62.13 A constructibility result

We “compute” the cohomology of a smooth projective family of curves with constant coefficients.

Lemma 62.13.1. Let $p$ be a prime number. Let $S$ be a scheme over $\mathbf{F}_ p$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ S$-module viewed as an $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$. Let $F : \mathcal{E} \to \mathcal{E}$ be a homomorphism of abelian sheaves on $S_{\acute{e}tale}$ such that $F(a e) = a^ pF(e)$ for local sections $a$, $e$ of $\mathcal{O}_ S$, $\mathcal{E}$ on $S_{\acute{e}tale}$. Then

$\mathop{\mathrm{Coker}}(F - 1 : \mathcal{E} \to \mathcal{E})$

is zero and

$\mathop{\mathrm{Ker}}(F - 1 : \mathcal{E} \to \mathcal{E})$

is a constructible abelian sheaf on $S_{\acute{e}tale}$.

This lemma is a generalization of Étale Cohomology, Lemma 59.63.2.

Proof. We may assume $S = \mathop{\mathrm{Spec}}(A)$ where $A$ is an $\mathbf{F}_ p$-algebra and that $\mathcal{E}$ is the quasi-coherent module associated to the free $A$-module $Ae_1 \oplus \ldots \oplus Ae_ n$. We write $F(e_ i) = \sum a_{ij} e_ j$.

Surjectivity of $F - 1$. It suffices to show that any element $\sum a_ i e_ i$, $a_ i \in A$ is in the image of $F - 1$ after replacing $A$ by a faithfully flat étale extension. Observe that

$F(\sum x_ ie_ i) - \sum x_ i e_ i = \sum x_ i^ p a_{ij} e_ j - \sum x_ i e_ i$

Consider the $A$-algebra

$A' = A[x_1, \ldots , x_ n]/(a_ i + x_ i - \sum \nolimits _ j a_{ji} x_ j^ p)$

A computation shows that $\text{d}x_ i$ is zero in $\Omega _{A'/A}$ and hence $\Omega _{A'/A} = 0$. Since $A'$ is of finite type over $A$, this implies that $\mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A)$ is unramified and hence is quasi-finite. Since $A'$ is generated by $n$ elements and cut out by $n$ equations, we conclude that $A'$ is a global relative complete intersection over $A$. Thus $A'$ is flat over $A$ and we conclude that $A \to A'$ is étale (as a flat and unramified ring map). Finally, the reader can show that $A \to A'$ is faithfully flat by verifying directly that all geometric fibres of $\mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A)$ are nonempty, however this also follows from Étale Cohomology, Lemma 59.63.2. Finally, the element $\sum x_ i e_ i \in A'e_1 \oplus \ldots \oplus A'e_ n$ maps to $\sum a_ i e_ i$ by $F - 1$.

Constructibility of the kernel. The calculations above show that $\mathop{\mathrm{Ker}}(F - 1)$ is represented by the scheme

$\mathop{\mathrm{Spec}}(A[x_1, \ldots , x_ n]/(x_ i - \sum \nolimits _ j a_{ji} x_ j^ p))$

over $S = \mathop{\mathrm{Spec}}(A)$. Since this is a scheme affine and étale over $S$ we obtain the result from Étale Cohomology, Lemma 59.73.1. $\square$

Lemma 62.13.2. Let $f : X \to S$ be a proper smooth morphism of schemes with geometrically connected fibres of dimension $1$. Let $\ell$ be a prime number. Then $R^ qf_*\underline{\mathbf{Z}/\ell \mathbf{Z}}$ is a constructible.

Proof. We may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(A)$. Then, if we write $A = \bigcup A_ i$ as the union of its finite type $\mathbf{Z}$-subalgebras, we can find an $i$ and a morphism $f_ i : X_ i \to S_ i = \mathop{\mathrm{Spec}}(A_ i)$ of finite type whose base change to $S$ is $f : X \to S$, see Limits, Lemma 32.10.1. After increasing $i$ we may assume $f_ i : X_ i \to S_ i$ is smooth, proper, and of relative dimension $1$, see Limits, Lemmas 32.13.1 32.8.9, and 32.17.3. By More on Morphisms, Lemma 37.52.8 we obtain an open subscheme $U_ i \subset S_ i$ such that the fibres of $f_ i : X_ i \to S_ i$ over $U_ i$ are geometrically connected. Then $S \to S_ i$ maps into $U_ i$. We may replace $X \to S$ by $f_ i : f_ i^{-1}(U_ i) \to U_ i$ to reduce to the case discussed in the next paragraph.

Assume $S$ is Noetherian. We may write $S = U \cup Z$ where $U$ is the open subscheme defined by the nonvanishing of $\ell$ and $Z = V(\ell ) \subset S$. Since the formation of $R^ qf_*\underline{\mathbf{Z}/\ell \mathbf{Z}}$ commutes with arbtrary base change (Étale Cohomology, Theorem 59.91.11), it suffices to prove the result over $U$ and over $Z$. Thus we reduce to the following two cases: (a) $\ell$ is invertible on $S$ and (b) $\ell$ is zero on $S$.

Case (a). We claim that in this case the sheaves $R^ qf_*\underline{\mathbf{Z}/\ell \mathbf{Z}}$ are finite locally constant on $S$. First, by proper base change (in the form of Étale Cohomology, Lemma 59.91.13) and by finiteness (Étale Cohomology, Theorem 59.83.10) we see that the stalks of $R^ qf_*\underline{\mathbf{Z}/\ell \mathbf{Z}}$ are finite. By Étale Cohomology, Lemma 59.94.4 all specialization maps are isomorphisms. We conclude the claim holds by Étale Cohomology, Lemma 59.75.6.

Case (b). Here $\ell = p$ is a prime and $S$ is a scheme over $\mathop{\mathrm{Spec}}(\mathbf{F}_ p)$. By the same references as above we already know that the stalks of $R^ qf_*\underline{\mathbf{Z}/p\mathbf{Z}}$ are finite and zero for $q \geq 2$. It follows from Étale Cohomology, Lemma 59.39.3 that $f_*\underline{\mathbf{Z}/p\mathbf{Z}} = \underline{\mathbf{Z}/p\mathbf{Z}}$. It remains to prove that $R^1f_*\underline{\mathbf{Z}/p\mathbf{Z}}$ is constructible. Consider the Artin-Schreyer sequence

$0 \to \underline{\mathbf{Z}/p\mathbf{Z}} \to \mathcal{O}_ X \xrightarrow {F - 1} \mathcal{O}_ X \to 0$

See Étale Cohomology, Section 59.63. Recall that $f_*\mathcal{O}_ X = \mathcal{O}_ S$ and $R^1f_*\mathcal{O}_ X$ is a finite locally free $\mathcal{O}_ S$-module of rank equal to the genera of the fibres of $X \to S$, see Algebraic Curves, Lemma 53.20.13. We conclude that we have a short exact sequence

$0 \to \mathop{\mathrm{Coker}}(F - 1 : \mathcal{O}_ S \to \mathcal{O}_ S) \to R^1f_*\underline{\mathbf{Z}/p\mathbf{Z}} \to \mathop{\mathrm{Ker}}(F - 1 : R^1f_*\mathcal{O}_ X \to R^1f_*\mathcal{O}_ X) \to 0$

Applying Lemma 62.13.1 we win. $\square$

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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