hcju/stacksqa huggingface dataset:
By adjointness of $Rf_*$ and $Lf^*$ such a map is the same thing as a map $\alpha : Lf^*\kappa[-1] \to \mathcal{O}_X$. Note that $$ H^i(Lf^*\kappa[-1]) = \left\{ \begin{matrix} 0 & \text{if} & i > 1 \\ \mathcal{O}_{X_s} & \text{if} & i = 1 \\ \text{some }\mathcal{O}_{X_s}\text{-module} & \text{if} & i \leq 0 \end{matrix} \right. $$ Since $\Hom(H^0(Lf^*\kappa[-1]), \mathcal{O}_X) = 0$ as $\mathcal{O}_X$ is torsion free, the spectral sequence for $\Ext$ (Cohomology on Sites, Example \ref{sites-cohomology-example-hom-complex-into-sheaf}) implies that $\Hom_{D(\mathcal{O}_X)}(Lf^*\kappa[-1], \mathcal{O}_X)$ is equal to $\Ext^1_{\mathcal{O}_X}(\mathcal{O}_{X_s}, \mathcal{O}_X)$. We conclude that $\alpha : Lf^*\kappa[-1] \to \mathcal{O}_X$ is given by an extension $$ 0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{O}_{X_s} \to 0 $$ By Lemma \ref{lemma-H1-injective} the pullback of this extension via the surjection $\mathcal{O}_X \to \mathcal{O}_{X_s}$ is zero (since this pullback is clearly split over $f^{-1}(U)$). Thus $1 \in \mathcal{O}_{X_s}$ lifts to a global section $s$ of $\mathcal{E}$. Multiplying $s$ by the ideal sheaf $\mathcal{I}$ of $X_s$ we obtain an $\mathcal{O}_X$-module map $c_s : \mathcal{I} \to \mathcal{O}_X$. Applying $f_*$ we obtain an $A$-linear map $f_*c_s : \mathfrak m \to A$. Since $A$ is a Noetherian normal local domain this map is given by multiplication by an element $a \in A$. Changing $s$ into $s -  a$ we find that $s$ is annihilated by $\mathcal{I}$ and the extension is trivial as desired.
We will use the observations made following Situation \ref{situation-vanishing} without further mention. As $X$ is normal and $g$ is dominant and birational, we have $g_*\mathcal{O}_{X'} = \mathcal{O}_X$, see for example More on Morphisms, Lemma \ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}. Since the fibres of $g$ have dimension $\leq 1$, we have $R^pg_*\mathcal{O}_{X'} = 0$ for $p > 1$, see for example Cohomology of Schemes, Lemma \ref{coherent-lemma-higher-direct-images-zero-above-dimension-fibre}. The support of $R^1g_*\mathcal{O}_{X'}$ is contained in the set of points of $X$ where the fibres of $g'$ have dimension $\geq 1$. Thus it is contained in the set of images of those irreducible components $C' \subset X'_s$ which map to points of $X_s$ which is a finite set of closed points (recall that $X'_s \to X_s$ is a morphism of proper $1$-dimensional schemes over $\kappa$). Then $R^1g_*\mathcal{O}_{X'}$ is globally generated by Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-support-dimension-0}. Using the morphism $f : X \to S$ and the references above we find that $H^p(X, \mathcal{F}) = 0$ for $p > 1$ for any coherent $\mathcal{O}_X$-module $\mathcal{F}$. Hence the short exact sequence of the lemma is a consequence of the Leray spectral sequence for $g$ and $\mathcal{O}_{X'}$, see Cohomology, Lemma \ref{cohomology-lemma-Leray}.
