The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

54.54 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 54.54.1. 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 54.32.4 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 54.54.2. 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 54.52.1 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 54.32.5 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 54.54.1 implies that $(R^ qf_*\mathcal{F})_{\overline{y}} = 0$ for $q > 0$ which implies (2) by Theorem 54.29.10. Part (1) follows from the corresponding statement of Lemma 54.54.1. $\square$

Lemma 54.54.3. 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 54.29.10). This is clear from the description of stalks in Proposition 54.54.2 and Lemma 54.36.2. $\square$

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 54.54.4. 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 54.54.2) 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 54.53.2 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 54.29.10). 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 54.54.2 and Lemma 54.36.2. 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 54.54.5. In the situation of Lemma 54.54.4 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 54.54.1.

Lemma 54.54.6. 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 54.51.3. Thus the higher direct images of $R^ p\epsilon _*\mathcal{F}$ are zero by Lemma 54.54.1 and we conclude by the Leray spectral sequence. $\square$

Lemma 54.54.7. 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 54.54.6 and the proof is done. $\square$

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