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## Tag 0AX7

### 50.7. Vanishing

In this section we will often work in the following setting. Recall that a modification is a proper birational morphism between integral schemes (Morphisms, Definition 28.48.11).

Situation 50.7.1. Here $(A, \mathfrak m, \kappa)$ be a local Noetherian normal domain of dimension $2$. Let $s$ be the closed point of $S = \mathop{\mathrm{Spec}}(A)$ and $U = S \setminus \{s\}$. Let $f : X \to S$ be a modification. We denote $C_1, \ldots, C_r$ the irreducible components of the special fibre $X_s$ of $f$.

By Varieties, Lemma 32.17.3 the morphism $f$ defines an isomorphism $f^{-1}(U) \to U$. The special fibre $X_s$ is proper over $\mathop{\mathrm{Spec}}(\kappa)$ and has dimension at most $1$ by Varieties, Lemma 32.19.3. By Stein factorization (More on Morphisms, Lemma 36.45.6) we have $f_*\mathcal{O}_X = \mathcal{O}_S$ and the special fibre $X_s$ is geometrically connected over $\kappa$. If $X_s$ has dimension $0$, then $f$ is finite (More on Morphisms, Lemma 36.38.5) and hence an isomorphism (Morphisms, Lemma 28.51.8). We will discard this uninteresting case and we conclude that $\dim(C_i) = 1$ for $i = 1, \ldots, r$.

Lemma 50.7.2. In Situation 50.7.1 there exists a $U$-admissible blowup $X' \to S$ which dominates $X$.

Proof. This is a special case of More on Flatness, Lemma 37.31.4. $\square$

Lemma 50.7.3. In Situation 50.7.1 there exists a nonzero $f \in \mathfrak m$ such that for every $i = 1, \ldots, r$ there exist

1. a closed point $x_i \in C_i$ with $x_i \not \in C_j$ for $j \not = i$,
2. a factorization $f = g_i f_i$ of $f$ in $\mathcal{O}_{X, x_i}$ such that $g_i \in \mathfrak m_{x_i}$ maps to a nonzero element of $\mathcal{O}_{C_i, x_i}$.

Proof. We will use the observations made following Situation 50.7.1 without further mention. Pick a closed point $x_i \in C_i$ which is not in $C_j$ for $j \not = i$. Pick $g_i \in \mathfrak m_{x_i}$ which maps to a nonzero element of $\mathcal{O}_{C_i, x_i}$. Since the fraction field of $A$ is the fraction field of $\mathcal{O}_{X_i, x_i}$ we can write $g_i = a_i/b_i$ for some $a_i, b_i \in A$. Take $f = \prod a_i$. $\square$

Lemma 50.7.4. In Situation 50.7.1 assume $X$ is normal. Let $Z \subset X$ be a nonempty effective Cartier divisor such that $Z \subset X_s$ set theoretically. Then the conormal sheaf of $Z$ is not trivial. More precisely, there exists an $i$ such that $C_i \subset Z$ and $\deg(\mathcal{C}_{Z/X}|_{C_i}) > 0$.

Proof. We will use the observations made following Situation 50.7.1 without further mention. Let $f$ be a function as in Lemma 50.7.3. 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 30.14.4 and 30.14.2, and since the degree gets multiplied by $e_i$, see Varieties, Lemma 32.43.7. 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 30.15.6. Thus the scheme $Z$ has no embedded associated points and we conclude that $\overline{f} = 0$ (Divisors, Lemmas 30.4.3 and 30.5.6). 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 32.43.12. $\square$

Lemma 50.7.5. In Situation 50.7.1 assume $X$ is normal and $A$ Nagata. The map $$H^1(X, \mathcal{O}_X) \longrightarrow H^1(f^{-1}(U), \mathcal{O}_X)$$ is injective.

Proof. 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 20.6.1). 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 30.29.6). 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 28.6.2). 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 28.42.11), hence an isomorphism (as $X$ is normal, see Morphisms, Lemma 28.51.8). This is impossible as mentioned above.

Let $X^\nu$ be the normalization of $X'$. Since $A$ is Nagata, we see that $X^\nu \to X'$ is finite (Morphisms, Lemmas 28.51.10 and 28.17.2). 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 30.22.7). 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 28.30.4. This is impossible by Lemma 50.7.4 and the proof is complete. $\square$

Lemma 50.7.6. In Situation 50.7.1 assume $X$ is normal and $A$ Nagata. Then $$\mathop{\mathrm{Hom}}\nolimits_{D(A)}(\kappa[-1], Rf_*\mathcal{O}_X)$$ is zero. This uses $D(A) = D_\mathit{QCoh}(\mathcal{O}_S)$ to think of $Rf_*\mathcal{O}_X$ as an object of $D(A)$.