By the short exact sequence $ 0 \to \mathcal{O}_X \xrightarrow{a} \mathcal{O}_X \to \mathcal{O}_Z \to 0 $ we see that \begin{equation} \label{equation-a-torsion} M_{X, a} = H^1(X, \mathcal{O}_X)[a] \end{equation} Here $N[a] = \{n \in N \mid an = 0\}$ for an $A$-module $N$. Thus if $a$ divides $b$, then $M_{X, a} \subset M_{X, b}$. Suppose that for some $c \in A$ the modules $M_{X, c}$ have bounded length. Then for every $X$ we have an exact sequence $$ 0 \to M_{X, c} \to M_{X, c^2} \to M_{X, c} $$ where the second arrow is given by multiplication by $c$. Hence we see that $M_{X, c^2}$ has bounded length as well. Thus it suffices to find a $c \in A$ for which the lemma is true such that $a$ divides $c^n$ for some $n > 0$. By More on Algebra, Lemma \ref{more-algebra-lemma-divides-radical} we may assume $A/(a)$ is a reduced ring. \medskip\noindent Assume that $A/(a)$ is reduced. Let $A/(a) \subset B$ be the normalization of $A/(a)$ in its quotient ring. Because $A$ is Nagata, we see that $\Coker(A \to B)$ is finite. We claim the length of this finite module is a bound. To see this, consider $f : X \to \Spec(A)$ as in the lemma and let $Z' \subset Z$ be the scheme theoretic closure of $Z \cap f^{-1}(U)$. Then $Z' \to \Spec(A/(a))$ is finite for example by Varieties, Lemma \ref{varieties-lemma-finite-in-codim-1}. Hence $Z' = \Spec(B')$ with $A/(a) \subset B' \subset B$. On the other hand, we claim the map $$ H^0(Z, \mathcal{O}_Z) \to H^0(Z', \mathcal{O}_{Z'}) $$ is injective. Namely, if $s \in H^0(Z, \mathcal{O}_Z)$ is in the kernel, then the restriction of $s$ to $f^{-1}(U) \cap Z$ is zero. Hence the image of $s$ in $H^1(X, \mathcal{O}_X)$ vanishes in $H^1(f^{-1}(U), \mathcal{O}_X)$. By Lemma \ref{lemma-H1-injective} we see that $s$ comes from an element $\tilde s$ of $A$. But by assumption $\tilde s$ maps to zero in $B'$ which implies that $s = 0$. Putting everything together we see that $M_{X, a}$ is a subquotient of $B'/A$, namely not every element of $B'$ extends to a global section of $\mathcal{O}_Z$, but in any case the length of $M_{X, a}$ is bounded by the length of $B/A$.
Choose a finite type $A$-algebra $C$ such that $B = C_\mathfrak q$ for some prime $\mathfrak q \subset C$. After replacing $C$ by the image of $C$ in $B$ we may assume that $C$ is a domain with fraction field equal to the fraction field of $A$. Then we can choose a closed immersion $\Spec(C) \to \mathbf{A}^n_A$ and take the closure in $\mathbf{P}^n_A$ to conclude that $B$ is isomorphic to $\mathcal{O}_{X, x}$ for some closed point $x \in X$ of a projective modification $X \to \Spec(A)$. (Morphisms, Lemma \ref{morphisms-lemma-dimension-formula}, shows that $\kappa(x)$ is finite over $\kappa$ and then Morphisms, Lemma \ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre} shows that $x$ is a closed point.) Let $\nu : X^\nu \to X$ be the normalization. Since $A$ is Nagata the morphism $\nu$ is finite (Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization}). Thus $X^\nu$ is projective over $A$ by More on Morphisms, Lemma \ref{more-morphisms-lemma-category-projective}. Since $B = \mathcal{O}_{X, x}$ is normal, we see that $\mathcal{O}_{X, x} = (\nu_*\mathcal{O}_{X^\nu})_x$. Hence there is a unique point $x^\nu \in X^\nu$ lying over $x$ and $\mathcal{O}_{X^\nu, x^\nu} = \mathcal{O}_{X, x}$. Thus we may assume $X$ is normal and projective over $A$. Let $Y \to \Spec(\mathcal{O}_{X, x}) = \Spec(B)$ be a modification with $Y$ normal. We have to show that $H^1(Y, \mathcal{O}_Y) = 0$. By Limits, Lemma \ref{limits-lemma-modifications} we can find a morphism of schemes $g : X' \to X$ which is an isomorphism over $X \setminus \{x\}$ such that $X' \times_X \Spec(\mathcal{O}_{X, x})$ is isomorphic to $Y$. Then $g$ is a modification as it is proper by Limits, Lemma \ref{limits-lemma-modifications-properties}. The local ring of $X'$ at a point of $x'$ is either isomorphic to the local ring of $X$ at $g(x')$ if $g(x') \not = x$ and if $g(x') = x$, then the local ring of $X'$ at $x'$ is isomorphic to the local ring of $Y$ at the corresponding point. Hence we see that $X'$ is normal as both $X$ and $Y$ are normal. Thus $H^1(X', \mathcal{O}_{X'}) = 0$ by our assumption on $A$. By Lemma \ref{lemma-exact-sequence} we have $R^1g_*\mathcal{O}_{X'} = 0$. Clearly this means that $H^1(Y, \mathcal{O}_Y) = 0$ as desired.
