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53.54. Vanishing of finite higher direct images

The next goal is to prove that the higher direct images of a finite morphism of schemes vanish.

Lemma 53.54.1. Let $R$ be a strictly henselian local ring. Set $S = \mathop{\mathrm{Spec}}(R)$ and let $\overline{s}$ be its closed point. Then the global sections functor $\Gamma(S, -) : \textit{Ab}(S_{\acute{e}tale}) \to \textit{Ab}$ is exact. In fact we have $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$ for any sheaf of sets $\mathcal{F}$. In particular $$ \forall p\geq 1, \quad H_{\acute{e}tale}^p(S, \mathcal{F})=0 $$ for all $\mathcal{F}\in \textit{Ab}(S_{\acute{e}tale})$.

Proof. If we show that $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$ the $\Gamma(S, -)$ is exact as the stalk functor is exact. Let $(U, \overline{u})$ be an étale neighbourhood of $\overline{s}$. Pick an affine open neighborhood $\mathop{\mathrm{Spec}}(A)$ of $\overline{u}$ in $U$. Then $R \to A$ is étale and $\kappa(\overline{s}) = \kappa(\overline{u})$. By Theorem 53.32.4 we see that $A \cong R \times A'$ as an $R$-algebra compatible with maps to $\kappa(\overline{s}) = \kappa(\overline{u})$. Hence we get a section $$ \xymatrix{ \mathop{\mathrm{Spec}}(A) \ar[r] & U \ar[d]\\ & S \ar[ul] } $$ It follows that in the system of étale neighbourhoods of $\overline{s}$ the identity map $(S, \overline{s}) \to (S, \overline{s})$ is cofinal. Hence $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$. The final statement of the lemma follows as the higher derived functors of an exact functor are zero, see Derived Categories, Lemma 13.17.9. $\square$

Proposition 53.54.2. Let $f : X \to Y$ be a finite morphism of schemes.

  1. For any geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ we have $$ (f_*\mathcal{F})_{\overline{y}} = \prod\nolimits_{\overline{x} : \mathop{\mathrm{Spec}}(k) \to X,~f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}}. $$ for $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits(X_{\acute{e}tale})$ and $$ (f_*\mathcal{F})_{\overline{y}} = \bigoplus\nolimits_{\overline{x} : \mathop{\mathrm{Spec}}(k) \to X,~f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}}. $$ for $\mathcal{F}$ in $\textit{Ab}(X_{\acute{e}tale})$.
  2. For any $q \geq 1$ we have $R^q f_*\mathcal{F} = 0$.

Proof. Let $X_{\overline{y}}^{sh}$ denote the fiber product $X \times_Y \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}^{sh})$. By Theorem 53.52.1 the stalk of $R^qf_*\mathcal{F}$ at $\overline{y}$ is computed by $H_{\acute{e}tale}^q(X_{\overline{y}}^{sh}, \mathcal{F})$. Since $f$ is finite, $X_{\bar y}^{sh}$ is finite over $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}^{sh})$, thus $X_{\bar y}^{sh} = \mathop{\mathrm{Spec}}(A)$ for some ring $A$ finite over $\mathcal{O}_{Y, \bar y}^{sh}$. Since the latter is strictly henselian, Lemma 53.32.5 implies that $A$ is a finite product of henselian local rings $A = A_1 \times \ldots \times A_r$. Since the residue field of $\mathcal{O}_{Y, \overline{y}}^{sh}$ is separably closed the same is true for each $A_i$. Hence $A_i$ is strictly henselian. This implies that $X_{\overline{y}}^{sh} = \coprod_{i = 1}^r \mathop{\mathrm{Spec}}(A_i)$. The vanishing of Lemma 53.54.1 implies that $(R^qf_*\mathcal{F})_{\overline{y}} = 0$ for $q > 0$ which implies (2) by Theorem 53.29.10. Part (1) follows from the corresponding statement of Lemma 53.54.1. $\square$

Lemma 53.54.3. Consider a cartesian square $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ of schemes with $f$ a finite morphism. For any sheaf of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $f'_*(g')^{-1}\mathcal{F} = g^{-1}f_*\mathcal{F}$.

