Situation 59.83.1. Here $k$ is an algebraically closed field, $X$ is a separated, finite type scheme of dimension $\leq 1$ over $k$, and $\mathcal{F}$ is a torsion abelian sheaf on $X_{\acute{e}tale}$.
59.83 Cohomology of torsion sheaves on curves
The goal of this section is to prove the basic finiteness and vanishing results for cohomology of torsion sheaves on curves, see Theorem 59.83.10. In Section 59.84 we will discuss constructible sheaves of torsion modules over a Noetherian ring.
In Situation 59.83.1 we want to prove the following statements
$H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 2$,
$H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 1$ if $X$ is affine,
$H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 1$ if $p = \text{char}(k) > 0$ and $\mathcal{F}$ is $p$-power torsion,
$H^ q_{\acute{e}tale}(X, \mathcal{F})$ is finite if $\mathcal{F}$ is constructible and torsion prime to $\text{char}(k)$,
$H^ q_{\acute{e}tale}(X, \mathcal{F})$ is finite if $X$ is proper and $\mathcal{F}$ constructible,
$H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k'/k$ of algebraically closed fields if $\mathcal{F}$ is torsion prime to $\text{char}(k)$,
$H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k'/k$ of algebraically closed fields if $X$ is proper,
$H^2_{\acute{e}tale}(X, \mathcal{F}) \to H^2_{\acute{e}tale}(U, \mathcal{F})$ is surjective for all $U \subset X$ open.
Given any Situation 59.83.1 we will say that “statements (1) – (8) hold” if those statements that apply to the given situation are true. We start the proof with the following consequence of our computation of cohomology with constant coefficients.
Lemma 59.83.2. In Situation 59.83.1 assume $X$ is smooth and $\mathcal{F} = \underline{\mathbf{Z}/\ell \mathbf{Z}}$ for some prime number $\ell $. Then statements (1) – (8) hold for $\mathcal{F}$.
Proof. Since $X$ is smooth, we see that $X$ is a finite disjoint union of smooth curves. Hence we may assume $X$ is a smooth curve.
Case I: $\ell $ different from the characteristic of $k$. This case follows from Lemma 59.69.1 (projective case) and Lemma 59.69.3 (affine case). Statement (6) on cohomology and extension of algebraically closed ground field follows from the fact that the genus $g$ and the number of “punctures” $r$ do not change when passing from $k$ to $k'$. Statement (8) follows as $H^2_{\acute{e}tale}(U, \mathcal{F})$ is zero as soon as $U \not= X$, because then $U$ is affine (Varieties, Lemmas 33.43.2 and 33.43.10).
Case II: $\ell $ is equal to the characteristic of $k$. Vanishing by Lemma 59.63.4. Statements (5) and (7) follow from Lemma 59.63.5. $\square$
Remark 59.83.3 (Invariance under extension of algebraically closed ground field). Let $k$ be an algebraically closed field of characteristic $p > 0$. In Section 59.63 we have seen that there is an exact sequence where the first arrow maps $f(x)$ to $f^ p - f$. A set of representatives for the cokernel is formed by the polynomials with $\lambda _ n \in k$. (If $k$ is not algebraically closed you have to add some constants to this as well.) In particular when $k'/k$ is an algebraically closed extension, then the map is not an isomorphism in general. In particular, the map $\pi _1(\mathbf{A}^1_{k'}) \to \pi _1(\mathbf{A}^1_ k)$ between étale fundamental groups (insert future reference here) is not an isomorphism either. Thus the étale homotopy type of the affine line depends on the algebraically closed ground field. From Lemma 59.83.2 above we see that this is a phenomenon which only happens in characteristic $p$ with $p$-power torsion coefficients.
\[ k[x] \to k[x] \to H^1_{\acute{e}tale}(\mathbf{A}^1_ k, \mathbf{Z}/p\mathbf{Z}) \to 0 \]
\[ \sum \nolimits _{p \not| n} \lambda _ n x^ n \]
\[ H^1_{\acute{e}tale}(\mathbf{A}^1_ k, \mathbf{Z}/p\mathbf{Z}) \to H^1_{\acute{e}tale}(\mathbf{A}^1_{k'}, \mathbf{Z}/p\mathbf{Z}) \]
Lemma 59.83.4. Let $k$ be an algebraically closed field. Let $X$ be a separated finite type scheme over $k$ of dimension $\leq 1$. Let $0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0$ be a short exact sequence of torsion abelian sheaves on $X$. If statements (1) – (8) hold for $\mathcal{F}_1$ and $\mathcal{F}_2$, then they hold for $\mathcal{F}$.