We choose a modification $X \to \Spec(A)$ with $X$ normal which maximizes the length of $H^1(X, \mathcal{O}_X)$. By Lemma \ref{lemma-exact-sequence} for any further modification $g : X' \to X$ with $X'$ normal we have $R^1g_*\mathcal{O}_{X'} = 0$ and $H^1(X, \mathcal{O}_X) = H^1(X', \mathcal{O}_{X'})$. \medskip\noindent Let $x \in X$ be a closed point. We will show that $\mathcal{O}_{X, x}$ defines a rational singularity. Let $Y \to \Spec(\mathcal{O}_{X, x})$ be a modification with $Y$ normal. We have to show that $H^1(Y, \mathcal{O}_Y) = 0$. By Limits, Lemma \ref{limits-lemma-modifications} we can find a morphism of schemes $g : X' \to X$ which is an isomorphism over $X \setminus \{x\}$ such that $X' \times_X \Spec(\mathcal{O}_{X, x})$ is isomorphic to $Y$. Then $g$ is a modification as it is proper by Limits, Lemma \ref{limits-lemma-modifications-properties}. The local ring of $X'$ at a point of $x'$ is either isomorphic to the local ring of $X$ at $g(x')$ if $g(x') \not = x$ and if $g(x') = x$, then the local ring of $X'$ at $x'$ is isomorphic to the local ring of $Y$ at the corresponding point. Hence we see that $X'$ is normal as both $X$ and $Y$ are normal. By maximality we have $R^1g_*\mathcal{O}_{X'} = 0$ (see first paragraph). Clearly this means that $H^1(Y, \mathcal{O}_Y) = 0$ as desired. \medskip\noindent The conclusion is that we've found one normal modification $X$ of $\Spec(A)$ such that the local rings of $X$ at closed points all define rational singularities. Then we choose a sequence of normalized blowups $X_n \to \ldots \to X_1 \to \Spec(A)$ such that $X_n$ dominates $X$, see Lemma \ref{lemma-dominate-by-normalized-blowing-up}. For a closed point $x' \in X_n$ mapping to $x \in X$ we can apply Lemma \ref{lemma-rational-propagates} to the ring map $\mathcal{O}_{X, x} \to \mathcal{O}_{X_n, x'}$ to see that $\mathcal{O}_{X_n, x'}$ defines a rational singularity.
