lemma-normalized-blowup-completion} there is a corresponding sequence of normalized blowing ups $$ X_{i, n_i} \to \ldots \to X_{i, 1} \to \Spec(\mathcal{O}_{Y, y_i}) $$ Then $X_{i, n_i}$ is a regular scheme by Lemma \ref{lemma-port-regularity-to-completion}. By Lemma \ref{lemma-equivalence-sequence-normalized-blowups} we can fit these normalized blowing ups into a corresponding sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to Y $$ and of course $X_n$ is regular too (look at the local rings). This completes the proof.
For $t = 0$ we use the composition $$ \Omega_{S/\Lambda} \to \Omega_{S/R} \to \Omega_R \to \Omega_{R/\Lambda} $$ where the second map is Lemma \ref{lemma-trace-well-defined}. There is an exact sequence $$ H_1(L_{S/R}) \xrightarrow{\delta} \Omega_{R/\Lambda} \otimes_R S \to \Omega_{S/\Lambda} \to \Omega_{S/R} \to 0 $$ (Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL}). The module $\Omega_{S/R}$ is free over $S$ with basis $\text{d}x$ and the module $H_1(L_{S/R})$ is free over $S$ with basis $x^p - a$ which $\delta$ maps to $-\text{d}a \otimes 1$ in $\Omega_{R/\Lambda} \otimes_R S$. In particular, if we set $$ M = \Coker(R \to \Omega_{R/\Lambda}, 1 \mapsto -\text{d}a) $$ then we see that $\Coker(\delta) = M \otimes_R S$. We obtain a canonical map $$ \Omega^{t + 1}_{S/\Lambda} \to \wedge_S^t(\Coker(\delta)) \otimes_S \Omega_{S/R} = \wedge^t_R(M) \otimes_R \Omega_{S/R} $$ Now, since the image of the map $\text{Tr} : \Omega_{S/R} \to \Omega_{R/\Lambda}$ of Lemma \ref{lemma-trace-well-defined} is contained in $R\text{d}a$ we see that wedging with an element in the image annihilates $\text{d}a$. Hence there is a canonical map $$ \wedge^t_R(M) \otimes_R \Omega_{S/R} \to \Omega_{R/\Lambda}^{t + 1} $$ mapping $\overline{\eta}_1 \wedge \ldots \wedge \overline{\eta}_t \wedge \omega$ to $\eta_1 \wedge \ldots \wedge \eta_t \wedge \text{Tr}(\omega)$.
The exact sequence $f^*\Omega_{X/S} \to \Omega_{Y/S} \to \Omega_{Y/X} \to 0$ shows that $\Omega_{Y/S}$ and hence $f_*\Omega_{Y/S}$ are coherent modules as well. Thus it suffices to prove the trace map in the generic point extends to stalks at $x \in X$ with $\dim(\mathcal{O}_{X, x}) = 1$, see Divisors, Lemma \ref{divisors-lemma-describe-reflexive-hull}. Thus we reduce to the case discussed in the next paragraph. \medskip\noindent Assume $X = \Spec(A)$ and $Y = \Spec(B)$ with $A$ a discrete valuation ring and $B$ finite over $A$. Since the induced extension $K \subset L$ of fraction fields is purely inseparable, we see that $B$ is local too. Hence $B$ is a discrete valuation ring too. Then either \begin{enumerate} \item $B/A$ has ramification index $p$ and hence $B = A[x]/(x^p - a)$ where $a \in A$ is a uniformizer, or \item $\mathfrak m_B = \mathfrak m_A B$ and the residue field $B/\mathfrak m_A B$ is purely inseparable of degree $p$ over $\kappa_A = A/\mathfrak m_A$. Choose any $x \in B$ whose residue class is not in $\kappa_A$ and then we'll have $B = A[x]/(x^p - a)$ where $a \in A$ is a unit. \end{enumerate} Let $\Spec(\Lambda) \subset S$ be an affine open such that $X$ maps into $\Spec(\Lambda)$. Then we can apply Lemma \ref{lemma-trace-higher} to see that the trace map extends to $\Omega^i_{B/\Lambda} \to \Omega^i_{A/\Lambda}$ for all $i \geq 1$.
