Lemma 84.4.1. Let $S$ be a scheme. Let $f : Y \to X$ be a surjective proper morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. Then $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ is injective with image the equalizer of the two maps $f_*f^{-1}\mathcal{F} \to g_*g^{-1}\mathcal{F}$ where $g$ is the structure morphism $g : Y \times _ X Y \to X$.
84.4 Proper base change
The proper base change theorem for algebraic spaces follows from the proper base change theorem for schemes and Chow's lemma with a little bit of work.
Proof. For any surjective morphism $f : Y \to X$ of algebraic spaces over $S$, the map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ is injective. Namely, if $\overline{x}$ is a geometric point of $X$, then we choose a geometric point $\overline{y}$ of $Y$ lying over $\overline{x}$ and we consider
\[ \mathcal{F}_{\overline{x}} \to (f_*f^{-1}\mathcal{F})_{\overline{x}} \to (f^{-1}\mathcal{F})_{\overline{y}} = \mathcal{F}_{\overline{x}} \]
See Properties of Spaces, Lemma 66.19.9 for the last equality.
The second statement is local on $X$ in the étale topology, hence we may and do assume $Y$ is an affine scheme.
Choose a surjective proper morphism $Z \to Y$ where $Z$ is a scheme, see Cohomology of Spaces, Lemma 69.18.1. The result for $Z \to X$ implies the result for $Y \to X$. Since $Z \to X$ is a surjective proper morphism of schemes and hence a ph covering (Topologies, Lemma 34.8.6) the result for $Z \to X$ follows from Étale Cohomology, Lemma 59.102.1 (in fact it is in some sense equivalent to this lemma). $\square$
Lemma 84.4.2. Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space over $A$ such that the structure morphism $f : X \to \mathop{\mathrm{Spec}}(A)$ is proper. Let $i : X_0 \to X$ be the inclusion of $X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I)$. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (X_0, i^{-1}\mathcal{F})$.
Proof. Choose a surjective proper morphism $Y \to X$ where $Y$ is a scheme, see Cohomology of Spaces, Lemma 69.18.1. Consider the diagram
\[ \xymatrix{ \Gamma (X_0, \mathcal{F}_0) \ar[r] & \Gamma (Y_0, \mathcal{G}_0) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma ((Y \times _ X Y)_0, \mathcal{H}_0) \\ \Gamma (X, \mathcal{F}) \ar[r] \ar[u] & \Gamma (Y, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] \ar[u] & \Gamma (Y \times _ X Y, \mathcal{H}) \ar[u] } \]
Here $\mathcal{G}$, resp. $\mathcal{H}$ is the pullbackf or $\mathcal{F}$ to $Y$, resp. $Y \times _ X Y$ and the index $0$ indicates base change to $\mathop{\mathrm{Spec}}(A/I)$. By the case of schemes (Étale Cohomology, Lemma 59.91.2) we see that the middle and right vertical arrows are bijective. By Lemma 84.4.1 it follows that the left one is too. $\square$
Lemma 84.4.3. Let $A$ be a henselian local ring. Let $X$ be an algebraic space over $A$ such that $f : X \to \mathop{\mathrm{Spec}}(A)$ is a proper morphism. Let $X_0 \subset X$ be the fibre of $f$ over the closed point. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (X_0, \mathcal{F}|_{X_0})$.
Proof. This is a special case of Lemma 84.4.2. $\square$
Lemma 84.4.4. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$ be a morphisms of algebraic spaces over $S$. Assume $f$ is proper. Set $X' = Y' \times _ Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$. Let $\mathcal{F}$ be any sheaf on $X_{\acute{e}tale}$. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.
Proof. The question is étale local on $Y'$. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $V'$ and a surjective étale morphism $V' \to V \times _ Y Y'$. Then we may replace $Y'$ by $V'$ and $Y$ by $V$. Hence we may assume $Y$ and $Y'$ are schemes. Then we may work Zariski locally on $Y$ and $Y'$ and hence we may assume $Y$ and $Y'$ are affine schemes.
Assume $Y$ and $Y'$ are affine schemes. Choose a surjective proper morphism $h_1 : X_1 \to X$ where $X_1$ is a scheme, see Cohomology of Spaces, Lemma 69.18.1. Set $X_2 = X_1 \times _ X X_1$ and denote $h_2 : X_2 \to X$ the structure morphism. Observe this is a scheme. By the case of schemes (Étale Cohomology, Lemma 59.91.5) we know the lemma is true for the cartesian diagrams
\[ \vcenter { \xymatrix{ X'_1 \ar[r] \ar[d] & X_1 \ar[d] \\ Y' \ar[r] & Y } } \quad \text{and}\quad \vcenter { \xymatrix{ X'_2 \ar[r] \ar[d] & X_2 \ar[d] \\ Y' \ar[r] & Y } } \]
and the sheaves $\mathcal{F}_ i = (X_ i \to X)^{-1}\mathcal{F}$. By Lemma 84.4.1 we have an exact sequence $0 \to \mathcal{F} \to h_{1, *}\mathcal{F}_1 \to h_{2, *}\mathcal{F}_2$ and similarly for $(g')^{-1}\mathcal{F}$ because $X'_2 = X'_1 \times _{X'} X'_1$. Hence we conlude that the lemma is true (some details omitted). $\square$
Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $\overline{x} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point. The fibre of $f$ at $\overline{x}$ is the algebraic space $Y_{\overline{x}} = \mathop{\mathrm{Spec}}(k) \times _{\overline{x}, X} Y$ over $\mathop{\mathrm{Spec}}(k)$. If $\mathcal{F}$ is a sheaf on $Y_{\acute{e}tale}$, then denote $\mathcal{F}_{\overline{x}} = p^{-1}\mathcal{F}$ the pullback of $\mathcal{F}$ to $(Y_{\overline{x}})_{\acute{e}tale}$. Here $p : Y_{\overline{x}} \to Y$ is the projection. In the following we will consider the set $\Gamma (Y_{\overline{x}}, \mathcal{F}_{\overline{x}})$.