Let $n$ be the degree of the fraction field extension $K \subset L$. Let $\text{Trace}_{L/K} : L \to K$ be the trace. Since the extension is finite separable the trace pairing $(h, g) \mapsto \text{Trace}_{L/K}(fg)$ is a nondegenerate bilinear form on $L$ over $K$. See Fields, Lemma \ref{fields-lemma-separable-trace-pairing}. Pick $b_1, \ldots, b_n \in B$ which form a basis of $L$ over $K$. By the above $d = \det(\text{Trace}_{L/K}(b_ib_j)) \in A$ is nonzero. \medskip\noindent Let $Y \to \Spec(B)$ be a modification with $Y$ normal. We can find a $U$-admissible blowup $X'$ of $\Spec(A)$ such that the strict transform $Y'$ of $Y$ is finite over $X'$, see More on Flatness, Lemma \ref{flat-lemma-finite-after-blowing-up}. Picture $$ \xymatrix{ Y' \ar[d] \ar[r] & Y \ar[r] & \Spec(B) \ar[d] \\ X' \ar[rr] & & \Spec(A) } $$ After replacing $X'$ and $Y'$ by their normalizations we may assume that $X'$ and $Y'$ are normal modifications of $\Spec(A)$ and $\Spec(B)$. In this way we reduce to the case where there exists a commutative diagram $$ \xymatrix{ Y \ar[d]_\pi \ar[r]_-g & \Spec(B) \ar[d] \\ X \ar[r]^-f & \Spec(A) } $$ with $X$ and $Y$ normal modifications of $\Spec(A)$ and $\Spec(B)$ and $\pi$ finite. \medskip\noindent The trace map on $L$ over $K$ extends to a map of $\mathcal{O}_X$-modules $\text{Trace} : \pi_*\mathcal{O}_Y \to \mathcal{O}_X$. Consider the map $$ \Phi : \pi_*\mathcal{O}_Y \longrightarrow \mathcal{O}_X^{\oplus n},\quad s \longmapsto (\text{Trace}(b_1s), \ldots, \text{Trace}(b_ns)) $$ This map is injective (because it is injective in the generic point) and there is a map $$ \mathcal{O}_X^{\oplus n} \longrightarrow \pi_*\mathcal{O}_Y,\quad (s_1, \ldots, s_n) \longmapsto \sum b_i s_i $$ whose composition with $\Phi$ has matrix $\text{Trace}(b_ib_j)$. Hence the cokernel of $\Phi$ is annihilated by $d$. Thus we see that we have an exact sequence $$ H^0(X, \Coker(\Phi)) \to H^1(Y, \mathcal{O}_Y) \to H^1(X, \mathcal{O}_X)^{\oplus n} $$ Since the right hand side is bounded by assumption, it suffices to show that the $d$-torsion in $H^1(Y, \mathcal{O}_Y)$ is bounded. This is the content of Lemma \ref{lemma-bound-a-torsion} and (\ref{equation-a-torsion}).
(The assumption that $A$ be Nagata is not necessary for this proof, but we've only defined the notion of rational singularity in the case of Nagata $2$-dimensional normal local domains.) Let $X \to \Spec(A)$ be a modification with $X$ normal. By Lemma \ref{lemma-dominate-by-blowing-up-in-points} we can dominate $X$ by a scheme $X_n$ which is the last in a sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A) $$ of blowing ups in closed points. By Lemma \ref{lemma-blowup-regular} the schemes $X_i$ are regular, in particular normal (Algebra, Lemma \ref{algebra-lemma-regular-normal}). By Lemma \ref{lemma-exact-sequence} we have $H^1(X, \mathcal{O}_X) \subset H^1(X_n, \mathcal{O}_{X_n})$. Thus it suffices to prove $H^1(X_n, \mathcal{O}_{X_n}) = 0$. Using Lemma \ref{lemma-exact-sequence} again, we see that it suffices to prove $R^1(X_i \to X_{i - 1})_*\mathcal{O}_{X_i} = 0$ for $i = 1, \ldots, n$. This follows from Lemma \ref{lemma-cohomology-of-blowup}.
