The Stacks project

Theorem 59.91.11. Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times _ Y X$ and consider the cartesian diagram

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

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. In the terminology introduced above, this means that cohomology commutes with base change for every proper morphism of schemes. By Lemma 59.91.10 it suffices to prove that cohomology commutes with base change for the morphism $\mathbf{P}^1_ S \to S$ for every scheme $S$.

Let $S$ be the spectrum of a strictly henselian local ring with closed point $s$. Set $X = \mathbf{P}^1_ S$ and $X_0 = X_ s = \mathbf{P}^1_ s$. Let $\mathcal{F}$ be a sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $X_{\acute{e}tale}$. The key to our proof is that

\[ H^ q_{\acute{e}tale}(X, \mathcal{F}) = H^ q_{\acute{e}tale}(X_0, \mathcal{F}|_{X_0}). \]

Namely, choose a resolution $\mathcal{F} \to \mathcal{I}^\bullet $ by injective sheaves of $\mathbf{Z}/\ell \mathbf{Z}$-modules. Then $\mathcal{I}^\bullet |_{X_0}$ is a resolution of $\mathcal{F}|_{X_0}$ by right $H^0_{\acute{e}tale}(X_0, -)$-acyclic objects, see Lemma 59.85.2. Leray's acyclicity lemma tells us the right hand side is computed by the complex $H^0_{\acute{e}tale}(X_0, \mathcal{I}^\bullet |_{X_0})$ which is equal to $H^0_{\acute{e}tale}(X, \mathcal{I}^\bullet )$ by Lemma 59.91.3. This complex computes the left hand side.

Assume $S$ is general and $\mathcal{F}$ is a sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $X_{\acute{e}tale}$. Let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point of $S$ lying over $s \in S$. We have

\[ (R^ qf_*\mathcal{F})_{\overline{s}} = H^ q_{\acute{e}tale}(\mathbf{P}^1_{\mathcal{O}_{S, \overline{s}}^{sh}}, \mathcal{F}|_{\mathbf{P}^1_{\mathcal{O}_{S, \overline{s}}^{sh}}}) = H^ q_{\acute{e}tale}(\mathbf{P}^1_{\kappa (s)^{sep}}, \mathcal{F}|_{\mathbf{P}^1_{\kappa (s)^{sep}}}) \]

where $\kappa (s)^{sep}$ is the residue field of $\mathcal{O}_{S, \overline{s}}^{sh}$, i.e., the separable algebraic closure of $\kappa (s)$ in $k$. The first equality by Theorem 59.53.1 and the second equality by the displayed formula in the previous paragraph.

Finally, consider any morphism of schemes $g : T \to S$ where $S$ and $\mathcal{F}$ are as above. Set $f' : \mathbf{P}^1_ T \to T$ the projection and let $g' : \mathbf{P}^1_ T \to \mathbf{P}^1_ S$ the morphism induced by $g$. Consider the base change map

\[ g^{-1}R^ qf_*\mathcal{F} \longrightarrow R^ qf'_*(g')^{-1}\mathcal{F} \]

Let $\overline{t}$ be a geometric point of $T$ with image $\overline{s} = g(\overline{t})$. By our discussion above the map on stalks at $\overline{t}$ is the map

\[ H^ q_{\acute{e}tale}(\mathbf{P}^1_{\kappa (s)^{sep}}, \mathcal{F}|_{\mathbf{P}^1_{\kappa (s)^{sep}}}) \longrightarrow H^ q_{\acute{e}tale}(\mathbf{P}^1_{\kappa (t)^{sep}}, \mathcal{F}|_{\mathbf{P}^1_{\kappa (t)^{sep}}}) \]

Since $\kappa (s)^{sep} \subset \kappa (t)^{sep}$ this map is an isomorphism by Lemma 59.83.12.

This proves cohomology commutes with base change for $\mathbf{P}^1_ S \to S$ and sheaves of $\mathbf{Z}/\ell \mathbf{Z}$-modules. In particular, for an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules the higher direct images of any base change are zero. In other words, condition (2) of Lemma 59.91.6 holds and the proof is complete. $\square$

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