As $A$ is regular of dimension $2$ we can write $\mathfrak m = (x, y)$. Then $x$ and $y$ placed in degree $1$ generate the Rees algebra $\bigoplus_{n \geq 0} \mathfrak m^n$ over $A$. Recall that $X = \text{Proj}(\bigoplus_{n \geq 0} \mathfrak m^n)$, see Divisors, Lemma \ref{divisors-lemma-blowing-up-affine}. Thus the surjection $$ A[T_0, T_1] \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathfrak m^n, \quad T_0 \mapsto x,\ T_1 \mapsto y $$ of graded $A$-algebras induces a closed immersion $r : X \to \mathbf{P}^1_S = \text{Proj}(A[T_0, T_1])$ such that $\mathcal{O}_X(1) = r^*\mathcal{O}_{\mathbf{P}^1_S}(1)$, see Constructions, Lemma \ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj}. This proves (2) because $\mathcal{O}_X(E) = \mathcal{O}_X(-1)$ by Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}. \medskip\noindent To prove (1) note that $$ \left(\bigoplus\nolimits_{n \geq 0} \mathfrak m^n\right) \otimes_A \kappa = \bigoplus\nolimits_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1} \cong \kappa[\overline{x}, \overline{y}] $$ a polynomial algebra, see Algebra, Lemma \ref{algebra-lemma-regular-graded}. This proves that the fibre of $X \to S$ over $\Spec(\kappa)$ is equal to $\text{Proj}(\kappa[\overline{x}, \overline{y}]) = \mathbf{P}^1_\kappa$, see Constructions, Lemma \ref{constructions-lemma-base-change-map-proj}. Recall that $E$ is the closed subscheme of $X$ defined by $\mathfrak m\mathcal{O}_X$, i.e., $E = X_\kappa$. By our choice of the morphism $r$ we see that $r|_E$ in fact produces the identification of $E = X_\kappa$ with the special fibre of $\mathbf{P}^1_S \to S$. \medskip\noindent Part (3) follows from (1) and (2) and Divisors, Lemma \ref{divisors-lemma-conormal-effective-Cartier-divisor}.
Observe that $X$ is integral by Divisors, Lemma \ref{divisors-lemma-blow-up-integral-scheme} and Algebra, Lemma \ref{algebra-lemma-regular-domain}. To see $X$ is regular it suffices to check that $\mathcal{O}_{X, x}$ is regular for closed points $x \in X$, see Properties, Lemma \ref{properties-lemma-characterize-regular}. Let $x \in X$ be a closed point. Since $f$ is proper $x$ maps to $\mathfrak m$, i.e., $x$ is a point of the exceptional divisor $E$. Then $E$ is an effective Cartier divisor and $E \cong \mathbf{P}^1_\kappa$. Thus if $g \in \mathfrak m_x \subset \mathcal{O}_{X, x}$ is a local equation for $E$, then $\mathcal{O}_{X, x}/(g) \cong \mathcal{O}_{\mathbf{P}^1_\kappa, x}$. Since $\mathbf{P}^1_\kappa$ is covered by two affine opens which are the spectrum of a polynomial ring over $\kappa$, we see that $\mathcal{O}_{\mathbf{P}^1_\kappa, x}$ is regular by Algebra, Lemma \ref{algebra-lemma-dim-affine-space}. We conclude by Algebra, Lemma \ref{algebra-lemma-regular-mod-x}.
Recall that $E = \mathbf{P}^1_\kappa$ has Picard group $\mathbf{Z}$ with generator $\mathcal{O}(1)$, see Divisors, Lemma \ref{divisors-lemma-Pic-projective-space-UFD}. By Lemma \ref{lemma-blowup} the invertible $\mathcal{O}_X$-module $\mathcal{O}_X(E)$ restricts to $\mathcal{O}(-1)$. Hence $\mathcal{O}_X(E)$ generates an infinite cyclic group in $\Pic(X)$. Since $A$ is regular it is a UFD, see More on Algebra,  Lemma \ref{more-algebra-lemma-regular-local-UFD}. Then the punctured spectrum $U = S \setminus \{\mathfrak m\} = X \setminus E$ has trivial Picard group, see Divisors, Lemma \ref{divisors-lemma-open-subscheme-UFD}. Hence for every invertible $\mathcal{O}_X$-module $\mathcal{L}$ there is an isomorphism $s : \mathcal{O}_U \to \mathcal{L}|_U$. Then $s$ is a regular meromorphic section of $\mathcal{L}$ and we see that $\text{div}_\mathcal{L}(s) = nE$ for some $n \in \mathbf{Z}$ (Divisors, Definition \ref{divisors-definition-divisor-invertible-sheaf}). By Divisors, Lemma \ref{divisors-lemma-normal-c1-injective} (and the fact that $X$ is normal by Lemma \ref{lemma-blowup-regular}) we conclude that $\mathcal{L} = \mathcal{O}_X(nE)$.