Proof. In great generality there is a pullback map $g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$, see Sites, Section 7.44. To check this map is an isomorphism it suffices to check on stalks (Theorem 53.29.10). This is clear from the description of stalks in Proposition 53.54.2 and Lemma 53.36.2. $\square$

The following lemma is a case of cohomological descent dealing with étale sheaves and finite surjective morphisms. We will significantly generalize this result once we prove the proper base change theorem.

Lemma 53.54.4. Let $f : X \to Y$ be a surjective finite morphism of schemes. Set $f_n : X_n \to Y$ equal to the $(n + 1)$-fold fibre product of $X$ over $Y$. For $\mathcal{F} \in \textit{Ab}(Y_{\acute{e}tale})$ set $\mathcal{F}_n = f_{n, *}f_n^{-1}\mathcal{F}$. There is an exact sequence $$ 0 \to \mathcal{F} \to \mathcal{F}_0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \ldots $$ on $X_{\acute{e}tale}$. Moreover, there is a spectral sequence $$ E_1^{p, q} = H^q_{\acute{e}tale}(X_p, f_p^{-1}\mathcal{F}) $$ converging to $H^{p + q}(Y_{\acute{e}tale}, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$.

Proof. If we prove the first statement of the lemma, then we obtain a spectral sequence with $E_1^{p, q} = H^q_{\acute{e}tale}(Y, \mathcal{F})$ converging to $H^{p + q}(Y_{\acute{e}tale}, \mathcal{F})$, see Derived Categories, Lemma 13.21.3. On the other hand, since $R^if_{p, *}f_p^{-1}\mathcal{F} = 0$ for $i > 0$ (Proposition 53.54.2) we get $$ H^q_{\acute{e}tale}(X_p, f_p^{-1}\mathcal{F}) = H^q_{\acute{e}tale}(Y, f_{p, *}f_p^{-1} \mathcal{F}) = H^q_{\acute{e}tale}(Y, \mathcal{F}_p) $$ by Proposition 53.53.2 and we get the spectral sequence of the lemma.

To prove the first statement of the lemma, observe that $X_n$ forms a simplicial scheme over $Y$, see Simplicial, Example 14.3.5. Observe moreover, that for each of the projections $d_j : X_{n + 1} \to X_n$ there is a map $d_j^{-1} f_n^{-1}\mathcal{F} \to f_{n + 1}^{-1}\mathcal{F}$. These maps induce maps $$ \delta_j : \mathcal{F}_n \to \mathcal{F}_{n + 1} $$ for $j = 0, \ldots, n + 1$. We use the alternating sum of these maps to define the differentials $\mathcal{F}_n \to \mathcal{F}_{n + 1}$. Similarly, there is a canonical augmentation $\mathcal{F} \to \mathcal{F}_0$, namely this is just the canonical map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$. To check that this sequence of sheaves is an exact complex it suffices to check on stalks at geometric points (Theorem 53.29.10). Thus we let $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ be a geometric point. Let $E = \{\overline{x} : \mathop{\mathrm{Spec}}(k) \to X \mid f(\overline{x}) = \overline{y}\}$. Then $E$ is a finite nonempty set and we see that $$ (\mathcal{F}_n)_{\overline{y}} = \bigoplus\nolimits_{e \in E^{n + 1}} \mathcal{F}_{\overline{y}} $$ by Proposition 53.54.2 and Lemma 53.36.2. Thus we have to see that given an abelian group $M$ the sequence $$ 0 \to M \to \bigoplus\nolimits_{e \in E} M \to \bigoplus\nolimits_{e \in E^2} M \to \ldots $$ is exact. Here the first map is the diagonal map and the map $\bigoplus_{e \in E^{n + 1}} M \to \bigoplus_{e \in E^{n + 2}} M$ is the alternating sum of the maps induced by the $(n + 2)$ projections $E^{n + 2} \to E^{n + 1}$. This can be shown directly or deduced by applying Simplicial, Lemma 14.26.9 to the map $E \to \{*\}$. $\square$