Proof. 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 $\mathop{\mathrm{Hom}}\nolimits(H^0(Lf^*\kappa[-1]), \mathcal{O}_X) = 0$ as $\mathcal{O}_X$ is torsion free, the spectral sequence for $\mathop{\mathrm{Ext}}\nolimits$ (Cohomology on Sites, Example 21.26.1) implies that $\mathop{\mathrm{Hom}}\nolimits_{D(\mathcal{O}_X)}(Lf^*\kappa[-1], \mathcal{O}_X)$ is equal to $\mathop{\mathrm{Ext}}\nolimits^1_{\mathcal{O}_X}(\mathcal{O}_{X_s}, \mathcal{O}_X)$. 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 50.7.5 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. $\square$

Remark 50.7.7. Let $X$ be an integral Noetherian normal scheme of dimension $2$. In this case the following are equivalent

1. $X$ has a dualizing complex $\omega_X^\bullet$,
2. there is a coherent $\mathcal{O}_X$-module $\omega_X$ such that $\omega_X[n]$ is a dualizing complex, where $n$ can be any integer.

This follows from the fact that $X$ is Cohen-Macaulay (Properties, Lemma 27.12.7) and Duality for Schemes, Lemma 46.24.1. In this situation we will say that $\omega_X$ is a dualizing module in accordance with Duality for Schemes, Section 46.23. In particular, when $A$ is a Noetherian normal local domain of dimension $2$, then we say $A$ has a dualizing module $\omega_A$ if the above is true. In this case, if $X \to \mathop{\mathrm{Spec}}(A)$ is a normal modification, then $X$ has a dualizing module too, see Duality for Schemes, Example 46.23.1. In this situation we always denote $\omega_X$ the dualizing module normalized with respect to $\omega_A$, i.e., such that $\omega_X[2]$ is the dualizing complex normalized relative to $\omega_A[2]$. See Duality for Schemes, Section 46.21.

The Grauert-Riemenschneider vanishing of the next proposition is a formal consequence of Lemma 50.7.6 and the general theory of duality.

Proposition 50.7.8 (Grauert-Riemenschneider). In Situation 50.7.1 assume

1. $X$ is a normal scheme,
2. $A$ is Nagata and has a dualizing complex $\omega_A^\bullet$.

Let $\omega_X$ be the dualizing module of $X$ (Remark 50.7.7). Then $R^1f_*\omega_X = 0$.

Proof. In this proof we will use the identification $D(A) = D_\mathit{QCoh}(\mathcal{O}_S)$ to identify quasi-coherent $\mathcal{O}_S$-modules with $A$-modules. Moreover, we may assume that $\omega_A^\bullet$ is normalized, see Dualizing Complexes, Section 45.16. Since $X$ is a Noetherian normal $2$-dimensional scheme it is Cohen-Macaulay (Properties, Lemma 27.12.7). Thus $\omega_X^\bullet = \omega_X[2]$ (Duality for Schemes, Lemma 46.24.1 and the normalization in Duality for Schemes, Example 46.23.1). If the proposition is false, then we can find a nonzero map $R^1f_*\omega_X \to \kappa$. In other words we obtain a nonzero map $\alpha : Rf_*\omega_X^\bullet \to \kappa[1]$. Applying $R\mathop{\mathrm{Hom}}\nolimits_A(-, \omega_A^\bullet)$ we get a nonzero map $$\beta : \kappa[-1] \longrightarrow Rf_*\mathcal{O}_X$$ which is impossible by Lemma 50.7.6. To see that $R\mathop{\mathrm{Hom}}\nolimits_A(-, \omega_A^\bullet)$ does what we said, first note that $$R\mathop{\mathrm{Hom}}\nolimits_A(\kappa[1], \omega_A^\bullet) = R\mathop{\mathrm{Hom}}\nolimits_A(\kappa, \omega_A^\bullet)[-1] = \kappa[-1]$$ as $\omega_A^\bullet$ is normalized and we have $$R\mathop{\mathrm{Hom}}\nolimits_A(Rf_*\omega_X^\bullet, \omega_A^\bullet) = Rf_*R\mathop{\mathcal{H}\!\mathit{om}}\nolimits_{\mathcal{O}_X}(\omega_X^\bullet, \omega_X^\bullet) = Rf_*\mathcal{O}_X$$ The first equality by Duality for Schemes, Lemma 46.3.6 and the fact that $\omega_X^\bullet = f^!\omega_A^\bullet$ by construction, and the second equality because $\omega_X^\bullet$ is a dualizing complex for $X$ (which goes back to Duality for Schemes, Lemma 46.18.6). $\square$