Since $I \subset \mathfrak m$ we see that every element of $I$ vanishes on $E$. Thus we see that $d \geq 1$. On the other hand, since $\mathfrak m^n \subset I$ we see that $d \leq n$. Consider the short exact sequence $$ 0 \to I\mathcal{O}_X \to \mathcal{O}_X \to \mathcal{O}_X/I\mathcal{O}_X \to 0 $$ Since $I\mathcal{O}_X$ is globally generated, we see that $H^1(X, I\mathcal{O}_X) = 0$ by Lemma \ref{lemma-cohomology-of-blowup}. Hence we obtain a surjection $A/I \to \Gamma(X, \mathcal{O}_X/I\mathcal{O}_X)$. Consider the short exact sequence $$ 0 \to \mathcal{O}_X(-dE)/I\mathcal{O}_X \to \mathcal{O}_X/I\mathcal{O}_X \to \mathcal{O}_X/\mathcal{O}_X(-dE) \to 0 $$ By Divisors, Lemma \ref{divisors-lemma-codim-1-part} we see that $\mathcal{O}_X(-dE)/I\mathcal{O}_X$ is supported in finitely many closed points of $X$. In particular, this coherent sheaf has vanishing higher cohomology groups (detail omitted). Thus in the following diagram $$ \xymatrix{ & & A/I \ar[d] \\ 0 \ar[r] & \Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) \ar[r] & \Gamma(X, \mathcal{O}_X/I\mathcal{O}_X) \ar[r] & \Gamma(X, \mathcal{O}_X/\mathcal{O}_X(-dE)) \ar[r] & 0 } $$ the bottom row is exact and the vertical arrow surjective. We have $$ \text{length}_A \Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) < \text{length}_A(A/I) $$ since $\Gamma(X, \mathcal{O}_X/\mathcal{O}_X(-dE))$ is nonzero. Namely, the image of $1 \in \Gamma(X, \mathcal{O}_X)$ is nonzero as $d > 0$. \medskip\noindent To finish the proof we translate the results above into the statements of the lemma. Since $\mathcal{O}_X(dE)$ is invertible we have $$ \mathcal{O}_X/\mathcal{I}' = \mathcal{O}_X(-dE)/I\mathcal{O}_X \otimes_{\mathcal{O}_X} \mathcal{O}_X(dE). $$ Thus $\mathcal{O}_X/\mathcal{I}'$ and $\mathcal{O}_X(-dE)/I\mathcal{O}_X$ are supported in the same set of finitely many closed points, say $x_1, \ldots, x_r \in E \subset X$. Moreover we obtain $$ \Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) = \bigoplus \mathcal{O}_X(-dE)_{x_i}/I\mathcal{O}_{X, x_i} \cong \bigoplus \mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i} = \Gamma(X, \mathcal{O}_X/\mathcal{I}') $$ because an invertible module over a local ring is trivial. Thus we obtain the strict inequality. We also get the second because $$ \text{length}_A(\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i}) \geq \text{length}_{\mathcal{O}_{X, x_i}}(\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i}) $$ as is immediate from the definition of length.
Writing $\mathbf{P}^1 = \mathbf{P}^1_S$, let $r : X \to \mathbf{P}^1$ be as in Lemma \ref{lemma-blowup}. Then we have an exact sequence $$ \mathcal{C}_{X/\mathbf{P}^1} \to r^*\Omega_{\mathbf{P}^1/S} \to \Omega_{X/S} \to 0 $$ see Morphisms, Lemma \ref{morphisms-lemma-differentials-relative-immersion}. Since $\Omega_{\mathbf{P}^1/S}|_E = \Omega_{E/\kappa}$ by Morphisms, Lemma \ref{morphisms-lemma-base-change-differentials} it suffices to see that the first arrow defines a surjection onto the kernel of the canonical map $r^*\Omega_{\mathbf{P}^1/S} \to i_*\Omega_{E/\kappa}$. This we can do locally. With notation as in the proof of Lemma \ref{lemma-blowup} on an affine open of $X$ the morphism $f$ corresponds to the ring map $$ A \to A[t]/(xt - y) $$ where $x, y \in \mathfrak m$ are generators. Thus $\text{d}(xt - y) = x\text{d}t$ and $y\text{d}t = t \cdot x \text{d}t$ which proves what we want.