Proof. This is mostly immediate from the definitions and the long exact sequence of cohomology. Also observe that $\mathcal{F}$ is constructible (resp. of torsion prime to the characteristic of $k$) if and only if both $\mathcal{F}_1$ and $\mathcal{F}_2$ are constructible (resp. of torsion prime to the characteristic of $k$). See Proposition 59.74.1. Some details omitted. $\square$
Lemma 59.83.5. 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 $\leq 1$. Let $\mathcal{F}$ be a torsion abelian sheaf on $X$. If statements (1) – (8) hold for $\mathcal{F}$, then they hold for $f_*\mathcal{F}$.
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). For (8) use that formation of $f_*$ commutes with arbitrary base change (Lemma 59.55.3). $\square$
Lemma 59.83.6. In Situation 59.83.1 assume $\mathcal{F}$ constructible. Let $j : X' \to X$ be the inclusion of a dense open subscheme. Then statements (1) – (8) hold for $\mathcal{F}$ if and only if they hold for $j_!j^{-1}\mathcal{F}$.
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 finite as $\mathcal{F}_{x_ i}$ is finite due to the assumption that $\mathcal{F}$ is constructible. 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 $H^ q_{\acute{e}tale}(X, j_!j^{-1}\mathcal{F}) \to H^ q_{\acute{e}tale}(X, \mathcal{F})$ for $q > 1$.
At this point it is easy to deduce each of (1) – (8) holds for $\mathcal{F}$ if and only if it holds for $j_!j^{-1}\mathcal{F}$. We make a few small remarks to help the reader: (a) if $\mathcal{F}$ is torsion prime to the characteristic of $k$, then so is $j_!j^{-1}\mathcal{F}$, (b) the sheaf $j_!j^{-1}\mathcal{F}$ is constructible, (c) we have $H^0_{\acute{e}tale}(Z, i^{-1}\mathcal{F}) = H^0_{\acute{e}tale}(Z_{k'}, i^{-1}\mathcal{F}|_{Z_{k'}})$, and (d) if $U \subset X$ is an open, then $U' = U \cap X'$ is dense in $U$. $\square$
Lemma 59.83.7. In Situation 59.83.1 assume $X$ is smooth. Let $j : U \to X$ an open immersion. Let $\ell $ be a prime number. Let $\mathcal{F} = j_!\underline{\mathbf{Z}/\ell \mathbf{Z}}$. Then statements (1) – (8) hold for $\mathcal{F}$.
Proof. Since $X$ is smooth, it is a disjoint union of smooth curves and hence we may assume $X$ is a curve (i.e., irreducible). Then either $U = \emptyset $ and there is nothing to prove or $U \subset X$ is dense. In this case the lemma follows from Lemmas 59.83.2 and 59.83.6. $\square$
Lemma 59.83.8. In Situation 59.83.1 assume $X$ reduced. Let $j : U \to X$ an open immersion. Let $\ell $ be a prime number and $\mathcal{F} = j_!\underline{\mathbf{Z}/\ell \mathbf{Z}}$. Then statements (1) – (8) hold for $\mathcal{F}$.