Remark 53.54.5. In the situation of Lemma 53.54.4 if $\mathcal{G}$ is a sheaf of sets on $Y_{\acute{e}tale}$, then we have $$ \Gamma(Y, \mathcal{G}) = \text{Equalizer}( \xymatrix{ \Gamma(X_0, f_0^{-1}\mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma(X_1, f_1^{-1}\mathcal{G}) } ) $$ This is proved in exactly the same way, by showing that the sheaf $\mathcal{G}$ is the equalizer of the two maps $f_{0, *}f_0^{-1}\mathcal{G} \to f_{1, *}f_1^{-1}\mathcal{G}$.

Here is a fun generalization of Lemma 53.54.1.

Lemma 53.54.6. Let $S$ be a scheme all of whose local rings are strictly henselian. Then for any abelian sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ we have $H^i(S_{\acute{e}tale}, \mathcal{F}) = H^i(S_{Zar}, \mathcal{F})$.

Proof. Let $\epsilon : S_{\acute{e}tale} \to S_{Zar}$ be the morphism of sites given by the inclusion functor. The Zariski sheaf $R^p\epsilon_*\mathcal{F}$ is the sheaf associated to the presheaf $U \mapsto H^p_{\acute{e}tale}(U, \mathcal{F})$. Thus the stalk at $x \in X$ is $\mathop{\mathrm{colim}}\nolimits H^p_{\acute{e}tale}(U, \mathcal{F}) = H^p_{\acute{e}tale}(\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}), \mathcal{G}_x)$ where $\mathcal{G}_x$ denotes the pullback of $\mathcal{F}$ to $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$, see Lemma 53.51.3. Thus the higher direct images of $R^p\epsilon_*\mathcal{F}$ are zero by Lemma 53.54.1 and we conclude by the Leray spectral sequence. $\square$

Lemma 53.54.7. Let $S$ be an affine scheme such that (1) all points are closed, and (2) all residue fields are separably algebraically closed. Then for any abelian sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ we have $H^i(S_{\acute{e}tale}, \mathcal{F}) = 0$ for $i > 0$.

Proof. Condition (1) implies that the underlying topological space of $S$ is profinite, see Algebra, Lemma 10.25.5. Thus the higher cohomology groups of an abelian sheaf on the topological space $S$ (i.e., Zariski cohomology) is trivial, see Cohomology, Lemma 20.23.3. The local rings are strictly henselian by Algebra, Lemma 10.148.10. Thus étale cohomology of $S$ is computed by Zariski cohomology by Lemma 53.54.6 and the proof is done. $\square$

    The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 7472–7750 (see updates for more information).

    \section{Vanishing of finite higher direct images}
    \label{section-vanishing-finite-morphism}
    
    \noindent
    The next goal is to prove that the higher direct images of a finite morphism of
    schemes vanish.
    
    \begin{lemma}
    \label{lemma-vanishing-etale-cohomology-strictly-henselian}
    Let $R$ be a strictly henselian local ring. Set $S = \Spec(R)$ and let
    $\overline{s}$ be its closed point. Then the global
    sections functor
    $\Gamma(S, -) : \textit{Ab}(S_\etale) \to \textit{Ab}$
    is exact. In fact we have $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$
    for any sheaf of sets $\mathcal{F}$. In particular
    $$
    \forall p\geq 1, \quad H_\etale^p(S, \mathcal{F})=0
    $$
    for all $\mathcal{F}\in \textit{Ab}(S_\etale)$.
    \end{lemma}
    