For $X \to \Spec(A)$ as in the statement we have to bound $H^1(X, \mathcal{O}_X)$. Let $\omega_X$ be the dualizing module of $X$ as in the statement of Grauert-Riemenschneider (Proposition \ref{proposition-Grauert-Riemenschneider}). The trace map is the map $Rf_*\omega_X \to \omega_A$ described in Duality for Schemes, Section \ref{duality-section-trace}. By Grauert-Riemenschneider we have $Rf_*\omega_X = f_*\omega_X$ thus the trace map indeed produces a map $\Gamma(X, \omega_X) \to \omega_A$. By duality we have $Rf_*\omega_X = R\Hom_A(Rf_*\mathcal{O}_X, \omega_A)$ (this uses that $\omega_X[2]$ is the dualizing complex on $X$ normalized relative to $\omega_A[2]$, see Duality for Schemes, Lemma \ref{duality-lemma-duality-bootstrap} or more directly Section \ref{duality-section-duality} or even more directly Example \ref{duality-example-iso-on-RSheafHom-noetherian}). The distinguished triangle $$ A \to Rf_*\mathcal{O}_X \to R^1f_*\mathcal{O}_X[-1] \to A[1] $$ is transformed by $R\Hom_A(-, \omega_A)$ into the short exact sequence $$ 0 \to f_*\omega_X \to \omega_A \to \Ext_A^2(R^1f_*\mathcal{O}_X, \omega_A) \to 0 $$ (and $\Ext_A^i(R^1f_*\mathcal{O}_X, \omega_A) = 0$ for $i \not = 2$; this will follow from the discussion below as well). Since $R^1f_*\mathcal{O}_X$ is supported in $\{\mathfrak m\}$, the local duality theorem tells us that $$ \Ext_A^2(R^1f_*\mathcal{O}_X, \omega_A) = \Ext_A^0(R^1f_*\mathcal{O}_X, \omega_A[2]) = \Hom_A(R^1f_*\mathcal{O}_X, E) $$ is the Matlis dual of $R^1f_*\mathcal{O}_X$ (and the other ext groups are zero), see Dualizing Complexes, Lemma \ref{dualizing-lemma-special-case-local-duality}. By the equivalence of categories inherent in Matlis duality (Dualizing Complexes, Proposition \ref{dualizing-proposition-matlis}), if $R^1f_*\mathcal{O}_X$ is not annihilated by $d$, then neither is the $\Ext^2$ above. Hence we see that $H^1(X, \mathcal{O}_X)$ is annihilated by $d$. Thus the required boundedness follows from Lemma \ref{lemma-bound-a-torsion} and (\ref{equation-a-torsion}).
Observe that $A$ is Gorenstein (Dualizing Complexes, Lemma \ref{dualizing-lemma-regular-gorenstein}) and hence the invertible module $\omega_A$ does indeed serve as a dualizing module. Moreover, any $X$ as in the lemma has an invertible dualizing module $\omega_X$ as $X$ is regular (hence Gorenstein) and proper over $A$, see Remark \ref{remark-dualizing-setup} and Lemma \ref{lemma-blowup-regular}. Suppose we have constructed the map $\varphi_X : (\Omega^r_{X/A_0})^{**} \to \omega_X$ and suppose that $b : X' \to X$ is a blowup in a closed point. Set $\Omega^r_X = (\Omega^r_{X/A_0})^{**}$ and $\Omega^r_{X'} = (\Omega^r_{X'/A_0})^{**}$. Since $\omega_{X'} = b^!(\omega_X)$ a map $\Omega^r_{X'} \to \omega_{X'}$ is the same thing as a map $Rb_*(\Omega^r_{X'}) \to \omega_X$. See discussion in Remark \ref{remark-dualizing-setup} and Duality for Schemes, Section \ref{duality-section-duality}. Thus in turn it suffices to produce a map $$ Rb_*(\Omega^r_{X'}) \longrightarrow \Omega^r_X $$ The sheaves $\Omega^r_{X'}$ and $\Omega^r_X$ are invertible, see Divisors, Lemma \ref{divisors-lemma-reflexive-over-regular-dim-2}. Consider the exact sequence $$ b^*\Omega_{X/A_0} \to \Omega_{X'/A_0} \to \Omega_{X'/X} \to 0 $$ A local calculation shows that $\Omega_{X'/X}$ is isomorphic to an invertible module on the exceptional divisor $E$, see Lemma \ref{lemma-differentials-of-blowup}. It follows that either $$ \Omega^r_{X'} \cong (b^*\Omega^r_X)(E) \quad\text{or}\quad \Omega^r_{X'} \cong b^*\Omega^r_X $$ see Divisors, Lemma \ref{divisors-lemma-wedge-product-ses}. (The second possibility never happens in characteristic zero, but can happen in characteristic $p$.) In both cases we see that $R^1b_*(\Omega^r_{X'}) = 0$ and $b_*(\Omega^r_{X'}) = \Omega^r_X$ by Lemma \ref{lemma-cohomology-of-blowup}.