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\section{Vanishing}
\label{section-vanishing}

\noindent
In this section we will often work in the following setting.
Recall that a modification is a proper birational morphism
between integral schemes (Morphisms, Definition
\ref{morphisms-definition-modification}).

\begin{situation}
\label{situation-vanishing}
Here $(A, \mathfrak m, \kappa)$ be a local Noetherian normal domain of
dimension $2$. Let $s$ be the closed point of $S = \Spec(A)$ and
$U = S \setminus \{s\}$. Let $f : X \to S$ be a modification.
We denote $C_1, \ldots, C_r$ the irreducible
components of the special fibre $X_s$ of $f$.
\end{situation}

\noindent
By Varieties, Lemma
\ref{varieties-lemma-modification-normal-iso-over-codimension-1}
the morphism $f$ defines an isomorphism $f^{-1}(U) \to U$.
The special fibre $X_s$ is proper over $\Spec(\kappa)$ and
has dimension at most $1$ by Varieties, Lemma
\ref{varieties-lemma-dimension-fibre-in-higher-codimension}.
By Stein factorization (More on Morphisms, Lemma
\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal})
we have $f_*\mathcal{O}_X = \mathcal{O}_S$ and
the special fibre $X_s$ is geometrically connected over $\kappa$.
If $X_s$ has dimension $0$, then $f$ is finite
(More on Morphisms, Lemma
\ref{more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood})
and hence an isomorphism
(Morphisms, Lemma \ref{morphisms-lemma-finite-birational-over-normal}).
We will discard this uninteresting case and we conclude that
$\dim(C_i) = 1$ for $i = 1, \ldots, r$.

\begin{lemma}
\label{lemma-dominate-by-scheme-modification}
In Situation \ref{situation-vanishing} there exists a $U$-admissible
blowup $X' \to S$ which dominates $X$.
\end{lemma}

\begin{proof}
This is a special case of
More on Flatness, Lemma \ref{flat-lemma-dominate-modification-by-blowup}.
\end{proof}

\begin{lemma}
\label{lemma-nice-meromorphic-function}
In Situation \ref{situation-vanishing} there exists a nonzero
$f \in \mathfrak m$ such that for every $i = 1, \ldots, r$ there exist
\begin{enumerate}
\item a closed point $x_i \in C_i$ with $x_i \not \in C_j$ for $j \not = i$,
\item a factorization $f = g_i f_i$ of $f$ in $\mathcal{O}_{X, x_i}$
such that $g_i \in \mathfrak m_{x_i}$ maps to a nonzero element
of $\mathcal{O}_{C_i, x_i}$.
\end{enumerate}
\end{lemma}

\begin{proof}
We will use the observations made following Situation \ref{situation-vanishing}
without further mention. Pick a closed point $x_i \in C_i$ which is not in
$C_j$ for $j \not = i$. Pick $g_i \in \mathfrak m_{x_i}$ which maps to a
nonzero element of $\mathcal{O}_{C_i, x_i}$. Since the fraction field of $A$
is the fraction field of $\mathcal{O}_{X_i, x_i}$ we can write
$g_i = a_i/b_i$ for some $a_i, b_i \in A$. Take $f = \prod a_i$.
\end{proof}

\begin{lemma}
\label{lemma-nontrivial-normal-bundle}
In Situation \ref{situation-vanishing} assume $X$ is normal.
Let $Z \subset X$ be a nonempty effective Cartier divisor such that
$Z \subset X_s$ set theoretically.
Then the conormal sheaf of $Z$ is not trivial.
More precisely, there exists an $i$ such that $C_i \subset Z$
and $\deg(\mathcal{C}_{Z/X}|_{C_i}) > 0$.
\end{lemma}

\begin{proof}
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}.
\end{proof}

\begin{lemma}
\label{lemma-H1-injective}
In Situation \ref{situation-vanishing} assume $X$ is normal
and $A$ Nagata. The map
$$H^1(X, \mathcal{O}_X) \longrightarrow H^1(f^{-1}(U), \mathcal{O}_X)$$
is injective.
\end{lemma}

\begin{proof}
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.
\end{proof}

\begin{lemma}
\label{lemma-R1-injective}
In Situation \ref{situation-vanishing} assume $X$ is normal and $A$ Nagata.
Then
$$\Hom_{D(A)}(\kappa[-1], Rf_*\mathcal{O}_X)$$
is zero. This uses $D(A) = D_\QCoh(\mathcal{O}_S)$ to think of
$Rf_*\mathcal{O}_X$ as an object of $D(A)$.
\end{lemma}