    \begin{proof}
    If we show that $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$
    the $\Gamma(S, -)$ is exact as the stalk functor is exact.
    Let $(U, \overline{u})$ be an \'etale neighbourhood of $\overline{s}$.
    Pick an affine open neighborhood $\Spec(A)$ of $\overline{u}$ in $U$.
    Then $R \to A$ is \'etale and $\kappa(\overline{s}) = \kappa(\overline{u})$.
    By Theorem \ref{theorem-henselian} we see that $A \cong R \times A'$
    as an $R$-algebra compatible with maps to
    $\kappa(\overline{s}) = \kappa(\overline{u})$.
    Hence we get a section
    $$
    \xymatrix{
    \Spec(A) \ar[r] & U \ar[d]\\
    & S \ar[ul]
    }
    $$
    It follows that in the system of \'etale neighbourhoods of $\overline{s}$
    the identity map $(S, \overline{s}) \to (S, \overline{s})$ is cofinal.
    Hence $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$.
    The final statement of the lemma follows as the higher derived
    functors of an exact functor are zero, see
    Derived Categories, Lemma \ref{derived-lemma-right-derived-exact-functor}.
    \end{proof}
    
    \begin{proposition}
    \label{proposition-finite-higher-direct-image-zero}
    Let $f : X \to Y$ be a finite morphism of schemes.
    \begin{enumerate}
    \item For any geometric point $\overline{y} : \Spec(k) \to Y$ we have
    $$
    (f_*\mathcal{F})_{\overline{y}} =
    \prod\nolimits_{\overline{x} : \Spec(k) \to X,\ f(\overline{x}) =
    \overline{y}} \mathcal{F}_{\overline{x}}.
    $$
    for $\mathcal{F}$ in $\Sh(X_\etale)$ and
    $$
    (f_*\mathcal{F})_{\overline{y}} =
    \bigoplus\nolimits_{\overline{x} : \Spec(k) \to X,\ f(\overline{x}) =
    \overline{y}} \mathcal{F}_{\overline{x}}.
    $$
    for $\mathcal{F}$ in $\textit{Ab}(X_\etale)$.
    \item For any $q \geq 1$ we have $R^q f_*\mathcal{F} = 0$.
    \end{enumerate}
    \end{proposition}
    
    \begin{proof}
    Let $X_{\overline{y}}^{sh}$ denote the fiber product
    $X \times_Y \Spec(\mathcal{O}_{Y, \overline{y}}^{sh})$.
    By Theorem \ref{theorem-higher-direct-images}
    the stalk of $R^qf_*\mathcal{F}$ at $\overline{y}$ is computed by
    $H_\etale^q(X_{\overline{y}}^{sh}, \mathcal{F})$.
    Since $f$ is finite, $X_{\bar y}^{sh}$ is finite over
    $\Spec(\mathcal{O}_{Y, \overline{y}}^{sh})$, thus
    $X_{\bar y}^{sh} = \Spec(A)$ for some ring $A$
    finite over $\mathcal{O}_{Y, \bar y}^{sh}$.
    Since the latter is strictly henselian,
    Lemma \ref{lemma-finite-over-henselian}
    implies that $A$ is a finite product of henselian local rings
    $A = A_1 \times \ldots \times A_r$. Since the residue field of
    $\mathcal{O}_{Y, \overline{y}}^{sh}$ is separably closed the
    same is true for each $A_i$. Hence $A_i$ is strictly henselian.
    This implies that $X_{\overline{y}}^{sh} = \coprod_{i = 1}^r \Spec(A_i)$.
    The vanishing of
    Lemma \ref{lemma-vanishing-etale-cohomology-strictly-henselian}
    implies that $(R^qf_*\mathcal{F})_{\overline{y}} = 0$ for $q > 0$
    which implies (2) by Theorem \ref{theorem-exactness-stalks}.
    Part (1) follows from the corresponding statement of
    Lemma \ref{lemma-vanishing-etale-cohomology-strictly-henselian}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-finite-pushforward-commutes-with-base-change}
    Consider a cartesian square
    $$
    \xymatrix{
    X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\
    Y' \ar[r]^g & Y
    }
    $$
    of schemes with $f$ a finite morphism. For any sheaf of sets
    $\mathcal{F}$ on $X_\etale$ we have
    $f'_*(g')^{-1}\mathcal{F} = g^{-1}f_*\mathcal{F}$.
    \end{lemma}
    
    \begin{proof}
    In great generality there is a pullback map
    $g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$, see
    Sites, Section \ref{sites-section-pullback}.
    To check this map is an isomorphism it suffices to check
    on stalks (Theorem \ref{theorem-exactness-stalks}).
    This is clear from the description of stalks
    in Proposition \ref{proposition-finite-higher-direct-image-zero} and
    Lemma \ref{lemma-stalk-pullback}.
    \end{proof}
    