Lemma 84.4.5. Let $S$ be a scheme. Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Let $\overline{x} \to X$ be a geometric point. For any sheaf $\mathcal{F}$ on $Y_{\acute{e}tale}$ the canonical map is bijective.
\[ (f_*\mathcal{F})_{\overline{x}} \longrightarrow \Gamma (Y_{\overline{x}}, \mathcal{F}_{\overline{x}}) \]
Proof. This is a special case of Lemma 84.4.4. $\square$
Theorem 84.4.6. Let $S$ be a scheme. Let be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$. Then the base change map is an isomorphism.
\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]
\[ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} \]
Proof. This proof repeats a few of the arguments given in the proof of the proper base change theorem for schemes. See Étale Cohomology, Section 59.91 for more details.
The statement is étale local on $Y'$ and $Y$, hence we may assume both $Y$ and $Y'$ are affine schemes. Observe that this in particular proves the theorem in case $f$ is representable (we will use this below).
For every $n \geq 1$ let $\mathcal{F}[n]$ be the subsheaf of sections of $\mathcal{F}$ annihilated by $n$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}[n]$. By Cohomology of Spaces, Lemma 69.5.2 the functors $g^{-1}R^ pf_*$ and $R^ pf'_*(g')^{-1}$ commute with filtered colimits. Hence it suffices to prove the theorem if $\mathcal{F}$ is killed by $n$.
Let $\mathcal{F} \to \mathcal{I}^\bullet $ be a resolution by injective sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. Observe that $g^{-1}f_*\mathcal{I}^\bullet = f'_*(g')^{-1}\mathcal{I}^\bullet $ by Lemma 84.4.4. Applying Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) we conclude it suffices to prove $R^ pf'_*(g')^{-1}\mathcal{I}^ m = 0$ for $p > 0$ and $m \in \mathbf{Z}$.
Choose a surjective proper morphism $h : Z \to X$ where $Z$ is a scheme, see Cohomology of Spaces, Lemma 69.18.1. Choose an injective map $h^{-1}\mathcal{I}^ m \to \mathcal{J}$ where $\mathcal{J}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules on $Z_{\acute{e}tale}$. Since $h$ is surjective the map $\mathcal{I}^ m \to h_*\mathcal{J}$ is injective (see Lemma 84.4.1). Since $\mathcal{I}^ m$ is injective we see that $\mathcal{I}^ m$ is a direct summand of $h_*\mathcal{J}$. Thus it suffices to prove the desired vanishing for $h_*\mathcal{J}$.
Denote $h'$ the base change by $g$ and denote $g'' : Z' \to Z$ the projection. There is a spectral sequence
\[ E_2^{p, q} = R^ pf'_* R^ qh'_* (g'')^{-1}\mathcal{J} \]
converging to $R^{p + q}(f' \circ h')_*(g'')^{-1}\mathcal{J}$. Since $h$ and $f \circ h$ are representable (by schemes) we know the result we want holds for them. Thus in the spectral sequence we see that $E_2^{p, q} = 0$ for $q > 0$ and $R^{p + q}(f' \circ h')_*(g'')^{-1}\mathcal{J} = 0$ for $p + q > 0$. It follows that $E_2^{p, 0} = 0$ for $p > 0$. Now
\[ E_2^{p, 0} = R^ pf'_* h'_* (g'')^{-1}\mathcal{J} = R^ pf'_* (g')^{-1}h_*\mathcal{J} \]
by Lemma 84.4.4. This finishes the proof. $\square$
Lemma 84.4.7. Let $S$ be a scheme. Let be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $E \in D^+(X_{\acute{e}tale})$ have torsion cohomology sheaves. Then the base change map $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism.
\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]
Proof. This is a simple consequence of the proper base change theorem (Theorem 84.4.6) using the spectral sequences
\[ E_2^{p, q} = R^ pf_*H^ q(E) \quad \text{and}\quad {E'}_2^{p, q} = R^ pf'_*(g')^{-1}H^ q(E) \]
converging to $R^ nf_*E$ and $R^ nf'_*(g')^{-1}E$. The spectral sequences are constructed in Derived Categories, Lemma 13.21.3. Some details omitted. $\square$
Lemma 84.4.8. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces. Let $\overline{y} \to Y$ be a geometric point.
For a torsion abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $(R^ nf_*\mathcal{F})_{\overline{y}} = H^ n_{\acute{e}tale}(X_{\overline{y}}, \mathcal{F}_{\overline{y}})$.
For $E \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves we have $(R^ nf_*E)_{\overline{y}} = H^ n_{\acute{e}tale}(X_{\overline{y}}, E_{\overline{y}})$.
Proof. In the statement, $\mathcal{F}_{\overline{y}}$ denotes the pullback of $\mathcal{F}$ to $X_{\overline{y}} = \overline{y} \times _ Y X$. Since pulling back by $\overline{y} \to Y$ produces the stalk of $\mathcal{F}$, the first statement of the lemma is a special case of Theorem 84.4.6. The second one is a special case of Lemma 84.4.7. $\square$
Lemma 84.4.9. Let $k'/k$ be an extension of separably closed fields. Let $X$ be a proper algebraic space over $k$. Let $\mathcal{F}$ be a torsion abelian sheaf on $X$. Then the map $H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for $q \geq 0$.
Proof. This is a special case of Theorem 84.4.6. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)