The Stacks project

Lemma 62.8.1. Consider a commutative diagram of schemes

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

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

\[ g_!Rf'_*K \longrightarrow Rf_*(g'_!K) \]

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

Proof. Represent $K$ by a K-injective complex $\mathcal{J}^\bullet $ of sheaves of $\Lambda $-modules on $X'_{\acute{e}tale}$. Choose a quasi-isomorphism $g'_!\mathcal{J}^\bullet \to \mathcal{I}^\bullet $ to a K-injective complex $\mathcal{I}^\bullet $ of sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$. Then we can consider the map

\[ g_!f'_*\mathcal{J}^\bullet = g_!f'_!\mathcal{J}^\bullet = f_!g'_!\mathcal{J}^\bullet = f_*g'_!\mathcal{J}^\bullet \to f_*\mathcal{I}^\bullet \]

where the first and third equality come from Lemma 62.3.4 and the second equality comes from Lemma 62.3.13 which tells us that both $g_! \circ f'_!$ and $f_! \circ g'_!$ are equal to $(g \circ f')_! = (f \circ g')_!$ as subsheaves of $(g \circ f')_* = (f \circ g')_*$.

Assume $\Lambda $ is torsion, i.e., we are in case (b). With notation as above, it suffices to show that $f_*g'_!\mathcal{J}^\bullet \to f_*\mathcal{I}^\bullet $ is an isomorphism. The question is local on $Y$. Hence we may assume that the dimension of fibres of $f$ is bounded, see Morphisms, Lemma 29.28.5. Then we see that $Rf_*$ has finite cohomological dimension, see Étale Cohomology, Lemma 59.92.2. Hence by Derived Categories, Lemma 13.32.2, if we show that $R^ qf_*(g'_!\mathcal{J}) = 0$ for $q > 0$ and any injective sheaf of $\Lambda $-modules $\mathcal{J}$ on $X'_{\acute{e}tale}$, then the result follows.

The stalk of $R^ qf_*(g'_!\mathcal{J})$ at a geometric point $\overline{y}$ is equal to $H^ q(X_{\overline{y}}, (g'_!\mathcal{J})|_{X_{\overline{y}}})$ by Étale Cohomology, Lemma 59.91.13. Since formation of $g'_!$ commutes with base change (Lemma 62.3.12) this is equal to

\[ H^ q(X_{\overline{y}}, g'_{\overline{y}, !}(\mathcal{J}|_{X'_{\overline{y}}})) \]

where $g'_{\overline{y}} : X'_{\overline{y}} \to X_{\overline{y}}$ is the induced morphism between geometric fibres. Since $Y' \to Y$ is locally quasi-finite, we see that $X'_{\overline{y}}$ is a disjoint union of the fibres $X'_{\overline{y}'}$ at geometric points $\overline{y}'$ of $Y'$ lying over $\overline{y}$. Denote $g'_{\overline{y}'} : X'_{\overline{y}'} \to X_{\overline{y}}$ the restriction of $g'_{\overline{y}}$ to $X'_{\overline{y}'}$. Thus the previous cohomology group is equal to

\[ H^ q(X_{\overline{y}}, \bigoplus \nolimits _{\overline{y}'/\overline{y}} g'_{\overline{y}', !}(\mathcal{J}|_{X'_{\overline{y}'}})) \]

for example by Lemma 62.3.15 (but it is also obvious from the definition of $g'_{\overline{y}, !}$ in Section 62.3). Since taking étale cohomology over $X_{\overline{y}}$ commutes with direct sums (Étale Cohomology, Theorem 59.51.3) we conclude it suffices to show that

\[ H^ q(X_{\overline{y}}, g'_{\overline{y}', !}(\mathcal{J}|_{X'_{\overline{y}'}})) \]

is zero. Observe that $g_{\overline{y}'} : X'_{\overline{y}'} \to X_{\overline{y}}$ is a morphism between proper scheme over $\overline{y}$ and hence is proper itself. As it is locally quasi-finite as well we conclude that $g_{\overline{y}'}$ is finite. Thus we see that $g'_{\overline{y}', !} = g'_{\overline{y}', *} = Rg'_{\overline{y}', *}$. By Leray we conlude that we have to show

\[ H^ q(X'_{\overline{y}'}, \mathcal{J}|_{X'_{\overline{y}'}}) \]

is zero. As $\Lambda $ is torsion, this follows from proper base change (Étale Cohomology, Lemma 59.91.13) as the higher direct images of $\mathcal{J}$ under $f'$ are zero.

Proof in case (a). We will deduce this from case (b) by standard arguments. We will show that the induced map $g_! R^ pf'_* K \to R^ pf_*(g'_!K)$ is an isomorphism for all $p \in \mathbf{Z}$. Fix an integer $p_0 \in \mathbf{Z}$. Let $a$ be an integer such that $H^ j(K) = 0$ for $j < a$. We will prove $g_! R^ pf'_* K \to R^ pf_*(g'_!K)$ is an isomorphism for $p \leq p_0$ by descending induction on $a$. If $a > p_0$, then we see that the left and right hand side of the map are zero for $p \leq p_0$ by trivial vanishing, see Derived Categories, Lemma 13.16.1 (and use that $g_!$ and $g'_!$ are exact functors). Assume $a \leq p_0$. Consider the distinguished triangle

\[ H^ a(K)[-a] \to K \to \tau _{\geq a + 1}K \]

By induction we have the result for $\tau _{\geq a + 1}K$. In the next paragraph, we will prove the result for $H^ a(K)[-a]$. Then five lemma applied to the map between long exact sequence of cohomology sheaves associated to the map of distinguished triangles

\[ \xymatrix{ g_! Rf'_*(H^ a(K)[-a]) \ar[d] \ar[r] & g_! Rf'_* K \ar[r] \ar[d] & g_! Rf'_* \tau _{\geq a + 1} K \ar[d] \\ Rf_*(g'_!(H^ a(K)[-a])) \ar[r] & Rf_*(g'_!K) \ar[r] & Rf_*(g'_!\tau _{|geq a + 1}K) } \]

gives the result for $K$. Some details omitted.

Let $\mathcal{F}$ be a torsion abelian sheaf on $X'_{\acute{e}tale}$. To finish the proof we show that $g_! Rf'_*\mathcal{F} \to R^ pf_*(g'_!\mathcal{F})$ is an isomorphism for all $p$. We can write $\mathcal{F} = \bigcup \mathcal{F}[n]$ where $\mathcal{F}[n] = \mathop{\mathrm{Ker}}(n : \mathcal{F} \to \mathcal{F})$. We have the isomorphism for $\mathcal{F}[n]$ by case (b). Since the functors $g_!$, $g'_!$, $R^ pf_*$, $R^ pf'_*$ commute with filtered colimits (follows from Lemma 62.3.17 and Étale Cohomology, Lemma 59.51.8) the proof is complete. $\square$


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