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. $\square$

Comments (0)

Post a comment

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F7B. Beware of the difference between the letter 'O' and the digit '0'.