Part (1) follows from  Cohomology of Schemes, Lemma \ref{coherent-lemma-higher-direct-images-zero-above-dimension-fibre}. If $\mathcal{F}$ is globally generated, then there is a surjection $\bigoplus_{i \in I} \mathcal{O}_X \to \mathcal{F}$. By part (1) and the long exact sequence of cohomology this induces a surjection on $H^1$. Since $H^1(X, \mathcal{O}_X) = 0$ as $S$ has a rational singularity, and since $H^1(X, -)$ commutes with direct sums (Cohomology, Lemma \ref{cohomology-lemma-quasi-separated-cohomology-colimit}) we conclude.
We have $H^0(X, \mathcal{O}_X) = A$, see discussion following Situation \ref{situation-vanishing}. Then $\mathfrak m \subset H^0(X, \mathcal{I}) \subset H^0(X, \mathcal{O}_X)$. The second inclusion is not an equality as $X_s \not = \emptyset$. Thus $H^0(X, \mathcal{I}) = \mathfrak m$. As $\mathcal{I}^n = \mathfrak m^n\mathcal{O}_X$ our Lemma \ref{lemma-globally-generated} shows that $H^1(X, \mathcal{I}^n) = 0$. \medskip\noindent Choose generators $x_1, \ldots, x_{\mu + 1}$ of $\mathfrak m$. These define global sections of $\mathcal{I}$ which generate it. Hence a short exact sequence $$ 0 \to \mathcal{F} \to \mathcal{O}_X^{\oplus \mu + 1} \to \mathcal{I} \to 0 $$ Then $\mathcal{F}$ is a finite locally free $\mathcal{O}_X$-module of rank $\mu$ and $\mathcal{F} \otimes \mathcal{I}$ is globally generated by Constructions, Lemma \ref{constructions-lemma-globally-generated-omega-twist-1}. Hence $\mathcal{F} \otimes \mathcal{I}^n$ is globally generated for all $n \geq 1$. Thus for $n \geq 2$ we can consider the exact sequence $$ 0 \to \mathcal{F} \otimes \mathcal{I}^{n - 1} \to (\mathcal{I}^{n - 1})^{\oplus \mu + 1} \to \mathcal{I}^n \to 0 $$ Applying the long exact sequence of cohomology using that $H^1(X, \mathcal{F} \otimes \mathcal{I}^{n - 1}) = 0$ by Lemma \ref{lemma-globally-generated} we obtain that every element of $H^0(X, \mathcal{I}^n)$ is of the form $\sum x_i a_i$ for some $a_i \in H^0(X, \mathcal{I}^{n - 1})$. This shows that $H^0(X, \mathcal{I}^n) = \mathfrak m^n$ by induction.
Let $X' \to \Spec(A)$ be the blowup, in other words $$ X' = \text{Proj}(A \oplus \mathfrak m \oplus \mathfrak m^2 \oplus \ldots). $$ is the Proj of the Rees algebra. This in particular shows that $X'$ is integral and that $X' \to \Spec(A)$ is a projective modification. Let $X$ be the normalization of $X'$. Since $A$ is Nagata, we see that $\nu : X \to X'$ is finite (Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization}). Let $E' \subset X'$ be the exceptional divisor and let $E \subset X$ be the inverse image. Let $\mathcal{I}' \subset \mathcal{O}_{X'}$ and $\mathcal{I} \subset \mathcal{O}_X$ be their ideal sheaves. Recall that $\mathcal{I}' = \mathcal{O}_{X'}(1)$ (Divisors, Lemma \ref{divisors-lemma-blowing-up-projective}). Observe that $\mathcal{I} = \nu^*\mathcal{I}'$ and that $E$ is an effective Cartier divisor (Divisors, Lemma \ref{divisors-lemma-pullback-effective-Cartier-defined}). We are trying to show that $\nu$ is an isomorphism. As $\nu$ is finite, it suffices to show that $\mathcal{O}_{X'} \to \nu_*\mathcal{O}_X$ is an isomorphism. If not, then we can find an $n \geq 0$ such that $$ H^0(X', (\mathcal{I}')^n) \not = H^0(X', (\nu_*\mathcal{O}_X) \otimes (\mathcal{I}')^n) $$ for example because we can recover quasi-coherent $\mathcal{O}_{X'}$-modules from their associated graded modules, see Properties, Lemma \ref{properties-lemma-ample-quasi-coherent}. By the projection formula we have $$ H^0(X', (\nu_*\mathcal{O}_X) \otimes (\mathcal{I}')^n) = H^0(X, \nu^*(\mathcal{I}')^n) = H^0(X, \mathcal{I}^n) = \mathfrak m^n $$ the last equality by Lemma \ref{lemma-sections-powers-I-rational}. On the other hand, there is clearly an injection $\mathfrak m^n \to H^0(X', (\mathcal{I}')^n)$. Since $H^0(X', (\mathcal{I}')^n)$ is torsion free we conclude equality holds for all $n$, hence $X = X'$.
