The Stacks project

59.84 Cohomology of torsion modules on curves

In this section we repeat the arguments of Section 59.83 for constructible sheaves of modules over a Noetherian ring which are torsion. We start with the most interesting step.

Lemma 59.84.1. Let $\Lambda $ be a Noetherian ring, let $M$ be a finite $\Lambda $-module which is annihilated by an integer $n > 0$, let $k$ be an algebraically closed field, and let $X$ be a separated, finite type scheme of dimension $\leq 1$ over $k$. Then

  1. $H^ q_{\acute{e}tale}(X, \underline{M})$ is a finite $\Lambda $-module if $n$ is prime to $\text{char}(k)$,

  2. $H^ q_{\acute{e}tale}(X, \underline{M})$ is a finite $\Lambda $-module if $X$ is proper.

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 of cohomology modules

\[ H^ q_{\acute{e}tale}(X, \underline{M[n]}) \to H^ q_{\acute{e}tale}(X, \underline{M}) \to H^ q_{\acute{e}tale}(X, \underline{M'}) \]

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

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 cohomology of $\mathcal{F}$ as a sheaf of $\Lambda $-modules is the same as the cohomology of $\mathcal{F}$ as a sheaf of $\Lambda /\ell \Lambda $-modules, for example by Cohomology on Sites, Lemma 21.12.4.

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 short exact sequence

\[ 0 \to N \to \Lambda ^{\oplus m} \to M \to 0 \]

This determines an exact sequence

\[ H^ q_{\acute{e}tale}(X, \underline{\Lambda ^{\oplus m}}) \to H^ q_{\acute{e}tale}(X, \underline{M}) \to H^{q + 1}_{\acute{e}tale}(X, \underline{N}) \]

By descending induction on $q$ we get the result for $M$ if we know the result for $\Lambda ^{\oplus m}$. Here we use that we know that our cohomology groups vanish in degrees $> 2$ by Theorem 59.83.10.

Let $\ell $ be a prime number and assume that $\ell $ annihilates $\Lambda $. It remains to show that the cohomology groups $H^ q_{\acute{e}tale}(X, \underline{\Lambda })$ are finite $\Lambda $-modules. We will use a trick to show this; the “correct” argument uses a coefficient theorem which we will show later. Choose a basis $\Lambda = \bigoplus _{i \in I} \mathbf{F}_\ell e_ i$ such that $e_0 = 1$ for some $0 \in I$. The choice of this basis determines an isomorphism

\[ \underline{\Lambda } = \bigoplus \underline{\mathbf{F}_\ell } e_ i \]

of sheaves on $X_{\acute{e}tale}$. Thus we see that

\[ H^ q_{\acute{e}tale}(X, \underline{\Lambda }) = H^ q_{\acute{e}tale}(X, \bigoplus \underline{\mathbf{F}_\ell } e_ i) = \bigoplus H^ q_{\acute{e}tale}(X, \underline{\mathbf{F}_\ell })e_ i \]

since taking cohomology over $X$ commutes with direct sums by Theorem 59.51.3 (or Lemma 59.51.4 or Lemma 59.52.2). Since we already know that $H^ q_{\acute{e}tale}(X, \underline{\mathbf{F}_\ell })$ is a finite dimensional $\mathbf{F}_\ell $-vector space (by Theorem 59.83.10), we see that $H^ q_{\acute{e}tale}(X, \underline{\Lambda })$ is free over $\Lambda $ of the same rank. Namely, given a basis $\xi _1, \ldots , \xi _ m$ of $H^ q_{\acute{e}tale}(X, \underline{\mathbf{F}_\ell })$ we see that $\xi _1 e_0, \ldots , \xi _ m e_0$ form a $\Lambda $-basis for $H^ q_{\acute{e}tale}(X, \underline{\Lambda })$. $\square$

Lemma 59.84.2. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $f : X \to Y$ be a finite morphism of separated finite type schemes over $k$ of dimension $\leq 1$, and let $\mathcal{F}$ be a sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$. If $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module, then so is $H^ q_{\acute{e}tale}(Y, f_*\mathcal{F})$.

