The Stacks project

Lemma 63.14.2. Let $S$ be a Noetherian affine scheme of finite dimension. Let $f : X \to S$ be a separated, affine, smooth morphism of relative dimension $1$. Let $\Lambda $ be a Noetherian ring which is torsion. Let $M$ be a finite $\Lambda $-module. Then $Rf_!\underline{M}$ has constructible cohomology sheaves.

Proof. We will prove the result by induction on $d = \dim (S)$.

Base case. If $d = 0$, then the only thing to show is that the stalks of $R^ qf_!\underline{M}$ are finite $\Lambda $-modules. If $\overline{s}$ is a geometric point of $S$, then we have $(R^ qf_!\underline{M})_{\overline{s}} = H^ q_ c(X_{\overline{s}}, \underline{M})$ by Lemma 63.12.2. This is a finite $\Lambda $-module by Lemma 63.12.4.

Induction step. It suffices to find a dense open $U \subset S$ such that $Rf_!\underline{M}|_ U$ has constructible cohomology sheaves. Namely, the restriction of $Rf_!\underline{M}$ to the complement $S \setminus U$ will have constructible cohomology sheaves by induction and the fact that formation of $Rf_!\underline{M}$ commutes with all base change (Lemma 63.9.4). In fact, let $\eta \in S$ be a generic point of an irreducible component of $S$. Then it suffices to find an open neighbourhood $U$ of $\eta $ such that the restriction of $Rf_!\underline{M}$ to $U$ is constructible. This is what we will do in the next paragraph.

Given a generic point $\eta \in S$ we choose a diagram

\[ \xymatrix{ \overline{Y}_1 \amalg \ldots \amalg \overline{Y}_ n \ar[rd] & Y_1 \amalg \ldots \amalg Y_ n \ar[r]_-\nu \ar[d] \ar[l]^ j & X_ V \ar[r] \ar[d] & X_ U \ar[r] \ar[d] & X \ar[d]^ f \\ & T_1 \amalg \ldots \amalg T_ n \ar[r] & V \ar[r] & U \ar[r] & S } \]

as in More on Morphisms, Lemma 37.56.1. We will show that $Rf_!\underline{M}|_ U$ is constructible. First, since $V \to U$ is finite and surjective, it suffices to show that the pullback to $V$ is constructible, see Étale Cohomology, Lemma 59.73.3. Since formation of $Rf_!$ commutes with base change, we see that it suffices to show that $R(X_ V \to V)_!\underline{M}$ is constructible. Let $W \subset X_ V$ be the open subscheme given to us by More on Morphisms, Lemma 37.56.1 part (4). Let $Z \subset X_ V$ be the reduced induced scheme structure on the complement of $W$ in $X_ V$. Then the fibres of $Z \to V$ have dimension $0$ (as $W$ is dense in the fibres) and hence $Z \to V$ is quasi-finite. From the distinguished triangle

\[ R(W \to V)_!\underline{M} \to R(X_ V \to V)_!\underline{M} \to R(Z \to V)_!\underline{M} \to \ldots \]

of Lemma 63.10.5 and from Lemma 63.14.1 we conclude that it suffices to show that $R(W \to V)_!\underline{M}$ has constructible cohomology sheaves. Next, we have

\[ R(W \to V)_!\underline{M} = R(\nu ^{-1}(W) \to V)_!\underline{M} \]

because the morphism $\nu : \nu ^{-1}(W) \to W$ is a thickening and we may apply Lemma 63.10.6. Next, we let $Z' \subset \coprod \overline{Y}_ i$ denote the complement of the open $j(\nu ^{-1}(W))$. Again $Z' \to V$ is quasi-finite. Again use the distinguished triangle

\[ R(\nu ^{-1}(W) \to V)_!\underline{M} \to R(\coprod \overline{Y}_ i \to V)_!\underline{M} \to R(Z' \to V)_!\underline{M} \to \ldots \]

to conclude that it suffices to prove

\[ R(\coprod \overline{Y}_ i \to V)_!\underline{M} = \bigoplus \nolimits _ i R(\overline{Y}_ i \to V)_!\underline{M} = \bigoplus \nolimits _ i R(T_ i \to V)_!R(\overline{Y}_ i \to T_ i)_!\underline{M} \]

has constructible cohomology sheaves (second equality by Lemma 63.9.2). The result for $R(\overline{Y}_ i \to T_ i)_!\underline{M}$ is Lemma 63.13.3 and we win because $T_ i \to V$ is finite étale and we can apply Lemma 63.14.1. $\square$

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