Say $T = \{x_1, \ldots, x_r\}$. Set $$ n_i = \text{length}_{\mathcal{O}_{X, x_i}}(\mathcal{O}_{X, x_i}/I_i) $$ This is finite as $\mathcal{O}_X/\mathcal{I}$ is supported on $T$ and hence $\mathcal{O}_{X, x_i}/I_i$ has support equal to $\{\mathfrak m_{x_i}\}$ (see Algebra, Lemma \ref{algebra-lemma-support-point}). We are going to use induction on $\sum n_i$. If $n_i = 0$ for all $i$, then $\mathcal{I} = \mathcal{O}_X$ and we are done. \medskip\noindent Suppose $n_i > 0$. Let $X' \to X$ be the blowing up of $X$ in $x_i$ (see discussion above the lemma). Since $\Spec(\mathcal{O}_{X, x_i}) \to X$ is flat we see that $X' \times_X \Spec(\mathcal{O}_{X, x_i})$ is the blowup of the ring $\mathcal{O}_{X, x_i}$ in the maximal ideal, see Divisors, Lemma \ref{divisors-lemma-flat-base-change-blowing-up}. Hence the square in the commutative diagram $$ \xymatrix{ \text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d) \ar[r] \ar[d] & X' \ar[d] \\ \Spec(\mathcal{O}_{X, x_i}) \ar[r] & X } $$ is cartesian. Let $E \subset X'$ and $E' \subset \text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d)$ be the exceptional divisors. Let $d \geq 1$ be the integer found in Lemma \ref{lemma-blowup-improve} for the ideal $\mathcal{I}_i \subset \mathcal{O}_{X, x_i}$. Since the horizontal arrows in the diagram are flat, since $E' \to E$ is surjective, and since $E'$ is the pullback of $E$, we see that $$ \mathcal{I}\mathcal{O}_{X'} \subset \mathcal{O}_{X'}(-dE) $$ (some details omitted). Set $\mathcal{I}' = \mathcal{I}\mathcal{O}_{X'}(dE) \subset \mathcal{O}_{X'}$. Then we see that $\mathcal{O}_{X'}/\mathcal{I}'$ is supported in finitely many closed points $T' \subset |X'|$ because this holds over $X \setminus \{x_i\}$ and for the pullback to $\text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d)$. The final assertion of Lemma \ref{lemma-blowup-improve} tells us that the sum of the lengths of the stalks $\mathcal{O}_{X', x'}/\mathcal{I}'\mathcal{O}_{X', x'}$ for $x'$ lying over $x_i$ is $< n_i$. Hence the sum of the lengths has decreased. \medskip\noindent By induction hypothesis, there exists a sequence $$ X'_n \to \ldots \to X'_1 \to X' $$ of blowups at closed points lying over $T'$ such that $\mathcal{I}'\mathcal{O}_{X'_n}$ is invertible. Since $\mathcal{I}'\mathcal{O}_{X'}(-dE) = \mathcal{I}\mathcal{O}_{X'}$, we see that $\mathcal{I}\mathcal{O}_{X'_n} = \mathcal{I}'\mathcal{O}_{X'_n}(-d(f')^{-1}E)$ where $f' : X'_n \to X'$ is the composition. Note that $(f')^{-1}E$ is an effective Cartier divisor by Divisors, Lemma \ref{divisors-lemma-blow-up-pullback-effective-Cartier}. Thus we are done by Divisors, Lemma \ref{divisors-lemma-sum-effective-Cartier-divisors}.
By More on Flatness, Lemma \ref{flat-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, Lemma \ref{divisors-lemma-universal-property-blowing-up}) we find the desired factorization.