Since $\mathcal{O}_X(1)$ is very ample by construction, we see that its restriction to the special fibre $E$ is very ample as well. By Lemma \ref{lemma-blow-up-normal-rational} the scheme $X$ is normal. Then $E$ is Cohen-Macaulay by Divisors, Lemma \ref{divisors-lemma-normal-effective-Cartier-divisor-S1}. Lemma \ref{lemma-sections-powers-I-rational} applies and we obtain (4) and (5) from the exact sequences $$ 0 \to \mathcal{I}^{n + 1} \to \mathcal{I}^n \to i_*\mathcal{O}_E(n) \to 0 $$ and the long exact cohomology sequence. In particular, we see that $$ \deg(\mathcal{O}_E(1)) = \chi(E, \mathcal{O}_E(1)) - \chi(E, \mathcal{O}_E) = \dim(\mathfrak m/\mathfrak m^2) - 1 $$ by Varieties, Definition \ref{varieties-definition-degree-invertible-sheaf}. Thus (3) follows as well.
By Grauert-Riemenschneider (Proposition \ref{proposition-Grauert-Riemenschneider}) we see that $Rf_*\omega_X = f_*\omega_X$. By duality we have a short exact sequence $$ 0 \to f_*\omega_X \to \omega_A \to \Ext^2_A(R^1f_*\mathcal{O}_X, \omega_A) \to 0 $$ (for example see proof of Lemma \ref{lemma-bound-dualizing-implies-bound}) and since $A$ defines a rational singularity we obtain $f_*\omega_X = \omega_A$.
We will use the results of Lemma \ref{lemma-cohomology-blow-up-rational} without further mention. Observe that $\omega_E = \omega_X|_E \otimes \mathcal{O}_E(-1)$ by Duality for Schemes, Lemmas \ref{duality-lemma-sheaf-with-exact-support-effective-Cartier} and \ref{duality-lemma-twisted-inverse-image-closed}. Thus $\omega_X|_E = \omega_E(1)$. Consider the short exact sequences $$ 0 \to \omega_X(n + 1) \to \omega_X(n) \to i_*\omega_E(n + 1) \to 0 $$ By Algebraic Curves, Lemma \ref{curves-lemma-vanishing-twist} we see that $H^1(E, \omega_E(n + 1)) = 0$ for $n \geq 0$. Thus we see that the maps $$ \ldots \to H^1(X, \omega_X(2)) \to H^1(X, \omega_X(1)) \to H^1(X, \omega_X) $$ are surjective. Since $H^1(X, \omega_X(n))$ is zero for $n \gg 0$ (Cohomology of Schemes, Lemma \ref{coherent-lemma-kill-by-twisting}) we conclude that (2) holds. \medskip\noindent By Algebraic Curves, Lemma \ref{curves-lemma-tensor-omega-with-globally-generated-invertible} we see that $\omega_X|_E = \omega_E \otimes \mathcal{O}_E(1)$ is globally generated. Since we seen above that $H^1(X, \omega_X(1)) = 0$ the map $H^0(X, \omega_X) \to H^0(E, \omega_X|_E)$ is surjective. We conclude that $\omega_X$ is globally generated hence (3) holds because $\Gamma(X, \omega_X) = \omega_A$ is used in Lemma \ref{lemma-dualizing-rational} to define the map.