    \noindent
    The following lemma is a case of cohomological descent dealing with
    \'etale sheaves and finite surjective morphisms. We will significantly
    generalize this result once we prove the proper base change theorem.
    
    \begin{lemma}
    \label{lemma-cohomological-descent-finite}
    Let $f : X \to Y$ be a surjective finite morphism of schemes.
    Set $f_n : X_n \to Y$ equal to the $(n + 1)$-fold fibre product
    of $X$ over $Y$. For $\mathcal{F} \in \textit{Ab}(Y_\etale)$ set
    $\mathcal{F}_n = f_{n, *}f_n^{-1}\mathcal{F}$. There is an exact
    sequence
    $$
    0 \to \mathcal{F} \to \mathcal{F}_0 \to \mathcal{F}_1 \to
    \mathcal{F}_2 \to \ldots
    $$
    on $X_\etale$. Moreover, there is a spectral sequence
    $$
    E_1^{p, q} = H^q_\etale(X_p, f_p^{-1}\mathcal{F})
    $$
    converging to $H^{p + q}(Y_\etale, \mathcal{F})$.
    This spectral sequence is functorial in $\mathcal{F}$.
    \end{lemma}
    
    \begin{proof}
    If we prove the first statement of the lemma, then we obtain a spectral
    sequence with $E_1^{p, q} = H^q_\etale(Y, \mathcal{F})$ converging
    to $H^{p + q}(Y_\etale, \mathcal{F})$, see
    Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}.
    On the other hand, since
    $R^if_{p, *}f_p^{-1}\mathcal{F} = 0$ for $i > 0$
    (Proposition \ref{proposition-finite-higher-direct-image-zero})
    we get
    $$
    H^q_\etale(X_p, f_p^{-1}\mathcal{F}) =
    H^q_\etale(Y, f_{p, *}f_p^{-1} \mathcal{F}) =
    H^q_\etale(Y, \mathcal{F}_p)
    $$
    by Proposition \ref{proposition-leray}
    and we get the spectral sequence of the lemma.
    
    \medskip\noindent
    To prove the first statement of the lemma, observe that
    $X_n$ forms a simplicial scheme over $Y$, see
    Simplicial, Example \ref{simplicial-example-fibre-products-simplicial-object}.
    Observe moreover, that for each of the projections
    $d_j : X_{n + 1} \to X_n$ there is a map
    $d_j^{-1} f_n^{-1}\mathcal{F} \to f_{n + 1}^{-1}\mathcal{F}$.
    These maps induce maps
    $$
    \delta_j : \mathcal{F}_n \to \mathcal{F}_{n + 1}
    $$
    for $j = 0, \ldots, n + 1$. We use the alternating sum of these maps
    to define the differentials $\mathcal{F}_n \to \mathcal{F}_{n + 1}$.
    Similarly, there is a canonical augmentation $\mathcal{F} \to \mathcal{F}_0$,
    namely this is just the canonical map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$.
    To check that this sequence of sheaves is an exact complex it suffices
    to check on stalks at geometric points (Theorem \ref{theorem-exactness-stalks}).
    Thus we let $\overline{y} : \Spec(k) \to Y$ be a geometric point. Let
    $E = \{\overline{x} : \Spec(k) \to X \mid f(\overline{x}) = \overline{y}\}$.
    Then $E$ is a finite nonempty set and we see that
    $$
    (\mathcal{F}_n)_{\overline{y}} =
    \bigoplus\nolimits_{e \in E^{n + 1}} \mathcal{F}_{\overline{y}}
    $$
    by Proposition \ref{proposition-finite-higher-direct-image-zero}
    and Lemma \ref{lemma-stalk-pullback}.
    Thus we have to see that given an abelian group $M$ the sequence
    $$
    0 \to M \to \bigoplus\nolimits_{e \in E} M \to
    \bigoplus\nolimits_{e \in E^2} M \to \ldots
    $$
    is exact. Here the first map is the diagonal map and the map
    $\bigoplus_{e \in E^{n + 1}} M  \to \bigoplus_{e \in E^{n + 2}} M$
    is the alternating sum of the maps induced by the $(n + 2)$
    projections $E^{n + 2} \to E^{n + 1}$. This can be shown directly
    or deduced by applying Simplicial, Lemma
    \ref{simplicial-lemma-fibre-products-simplicial-object-w-section}
    to the map $E \to \{*\}$.
    \end{proof}
    