We may assume $U$ contains every point of codimension $1$, see Morphisms, Lemma \ref{morphisms-lemma-extend-across}. Hence the complement $T \subset X$ of $U$ is a finite set of closed points whose local rings are regular of dimension $2$. Applying Divisors, Lemma \ref{divisors-lemma-extend-rational-map-after-modification} we find a proper morphism $p : X' \to X$ which is an isomorphism over $U$ and a morphism $f' : X' \to Y$ agreeing with $f$ over $U$. Apply Lemma \ref{lemma-dominate-by-blowing-up-in-points} to the morphism $p : X' \to X$. The composition $X_n \to X' \to Y$ is the desired morphism.
The blowup morphism $X' \to X$ is proper (as $X$ is locally Noetherian we may apply Divisors, Lemma \ref{divisors-lemma-blowing-up-projective}). Thus $X'$ is Nagata (Morphisms, Lemma \ref{morphisms-lemma-finite-type-nagata}). Therefore the normalization $X'' \to X'$ is finite (Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization}) and we conclude that $X'' \to X$ is proper as well (Morphisms, Lemmas \ref{morphisms-lemma-finite-proper} and \ref{morphisms-lemma-composition-proper}). It follows that the normalized blowing up is a normal (Morphisms, Lemma \ref{morphisms-lemma-normalization-normal}) Nagata algebraic space.
Applying Divisors, Lemma \ref{divisors-lemma-extend-rational-map-after-modification} we find a proper morphism $p : X' \to X$ which is an isomorphism over $U$ and a morphism $f' : X' \to Y$ agreeing with $f$ over $U$. Apply Lemma \ref{lemma-dominate-by-normalized-blowing-up} to the morphism $p : X' \to X$. The composition $X_n \to X' \to Y$ is the desired morphism.
For $n = 1$ this is Limits, Lemma \ref{limits-lemma-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 $g_i : Y_i \to S_i$ are objects of $FP_{S_i, s_i}$. Then by the case $n = 1$ we can find $f'_i : X'_i \to S$ of finite presentation which are isomorphisms over $S \setminus \{s_i\}$ and whose base change to $S_i$ is $g_i$. Then we can set $$ f : X = X'_1 \times_S \ldots \times_S X'_n \to S $$ This is an object of $FP_{S, \{s_1, \ldots, s_n\}}$ whose base change by $S_i \to S$ recovers $g_i$. Thus the functor is essentially surjective. We omit the proof of fully faithfulness.
Let's start with a sequence of blowups $Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S$. The first morphism $Z_1 \to S$ is given by blowing up one of the $s_i$, say $s_1$. Applying $F$ to $Z_1 \to S$ we find a blowup $Z_{1, 1} \to S_1$ at $s_1$ is the blowing up at $s_1$ and otherwise $Z_{i, 0} = S_i$ for $i > 1$. In the next step, we either blow up one of the $s_i$, $i \geq 2$ on $Z_1$ or we pick a closed point $z_1$ of the fibre of $Z_1 \to S$ over $s_1$. In the first case it is clear what to do and in the second case we use that $(Z_1)_{s_1} \cong (Z_{1, 1})_{s_1}$ (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 $Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i$ we construct the sequence of blowing ups of $S$ in exactly the same manner.
We will use the observations made following Situation \ref{situation-vanishing} without further mention. Let $f$ be a function as in Lemma \ref{lemma-nice-meromorphic-function}. Let $\xi_i \in C_i$ be the generic point. Let $\mathcal{O}_i$ be the local ring of $X$ at $\xi_i$. Then $\mathcal{O}_i$ is a discrete valuation ring. Let $e_i$ be the valuation of $f$ in $\mathcal{O}_i$, so $e_i > 0$. Let $h_i \in \mathcal{O}_i$ be a local equation for $Z$ and let $d_i$ be its valuation. Then $d_i \geq 0$. Choose and fix $i$ with $d_i/e_i$ maximal (then $d_i > 0$ as $Z$ is not empty). Replace $f$ by $f^{d_i}$ and $Z$ by $e_iZ$. This is permissible, by the relation $\mathcal{O}_X(e_i Z) = \mathcal{O}_X(Z)^{\otimes e_i}$, the relation between the conormal sheaf and $\mathcal{O}_X(Z)$ (see Divisors, Lemmas \ref{divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors} and \ref{divisors-lemma-conormal-effective-Cartier-divisor}, and since the degree gets multiplied by $e_i$, see Varieties, Lemma \ref{varieties-lemma-degree-tensor-product}. Let $\mathcal{I}$ be the ideal sheaf of $Z$ so that $\mathcal{C}_{Z/X} = \mathcal{I}|_Z$. Consider the image $\overline{f}$ of $f$ in $\Gamma(Z, \mathcal{O}_Z)$. By our choices above we see that $\overline{f}$ vanishes in the generic points of irreducible components of $Z$ (these are all generic points of $C_j$ as $Z$ is contained in the special fibre). On the other hand, $Z$ is $(S_1)$ by Divisors, Lemma \ref{divisors-lemma-normal-effective-Cartier-divisor-S1}. Thus the scheme $Z$ has no embedded associated points and we conclude that $\overline{f} = 0$ (Divisors, Lemmas \ref{divisors-lemma-S1-no-embedded} and \ref{divisors-lemma-restriction-injective-open-contains-weakly-ass}). Hence $f$ is a global section of $\mathcal{I}$ which generates $\mathcal{I}_{\xi_i}$ by construction. Thus the image $s_i$ of $f$ in $\Gamma(C_i, \mathcal{I}|_{C_i})$ is nonzero. However, our choice of $f$ guarantees that $s_i$ has a zero at $x_i$. Hence the degree of $\mathcal{I}|_{C_i}$ is $> 0$ by Varieties, Lemma \ref{varieties-lemma-check-invertible-sheaf-trivial}.
Let $0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{O}_X \to 0$ be the extension corresponding to a nontrivial element $\xi$ of $H^1(X, \mathcal{O}_X)$ (Cohomology, Lemma \ref{cohomology-lemma-h1-extensions}). Let $\pi : P = \mathbf{P}(\mathcal{E}) \to X$ be the projective bundle associated to $\mathcal{E}$. The surjection $\mathcal{E} \to \mathcal{O}_X$ defines a section $\sigma : X \to P$ whose conormal sheaf is isomorphic to $\mathcal{O}_X$ (Divisors, Lemma \ref{divisors-lemma-conormal-sheaf-section-projective-bundle}). If the restriction of $\xi$ to $f^{-1}(U)$ is trivial, then we get a map $\mathcal{E}|_{f^{-1}(U)} \to \mathcal{O}_{f^{-1}(U)}$ splitting the injection $\mathcal{O}_X \to \mathcal{E}$. This defines a second section $\sigma' : f^{-1}(U) \to P$ disjoint from $\sigma$. Since $\xi$ is nontrivial we conclude that $\sigma'$ cannot extend to all of $X$ and be disjoint from $\sigma$. Let $X' \subset P$ be the scheme theoretic image of $\sigma'$ (Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-image}). Picture $$ \xymatrix{ & X' \ar[rd]_g \ar[r] & P \ar[d]_\pi \\ f^{-1}(U) \ar[ru]_{\sigma'} \ar[rr] & & X \ar@/_/[u]_\sigma } $$ The morphism $P \setminus \sigma(X) \to X$ is affine. If $X' \cap \sigma(X) = \emptyset$, then $X' \to X$ is both affine and proper, hence finite (Morphisms, Lemma \ref{morphisms-lemma-finite-proper}), hence an isomorphism (as $X$ is normal, see Morphisms, Lemma \ref{morphisms-lemma-finite-birational-over-normal}). This is impossible as mentioned above. \medskip\noindent Let $X^\nu$ be the normalization of $X'$. Since $A$ is Nagata, we see that $X^\nu \to X'$ is finite (Morphisms, Lemmas \ref{morphisms-lemma-nagata-normalization} and \ref{morphisms-lemma-ubiquity-nagata}). Let $Z \subset X^\nu$ be the pullback of the effective Cartier divisor $\sigma(X) \subset P$. By the above we see that $Z$ is not empty and is contained in the closed fibre of $X^\nu \to S$. Since $P \to X$ is smooth, we see that $\sigma(X)$ is an effective Cartier divisor (Divisors, Lemma \ref{divisors-lemma-section-smooth-regular-immersion}). Hence $Z \subset X^\nu$ is an effective Cartier divisor too. Since the conormal sheaf of $\sigma(X)$ in $P$ is $\mathcal{O}_X$, the conormal sheaf of $Z$ in $X^\nu$ (which is a priori invertible) is $\mathcal{O}_Z$ by Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial-flat}. This is impossible by Lemma \ref{lemma-nontrivial-normal-bundle} and the proof is complete.