The Stacks project

62.14 Complexes with constructible cohomology

We continue the discussion started in Étale Cohomology, Section 59.76. In particular, for a scheme $X$ and a Noetherian ring $\Lambda $ we denote $D_ c(X_{\acute{e}tale}, \Lambda )$ the strictly full saturated triangulated subcategory of $D(X_{\acute{e}tale}, \Lambda )$ consisting of objects whose cohomology sheaves are constructible sheaves of $\Lambda $-modules.

Lemma 62.14.1. Let $f : X \to Y$ be a morphism of schemes which is locally quasi-finite and of finite presentation. The functor $f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ of Lemma 62.7.1 sends $D_ c(X_{\acute{e}tale}, \Lambda )$ into $D_ c(Y_{\acute{e}tale}, \Lambda )$.

Proof. Since the functor $f_!$ is exact, it suffices to show that $f_!\mathcal{F}$ is constructible for any constructible sheaf $\mathcal{F}$ of $\Lambda $-modules on $X_{\acute{e}tale}$. The question is local on $Y$ and hence we may and do assume $Y$ is affine. Then $X$ is quasi-compact and quasi-separated, see Morphisms, Definition 29.21.1. Say $X = \bigcup _{i = 1, \ldots , n} X_ i$ is a finite affine open covering. By Lemma 62.4.7 we see that it suffices to show that $f_{i, !}\mathcal{F}|_{X_ i}$ and $f_{ii', !}\mathcal{F}|_{X_ i \cap X_{i'}}$ are constructible where $f_ i : X_ i \to Y$ and $f_{ii'} : X_ i \cap X_{i'} \to Y$ are the restrictions of $f$. Since $X_ i$ and $X_ i \cap X_{i'}$ are quasi-compact and separated this means we may assume $f$ is separated. By Zariski's main theorem (in the form of More on Morphisms, Lemma 37.42.4) we can choose a factorization $f = g \circ j$ where $j : X \to X'$ is an open immersion and $g : X' \to Y$ is finite and of finite presentation. Then $f_! = g_! \circ j_!$ by Lemma 62.3.13. By Étale Cohomology, Lemma 59.73.1 we see that $j_!\mathcal{F}$ is constructible on $X'$. The morphism $g$ is finite hence $g_! = g_*$ by Lemma 62.3.4. Thus $f_!\mathcal{F} = g_!j_!\mathcal{F} = g_*j_!\mathcal{F}$ is constructible by Étale Cohomology, Lemma 59.73.9. $\square$

Lemma 62.14.2. 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 62.12.2. This is a finite $\Lambda $-module by Lemma 62.12.4.

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 62.9.4). 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 37.53.1. 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 37.53.1 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 62.10.5 and from Lemma 62.14.1 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 62.10.6. 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 62.9.2). The result for $R(\overline{Y}_ i \to T_ i)_!\underline{M}$ is Lemma 62.13.3 and we win because $T_ i \to V$ is finite étale and we can apply Lemma 62.14.1. $\square$

Lemma 62.14.3. Let $Y$ be a Noetherian affine scheme of finite dimension. Let $\Lambda $ be a Noetherian ring which is torsion. Let $\mathcal{F}$ be a finite type, locally constant sheaf of $\Lambda $-modules on an open subscheme $U \subset \mathbf{A}^1_ Y$. Then $Rf_!\mathcal{F}$ has constructible cohomology sheaves where $f : U \to Y$ is the structure morphism.

Proof. We may decompose $\Lambda $ as a product $\Lambda = \Lambda _1 \times \ldots \times \Lambda _ r$ where $\Lambda _ i$ is $\ell _ i$-primary for some prime $\ell _ i$. Thus we may assume there exists a prime $\ell $ and an integer $n > 0$ such that $\ell ^ n$ annihilates $\Lambda $ (and hence $\mathcal{F}$).

Since $U$ is Noetherian, we see that $U$ has finitely many connected components. Thus we may assume $U$ is connected. Let $g : U' \to U$ be the finite étale covering constructed in Étale Cohomology, Lemma 59.66.4. The discussion in Étale Cohomology, Section 59.66 gives maps

\[ \mathcal{F} \to g_*g^{-1}\mathcal{F} \to \mathcal{F} \]

whose composition is an isomorphism. Hence it suffices to prove the result for $g_*g^{-1}\mathcal{F}$. On the other hand, we have $Rf_!g_*g^{-1}\mathcal{F} = R(f \circ g)_!g^{-1}\mathcal{F}$ by Lemma 62.9.2. Since $g^{-1}\mathcal{F}$ has a finite filtration by constant sheaves of $\Lambda $-modules of the form $\underline{M}$ for some finite $\Lambda $-module $M$ (by our choice of $g$) this reduces us to the case proved in Lemma 62.14.2. $\square$

Lemma 62.14.4. Let $Y$ be an affine scheme. Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda $-modules on $\mathbf{A}^1_ Y$ which is torsion. Then $Rf_!\mathcal{F}$ has constructible cohomology sheaves where $f : \mathbf{A}^1_ Y \to Y$ is the structure morphism.