    \begin{remark}
    \label{remark-cohomological-descent-finite}
    In the situation of Lemma \ref{lemma-cohomological-descent-finite}
    if $\mathcal{G}$ is a sheaf of sets on $Y_\etale$, then we have
    $$
    \Gamma(Y, \mathcal{G}) =
    \text{Equalizer}(
    \xymatrix{
    \Gamma(X_0, f_0^{-1}\mathcal{G})
    \ar@<1ex>[r] \ar@<-1ex>[r] &
    \Gamma(X_1, f_1^{-1}\mathcal{G})
    }
    )
    $$
    This is proved in exactly the same way, by showing that
    the sheaf $\mathcal{G}$ is the equalizer of the two maps
    $f_{0, *}f_0^{-1}\mathcal{G} \to f_{1, *}f_1^{-1}\mathcal{G}$.
    \end{remark}
    
    
    
    
    
    \noindent
    Here is a fun generalization of
    Lemma \ref{lemma-vanishing-etale-cohomology-strictly-henselian}.
    
    \begin{lemma}
    \label{lemma-local-rings-strictly-henselian}
    Let $S$ be a scheme all of whose local rings are strictly henselian.
    Then for any abelian sheaf $\mathcal{F}$ on $S_\etale$ we have
    $H^i(S_\etale, \mathcal{F}) = H^i(S_{Zar}, \mathcal{F})$.
    \end{lemma}
    
    \begin{proof}
    Let $\epsilon : S_\etale \to S_{Zar}$ be the morphism of sites given
    by the inclusion functor. The Zariski sheaf $R^p\epsilon_*\mathcal{F}$
    is the sheaf associated to the presheaf $U \mapsto H^p_\etale(U, \mathcal{F})$.
    Thus the stalk at $x \in X$ is
    $\colim H^p_\etale(U, \mathcal{F}) =
    H^p_\etale(\Spec(\mathcal{O}_{X, x}), \mathcal{G}_x)$
    where $\mathcal{G}_x$ denotes the pullback of $\mathcal{F}$
    to $\Spec(\mathcal{O}_{X, x})$, see
    Lemma \ref{lemma-directed-colimit-cohomology}.
    Thus the higher direct images of $R^p\epsilon_*\mathcal{F}$ are
    zero by
    Lemma \ref{lemma-vanishing-etale-cohomology-strictly-henselian}
    and we conclude by the Leray spectral sequence.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-affine-only-closed-points}
    Let $S$ be an affine scheme such that
    (1) all points are closed, and (2) all residue fields are separably
    algebraically closed. Then
    for any abelian sheaf $\mathcal{F}$ on $S_\etale$ we have
    $H^i(S_\etale, \mathcal{F}) = 0$ for $i > 0$.
    \end{lemma}
    
    \begin{proof}
    Condition (1) implies that the underlying topological space
    of $S$ is profinite, see
    Algebra, Lemma \ref{algebra-lemma-ring-with-only-minimal-primes}.
    Thus the higher cohomology groups of an abelian sheaf on the topological
    space $S$ (i.e., Zariski cohomology) is trivial, see
    Cohomology, Lemma \ref{cohomology-lemma-vanishing-for-profinite}.
    The local rings are strictly henselian by
    Algebra, Lemma \ref{algebra-lemma-local-dimension-zero-henselian}.
    Thus \'etale cohomology of $S$ is computed by Zariski cohomology
    by Lemma \ref{lemma-local-rings-strictly-henselian}
    and the proof is done.
    \end{proof}
    
    
    
    
    
    
    
    
    
    %10.06.09

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