Proof. Namely, we have $H^ q_{\acute{e}tale}(X, \mathcal{F}) = H^ q_{\acute{e}tale}(Y, f_*\mathcal{F})$ by the vanishing of $R^ qf_*$ for $q > 0$ (Proposition 59.55.2) and the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.6). $\square$

Lemma 59.84.3. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $X$ be a separated finite type scheme over $k$ of dimension $\leq 1$, let $\mathcal{F}$ be a constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$, and let $j : X' \to X$ be the inclusion of a dense open subscheme. Then $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module if and only if $H^ q_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F})$ is a finite $\Lambda $-module.

Proof. Since $X'$ is dense, we see that $Z = X \setminus X'$ has dimension $0$ and hence is a finite set $Z = \{ x_1, \ldots , x_ n\} $ of $k$-rational points. Consider the short exact sequence

\[ 0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0 \]

of Lemma 59.70.8. Observe that $H^ q_{\acute{e}tale}(X, i_*i^{-1}\mathcal{F}) = H^ q_{\acute{e}tale}(Z, i^*\mathcal{F})$. Namely, $i : Z \to X$ is a closed immersion, hence finite, hence we have the vanishing of $R^ qi_*$ for $q > 0$ by Proposition 59.55.2, and hence the equality follows from the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.6). Since $Z$ is a disjoint union of spectra of algebraically closed fields, we conclude that $H^ q_{\acute{e}tale}(Z, i^*\mathcal{F}) = 0$ for $q > 0$ and

\[ H^0_{\acute{e}tale}(Z, i^{-1}\mathcal{F}) = \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{F}_{x_ i} \]

which is a finite $\Lambda $-module $\mathcal{F}_{x_ i}$ is finite due to the assumption that $\mathcal{F}$ is a constructible sheaf of $\Lambda $-modules. The long exact cohomology sequence gives an exact sequence

\[ 0 \to H^0_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^0_{\acute{e}tale}(X, \mathcal{F}) \to H^0_{\acute{e}tale}(Z, i^{-1}\mathcal{F}) \to H^1_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^1_{\acute{e}tale}(X, \mathcal{F}) \to 0 \]

and isomorphisms $H^0_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^0_{\acute{e}tale}(X, \mathcal{F})$ for $q > 1$. The lemma follows easily from this. $\square$

Lemma 59.84.4. Let $\Lambda $ be a Noetherian ring, let $M$ be a finite $\Lambda $-module which is annihilated by an integer $n > 0$, let $k$ be an algebraically closed field, let $X$ be a separated, finite type scheme of dimension $\leq 1$ over $k$, and let $j : U \to X$ be an open immersion. Then

  1. $H^ q_{\acute{e}tale}(X, j_!\underline{M})$ is a finite $\Lambda $-module if $n$ is prime to $\text{char}(k)$,

  2. $H^ q_{\acute{e}tale}(X, j_!\underline{M})$ is a finite $\Lambda $-module if $X$ is proper.

Proof. Since $\dim (X) \leq 1$ there is an open $V \subset X$ which is disjoint from $U$ such that $X' = U \cup V$ is dense open in $X$ (details omitted). If $j' : X' \to X$ denotes the inclusion morphism, then we see that $j_!\underline{M}$ is a direct summand of $j'_!\underline{M}$. Hence it suffices to prove the lemma in case $U$ is open and dense in $X$. This case follows from Lemmas 59.84.3 and 59.84.1. $\square$

Lemma 59.84.5. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $X$ be a separated finite type scheme over $k$ of dimension $\leq 1$, and let $0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0$ be a short exact sequence of sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$. If $H^ q_{\acute{e}tale}(X, \mathcal{F}_ i)$, $i = 1, 2$ are finite $\Lambda $-modules then $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module.