Proof. Say $\mathcal{F}$ is annihilated by $n > 0$. Then we can replace $\Lambda $ by $\Lambda /n\Lambda $ without changing $Rf_!\mathcal{F}$. Thus we may and do assume $\Lambda $ is a torsion ring.

Say $Y = \mathop{\mathrm{Spec}}(R)$. Then, if we write $R = \bigcup R_ i$ as the union of its finite type $\mathbf{Z}$-subalgebras, we can find an $i$ such that $\mathcal{F}$ is the pullback of a constructible sheaf of $\Lambda $-modules on $\mathbf{A}^1_{R_ i}$, see Étale Cohomology, Lemma 59.73.10. Hence we may assume $Y$ is a Noetherian scheme of finite dimension.

Assume $Y$ is a Noetherian scheme of finite dimension $d = \dim (Y)$ and $\Lambda $ is torsion. We will prove the result by induction on $d$.

Base case. If $d = 0$, then the only thing to show is that the stalks of $R^ qf_!\mathcal{F}$ are finite $\Lambda $-modules. If $\overline{y}$ is a geometric point of $Y$, then we have $(R^ qf_!\mathcal{F})_{\overline{y}} = H^ q_ c(X_{\overline{y}}, \mathcal{F})$ by Lemma 62.12.2. This is a finite $\Lambda $-module by Lemma 62.12.4.

Induction step. It suffices to find a dense open $V \subset Y$ such that $Rf_!\mathcal{F}|_ V$ has constructible cohomology sheaves. Namely, the restriction of $Rf_!\mathcal{F}$ to the complement $Y \setminus V$ will have constructible cohomology sheaves by induction and the fact that formation of $Rf_!\mathcal{F}$ commutes with all base change (Lemma 62.9.4). By definition of constructible sheaves of $\Lambda $-modules, there is a dense open subscheme $U \subset \mathbf{A}^1_ Y$ such that $\mathcal{F}|_ U$ is a finite type, locally constant sheaf of $\Lambda $-modules. Denote $Z \subset \mathbf{A}^1_ Y$ the complement (viewed as a reduced closed subscheme). Note that $U$ contains all the generic points of the fibres of $\mathbf{A}^1_ Y \to Y$ over the generic points $\xi _1, \ldots , \xi _ n$ of the irreducible components of $Y$. Hence $Z \to Y$ has finite fibres over $\xi _1, \ldots , \xi _ n$. After replacing $Y$ by a dense open (which is allowed), we may assume $Z \to Y$ is finite, see Morphisms, Lemma 29.51.1. By the distinguished triangle of Lemma 62.10.5 and the result for $Z \to Y$ (Lemma 62.14.1) we reduce to showing that $R(U \to Y)_!\mathcal{F}$ has constructible cohomology sheaves. This is Lemma 62.14.3. $\square$

Theorem 62.14.5. Let $f : X \to Y$ be a separated morphism of finite presentation of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a Noetherian ring. Let $K$ be an object of $D^+_{tors, c}(X_{\acute{e}tale}, \Lambda )$ or of $D_ c(X_{\acute{e}tale}, \Lambda )$ in case $\Lambda $ is torsion. Then $Rf_!K$ has constructible cohomology sheaves, i.e., $Rf_!K$ is in $D^+_{tors, c}(Y_{\acute{e}tale}, \Lambda )$ or in $D_ c(Y_{\acute{e}tale}, \Lambda )$ in case $\Lambda $ is torsion.

Proof. The question is local on $Y$ hence we may and do assume $Y$ is affine. By the induction principle and Lemma 62.10.4 we reduce to the case where $X$ is also affine.

Assume $X$ and $Y$ are affine. Since $X$ is of finite presentation, we can choose a closed immersion $i : X \to \mathbf{A}^ n_ Y$ which is of finite presentation. If $p : \mathbf{A}^ n_ Y \to Y$ denotes the structure morphism, then we see that $Rf_! = Rp_! \circ Ri_!$ by Lemma 62.9.2. By Lemma 62.14.1 we have the result for $Ri_! = i_!$. Hence we may assume $f$ is the projection morphism $\mathbf{A}^ n_ Y \to Y$. Since we can view $f$ as the composition

\[ X = \mathbf{A}^ n_ Y \to \mathbf{A}^{n - 1}_ Y \to \mathbf{A}^{n - 2}_ S \to \ldots \to \mathbf{A}^1_ Y \to Y \]

we may assume $n = 1$.

Assume $Y$ is affine and $X = \mathbf{A}^1_ Y$. Since $Rf_!$ has finite cohomological dimension (Lemma 62.10.2) we may assume $K$ is bounded below. Using the first spectral sequence of Derived Categories, Lemma 13.21.3 (or alternatively using an argument with truncations), we reduce to showing the result of Lemma 62.14.4. $\square$


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