The Stacks project

Lemma 62.8.6. Consider a commutative diagram of schemes

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

with $f$ and $g$ locally quasi-finite and $h$ proper. Let $\Lambda $ be a ring. Funtorially in $K \in D(X_{\acute{e}tale}, \Lambda )$ there is a canonical map

\[ g_!K \longrightarrow Rh_*(f_!K) \]

in $D(Z_{\acute{e}tale}, \Lambda )$. This map is an isomorphism if (a) $K$ is bounded below and has torsion cohomology sheaves, or (b) $\Lambda $ is a torsion ring.

Proof. This is a special case of Lemma 62.8.1 if $f$ and $g$ are separated. We urge the reader to skip the proof in the general case as we'll mainly use the case where $f$ and $g$ are separated.

Represent $K$ by a complex $\mathcal{K}^\bullet $ of sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$. Choose a quasi-isomorphism $f_!\mathcal{K}^\bullet \to \mathcal{I}^\bullet $ into a K-injective complex $\mathcal{I}^\bullet $ of sheaves of $\Lambda $-modules on $Y_{\acute{e}tale}$. Consider the map

\[ g_!\mathcal{K}^\bullet = h_!f_!\mathcal{K}^\bullet = h_*f_!\mathcal{K}^\bullet \longrightarrow h_*\mathcal{I}^\bullet \]

where the equalities are Lemmas 62.4.11 and 62.3.4. This map of complexes determines the map $g_!K \to Rh_*(f_!K)$ of the statement of the lemma.

Assume $\Lambda $ is torsion, i.e., we are in case (b). To check the map is an isomorphism we may work locally on $Z$. Hence we may assume that the dimension of fibres of $h$ is bounded, see Morphisms, Lemma 29.28.5. Then we see that $Rh_*$ has finite cohomological dimension, see Étale Cohomology, Lemma 59.92.2. Hence by Derived Categories, Lemma 13.32.2, if we show that $R^ qh_*(f_!\mathcal{F}) = 0$ for $q > 0$ and any sheaf $\mathcal{F}$ of $\Lambda $-modules on $X_{\acute{e}tale}$, then $h_*f_!\mathcal{K}^\bullet \to h_*\mathcal{I}^\bullet $ is a quasi-isomorphism.

Observe that $\mathcal{G} = f_!\mathcal{F}$ is a sheaf of $\Lambda $-modules on $Y$ whose stalks are nonzero only at points $y \in Y$ such that $\kappa (y)/\kappa (h(y))$ is a finite extension. This follows from the description of stalks of $f_!\mathcal{F}$ in Lemma 62.4.5 and the fact that both $f$ and $g$ are locally quasi-finite. Hence by the proper base change theorem (Étale Cohomology, Lemma 59.91.13) it suffices to show that $H^ q(Y_{\overline{z}}, \mathcal{H}) = 0$ where $\mathcal{H}$ is a sheaf on the proper scheme $Y_{\overline{z}}$ over $\kappa (\overline{z})$ whose support is contained in the set of closed points. Thus the required vanishing by Étale Cohomology, Lemma 59.97.3.

Case (a) follows from case (b) by the exact same argument as used in the proof of Lemma 62.8.1 (using Lemma 62.4.5 instead of Lemma 62.3.17). $\square$


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