The Stacks project

Theorem 84.4.6. 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 √Čtale 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. $\square$

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