The Stacks project

Lemma 62.13.3. Let $f : X \to S$ be a proper smooth morphism of schemes with geometrically connected fibres of dimension $1$. Let $\Lambda $ be a Noetherian ring. Let $M$ be a finite $\Lambda $-module annihilated by an integer $n > 0$. Then $R^ qf_*\underline{M}$ is a constructible sheaf of $\Lambda $-modules on $S$.

Proof. If $n = \ell n'$ for some prime number $\ell $, then we get a short exact sequence $0 \to M[\ell ] \to M \to M' \to 0$ of finite $\Lambda $-modules and $M'$ is annihilated by $n'$. This produces a corresponding short exact sequence of constant sheaves, which in turn gives rise to an exact sequence

\[ R^{q - 1}f_*\underline{M'} \to R^ qf_*\underline{M[n]} \to R^ qf_*\underline{M} \to R^ qf_*\underline{M'} \to R^{q + 1}f_*\underline{M[n]} \]

Thus, if we can show the result in case $M$ is annihilated by a prime number, then by induction on $n$ we win by Étale Cohomology, Lemma 59.71.6.

Let $\ell $ be a prime number such that $\ell $ annihilates $M$. Then we can replace $\Lambda $ by the $\mathbf{F}_\ell $-algebra $\Lambda /\ell \Lambda $. Namely, the sheaf $R^ qf_*\underline{M}$ where $\underline{M}$ is viewed as a sheaf of $\Lambda $-modules is the same as the sheaf $R^ qf_*\underline{M}$ computed by viewing $\underline{M}$ as a sheaf of $\Lambda /\ell \Lambda $-modules, see Cohomology on Sites, Lemma 21.20.7.

Assume $\ell $ be a prime number such that $\ell $ annihilates $M$ and $\Lambda $. Let us reduce to the case where $M$ is a finite free $\Lambda $-module. Namely, choose a resolution

\[ \ldots \to \Lambda ^{\oplus m_2} \to \Lambda ^{\oplus m_1} \to \Lambda ^{\oplus m_0} \to M \to 0 \]

Recall that $f_*$ has finite cohomological dimension on sheaves of $\Lambda $-modules, see Étale Cohomology, Lemma 59.92.2 and Derived Categories, Lemma 13.32.2. Thus we see that $R^ qf_*\underline{M}$ is the $q$th cohomology sheaf of the object

\[ Rf_*(\underline{\Lambda ^{\oplus m_ a}} \to \ldots \to \underline{\Lambda ^{\oplus m_0}}) \]

in $D(S_{\acute{e}tale}, \Lambda )$ for some integer $a$ large enough. Using the first spectral sequence of Derived Categories, Lemma 13.21.3 (or alternatively using an argument with truncations) we conclude that it suffices to prove that $R^ qf_*\underline{\Lambda })$ is constructible.

At this point we can finally use that

\[ (Rf_*\underline{\mathbf{Z}/\ell \mathbf{Z}}) \otimes _{\mathbf{Z}/\ell \mathbf{Z}}^\mathbf {L} \underline{\Lambda } = Rf_*\underline{\Lambda } \]

by Étale Cohomology, Lemma 59.96.6. Since any module over the field $\mathbf{Z}/\ell \mathbf{Z}$ is flat we obtain

\[ (R^ qf_*\underline{\mathbf{Z}/\ell \mathbf{Z}}) \otimes _{\mathbf{Z}/\ell \mathbf{Z}} \underline{\Lambda } = R^ qf_*\underline{\Lambda } \]

Hence it suffices to prove the result for $R^ qf_*\underline{\mathbf{Z}/\ell \mathbf{Z}}$ by Étale Cohomology, Lemma 59.71.10. This case is Lemma 62.13.2. $\square$


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