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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