Proof. Immediate from the long exact sequence of cohomology. $\square$

Lemma 59.84.6. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $X$ be a separated, finite type scheme of dimension $\leq 1$ over $k$, let $j : U \to X$ be an open immersion with $U$ connected, let $\ell $ be a prime number, let $n > 0$, and let $\mathcal{G}$ be a finite type, locally constant sheaf of $\Lambda $-modules on $U_{\acute{e}tale}$ annihilated by $\ell ^ n$. Then

  1. $H^ q_{\acute{e}tale}(X, j_!\mathcal{G})$ is a finite $\Lambda $-module if $\ell $ is prime to $\text{char}(k)$,

  2. $H^ q_{\acute{e}tale}(X, j_!\mathcal{G})$ is a finite $\Lambda $-module if $X$ is proper.

Proof. Let $f : V \to U$ be a finite étale morphism of degree prime to $\ell $ as in Lemma 59.66.4. The discussion in Section 59.66 gives maps

\[ \mathcal{G} \to f_*f^{-1}\mathcal{G} \to \mathcal{G} \]

whose composition is an isomorphism. Hence it suffices to prove the finiteness of $H^ q_{\acute{e}tale}(X, j_!f_*f^{-1}\mathcal{G})$. By Zariski's Main theorem (More on Morphisms, Lemma 37.42.3) we can choose a diagram

\[ \xymatrix{ V \ar[r]_{j'} \ar[d]_ f & Y \ar[d]^{\overline{f}} \\ U \ar[r]^ j & X } \]

with $\overline{f} : Y \to X$ finite and $j'$ an open immersion with dense image. Since $f$ is finite and $V$ dense in $Y$ we have $V = U \times _ X Y$. By Lemma 59.70.9 we have

\[ j_!f_*f^{-1}\mathcal{G} = \overline{f}_*j'_!f^{-1}\mathcal{G} \]

By Lemma 59.84.2 it suffices to consider $j'_!f^{-1}\mathcal{G}$. The existence of the filtration given by Lemma 59.66.4, the fact that $j'_!$ is exact, and Lemma 59.84.5 reduces us to the case $\mathcal{F} = j'_!\underline{M}$ for a finite $\Lambda $-module $M$ which is Lemma 59.84.4. $\square$

Theorem 59.84.7. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $X$ be a separated, finite type scheme of dimension $\leq 1$ over $k$, and let $\mathcal{F}$ be a constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$ which is torsion. Then

  1. $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module if $\mathcal{F}$ is torsion prime to $\text{char}(k)$,

  2. $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module if $X$ is proper.

Proof. without further mention. Write $\mathcal{F} = \mathcal{F}_1 \oplus \ldots \oplus \mathcal{F}_ r$ where $\mathcal{F}_ i$ is annihilated by $\ell _ i^{n_ i}$ for some prime $\ell _ i$ and integer $n_ i > 0$. By Lemma 59.84.5 it suffices to prove the theorem for $\mathcal{F}_ i$. Thus we may and do assume that $\ell ^ n$ kills $\mathcal{F}$ for some prime $\ell $ and integer $n > 0$.

Since $\mathcal{F}$ is constructible as a sheaf of $\Lambda $-modules, there is a dense open $U \subset X$ such that $\mathcal{F}|_ U$ is a finite type, locally constant sheaf of $\Lambda $-modules. Since $\dim (X) \leq 1$ we may assume, after shrinking $U$, that $U = U_1 \amalg \ldots \amalg U_ n$ is a disjoint union of irreducible schemes (just remove the closed points which lie in the intersections of $\geq 2$ components of $U$). By Lemma 59.84.3 we reduce to the case $\mathcal{F} = j_!\mathcal{G}$ where $\mathcal{G}$ is a finite type, locally constant sheaf of $\Lambda $-modules on $U$ (and annihilated by $\ell ^ n$).

Since we chose $U = U_1 \amalg \ldots \amalg U_ n$ with $U_ i$ irreducible we have

\[ j_!\mathcal{G} = j_{1!}(\mathcal{G}|_{U_1}) \oplus \ldots \oplus j_{n!}(\mathcal{G}|_{U_ n}) \]

where $j_ i : U_ i \to X$ is the inclusion morphism. The case of $j_{i!}(\mathcal{G}|_{U_ i})$ is handled in Lemma 59.84.6. $\square$


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