Lemma 63.8.1 (0F7B)—The Stacks project

Lemma 63.8.1. 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 63.3.4 and the second equality comes from Lemma 63.3.13 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 63.3.12) 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 63.3.15 (but it is also obvious from the definition of $g'_{\overline{y}, !}$ in Section 63.3). 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 63.3.17 and Étale Cohomology, Lemma 59.51.8) the proof is complete. $\square$

The Stacks project

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