Theorem 83.4.6. 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.

Theorem 83.4.6. 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 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 68.5.2 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 83.4.4. 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 68.18.1. 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 83.4.1). 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 83.4.4. This finishes the proof. $\square$

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)