The Stacks project

82.4 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.

Lemma 82.4.1. 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 $g : Y \times _ X Y \to X$.

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 64.19.9 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 67.18.1. 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 58.96.1 (in fact it is in some sense equivalent to this lemma). $\square$

Lemma 82.4.2. 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 $\Gamma (X, \mathcal{F}) = \Gamma (Z, i^{-1}\mathcal{F})$.

Proof. Choose a surjective proper morphism $Y \to X$ where $Y$ is a scheme, see Cohomology of Spaces, Lemma 67.18.1. Consider the diagram

\[ \xymatrix{ \Gamma (X_0, \mathcal{F}_0) \ar[r] \ar[d] & \Gamma (Y_0, \mathcal{G}_0) \ar@<1ex>[r] \ar@<-1ex>[r] \ar[d] & \Gamma ((Y \times _ X Y)_0, \mathcal{H}_0) \ar[d] \\ \Gamma (X, \mathcal{F}) \ar[r] & \Gamma (Y, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma (Y \times _ X Y, \mathcal{H}) } \]

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 58.87.2) we see that the middle and right vertical arrows are bijective. By Lemma 82.4.1 it follows that the left one is too. $\square$

Lemma 82.4.3. Let $A$ be a henselian local ring. Let $X$ be an algebraic space over $A$ such that $f : X \to \mathop{\mathrm{Spec}}(A)$ be 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 82.4.2. $\square$

Lemma 82.4.4. 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 67.18.1. 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 58.87.5) 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 82.4.1 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 82.4.5. 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

\[ (f_*\mathcal{F})_{\overline{x}} \longrightarrow \Gamma (Y_{\overline{x}}, \mathcal{F}_{\overline{x}}) \]

is bijective.

Proof. This is a special case of Lemma 82.4.4. $\square$

Theorem 82.4.6. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

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

\[ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} \]

is an isomorphism.

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 58.87 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 67.5.2 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 82.4.4. 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 67.18.1. 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 82.4.1). 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 82.4.4. This finishes the proof. $\square$

Lemma 82.4.7. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

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.

Proof. This is a simple consequence of the proper base change theorem (Theorem 82.4.6) 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 82.4.8. 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.

  1. For a torsion abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $(R^ nf_*\mathcal{F})_{\overline{y}} = H^ n_{\acute{e}tale}(X_{\overline{y}}, \mathcal{F}_{\overline{y}})$.

  2. 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 82.4.6. The second one is a special case of Lemma 82.4.7. $\square$

Lemma 82.4.9. Let $k \subset k'$ 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 82.4.6. $\square$


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