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 37.43.3) 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GJB. Beware of the difference between the letter 'O' and the digit '0'.