\begin{proof}
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.
\end{proof}

\begin{remark}
\label{remark-dualizing-setup}
Let $X$ be an integral Noetherian normal scheme of dimension $2$.
In this case the following are equivalent
\begin{enumerate}
\item $X$ has a dualizing complex $\omega_X^\bullet$,
\item there is a coherent $\mathcal{O}_X$-module $\omega_X$ such that
$\omega_X[n]$ is a dualizing complex, where $n$ can be any integer.
\end{enumerate}
This follows from the fact that $X$ is Cohen-Macaulay
(Properties, Lemma \ref{properties-lemma-normal-dimension-2-Cohen-Macaulay}) and
Duality for Schemes, Lemma \ref{duality-lemma-dualizing-module-CM-scheme}.
In this situation we will say that $\omega_X$ is a {\it dualizing module}
in accordance with
Duality for Schemes, Section \ref{duality-section-dualizing-module}.
In particular, when $A$ is a Noetherian normal local domain of dimension
$2$, then we say {\it $A$ has a dualizing module $\omega_A$}
if the above is true. In this case, if $X \to \Spec(A)$ is a normal
modification, then $X$ has a dualizing module too, see
Duality for Schemes, Example \ref{duality-example-proper-over-local}.
In this situation we always denote $\omega_X$ the dualizing
module normalized with respect to $\omega_A$, i.e., such that
$\omega_X[2]$ is the dualizing complex normalized relative to
$\omega_A[2]$. See Duality for Schemes, Section \ref{duality-section-glue}.
\end{remark}

\noindent
The Grauert-Riemenschneider vanishing of the next proposition is a formal
consequence of Lemma \ref{lemma-R1-injective} and the general theory of
duality.

\begin{proposition}[Grauert-Riemenschneider]
\label{proposition-Grauert-Riemenschneider}
In Situation \ref{situation-vanishing} assume
\begin{enumerate}
\item $X$ is a normal scheme,
\item $A$ is Nagata and has a dualizing complex $\omega_A^\bullet$.
\end{enumerate}
Let $\omega_X$ be the dualizing module of $X$
(Remark \ref{remark-dualizing-setup}). Then $R^1f_*\omega_X = 0$.
\end{proposition}

\begin{proof}
In this proof we will use the identification $D(A) = D_\QCoh(\mathcal{O}_S)$
to identify quasi-coherent $\mathcal{O}_S$-modules with $A$-modules.
Moreover, we may assume that $\omega_A^\bullet$ is normalized, see
Dualizing Complexes, Section \ref{dualizing-section-dualizing-local}.
Since $X$ is a Noetherian normal $2$-dimensional scheme
it is Cohen-Macaulay (Properties, Lemma
\ref{properties-lemma-normal-dimension-2-Cohen-Macaulay}).
Thus $\omega_X^\bullet = \omega_X[2]$ (Duality for Schemes, Lemma
\ref{duality-lemma-dualizing-module-CM-scheme} and the
normalization in Duality for Schemes, Example
\ref{duality-example-proper-over-local}).
If the proposition is false, then we can find a nonzero map
$R^1f_*\omega_X \to \kappa$. In other words we obtain a nonzero map
$\alpha : Rf_*\omega_X^\bullet \to \kappa[1]$.
Applying $R\Hom_A(-, \omega_A^\bullet)$ we get a nonzero map
$$\beta : \kappa[-1] \longrightarrow Rf_*\mathcal{O}_X$$
which is impossible by Lemma \ref{lemma-R1-injective}.
To see that $R\Hom_A(-, \omega_A^\bullet)$ does what we said, first
note that
$$R\Hom_A(\kappa[1], \omega_A^\bullet) = R\Hom_A(\kappa, \omega_A^\bullet)[-1] = \kappa[-1]$$
as $\omega_A^\bullet$ is normalized and we have
$$R\Hom_A(Rf_*\omega_X^\bullet, \omega_A^\bullet) = Rf_*R\SheafHom_{\mathcal{O}_X}(\omega_X^\bullet, \omega_X^\bullet) = Rf_*\mathcal{O}_X$$
The first equality by
Duality for Schemes, Lemma \ref{duality-lemma-iso-on-RSheafHom}
and the fact that $\omega_X^\bullet = f^!\omega_A^\bullet$
by construction, and the second equality because $\omega_X^\bullet$
is a dualizing complex for $X$ (which goes back to
Duality for Schemes, Lemma \ref{duality-lemma-shriek-dualizing}).
\end{proof}

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