Proof. The difference with Lemma 59.83.7 is that here we do not assume $X$ is smooth. Let $\nu : X^\nu \to X$ be the normalization morphism. Then $\nu $ is finite (Varieties, Lemma 33.27.1) and $X^\nu $ is smooth (Varieties, Lemma 33.43.8). Let $j^\nu : U^\nu \to X^\nu $ be the inverse image of $U$. By Lemma 59.83.7 the result holds for $j^\nu _!\underline{\mathbf{Z}/\ell \mathbf{Z}}$. By Lemma 59.83.5 the result holds for $\nu _*j^\nu _!\underline{\mathbf{Z}/\ell \mathbf{Z}}$. In general it won't be true that $\nu _*j^\nu _!\underline{\mathbf{Z}/\ell \mathbf{Z}}$ is equal to $j_!\underline{\mathbf{Z}/\ell \mathbf{Z}}$ but we can work around this as follows. As $X$ is reduced the morphism $\nu : X^\nu \to X$ is an isomorphism over a dense open $j' : X' \to X$ (Varieties, Lemma 33.27.1). Over this open we have agreement
\[ (j')^{-1}(\nu _*j^\nu _!\underline{\mathbf{Z}/\ell \mathbf{Z}}) = (j')^{-1}(j_!\underline{\mathbf{Z}/\ell \mathbf{Z}}) \]
Using Lemma 59.83.6 twice for $j' : X' \to X$ and the sheaves above we conclude. $\square$
Lemma 59.83.9. In Situation 59.83.1 assume $X$ reduced. Let $j : U \to X$ an open immersion with $U$ connected. Let $\ell $ be a prime number. Let $\mathcal{G}$ a finite locally constant sheaf of $\mathbf{F}_\ell $-vector spaces on $U$. Let $\mathcal{F} = j_!\mathcal{G}$. Then statements (1) – (8) hold for $\mathcal{F}$.
Proof. Let $f : V \to U$ be a finite étale morphism of degree prime to $\ell $ as in Lemma 59.66.2. 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 lemma with $\mathcal{F} = j_!f_*f^{-1}\mathcal{G}$. 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. We may replace $Y$ by its reduction (this does not change $V$ as $V$ is reduced being étale over $U$). 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 59.83.5 it suffices to consider $j'_!f^{-1}\mathcal{G}$. The existence of the filtration given by Lemma 59.66.2, the fact that $j'_!$ is exact, and Lemma 59.83.4 reduces us to the case $\mathcal{F} = j'_!\underline{\mathbf{Z}/\ell \mathbf{Z}}$ which is Lemma 59.83.8. $\square$
Theorem 59.83.10. If $k$ is an algebraically closed field, $X$ is a separated, finite type scheme of dimension $\leq 1$ over $k$, and $\mathcal{F}$ is a torsion abelian sheaf on $X_{\acute{e}tale}$, then
$H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 2$,
$H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 1$ if $X$ is affine,
$H^ q_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $q > 1$ if $p = \text{char}(k) > 0$ and $\mathcal{F}$ is $p$-power torsion,
$H^ q_{\acute{e}tale}(X, \mathcal{F})$ is finite if $\mathcal{F}$ is constructible and torsion prime to $\text{char}(k)$,
$H^ q_{\acute{e}tale}(X, \mathcal{F})$ is finite if $X$ is proper and $\mathcal{F}$ constructible,
$H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k'/k$ of algebraically closed fields if $\mathcal{F}$ is torsion prime to $\text{char}(k)$,
$H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k'/k$ of algebraically closed fields if $X$ is proper,
$H^2_{\acute{e}tale}(X, \mathcal{F}) \to H^2_{\acute{e}tale}(U, \mathcal{F})$ is surjective for all $U \subset X$ open.
Proof. The theorem says that in Situation 59.83.1 statements (1) – (8) hold. Our first step is to replace $X$ by its reduction, which is permissible by Proposition 59.45.4. By Lemma 59.73.2 we can write $\mathcal{F}$ as a filtered colimit of constructible abelian sheaves. Taking cohomology commutes with colimits, see Lemma 59.51.4. Moreover, pullback via $X_{k'} \to X$ commutes with colimits as a left adjoint. Thus it suffices to prove the statements for a constructible sheaf.
In this paragraph we use Lemma 59.83.4 without further mention. Writing $\mathcal{F} = \mathcal{F}_1 \oplus \ldots \oplus \mathcal{F}_ r$ where $\mathcal{F}_ i$ is $\ell _ i$-primary for some prime $\ell _ i$, we may assume that $\ell ^ n$ kills $\mathcal{F}$ for some prime $\ell $. Now consider the exact sequence
\[ 0 \to \mathcal{F}[\ell ] \to \mathcal{F} \to \mathcal{F}/\mathcal{F}[\ell ] \to 0. \]
Thus we see that it suffices to assume that $\mathcal{F}$ is $\ell $-torsion. This means that $\mathcal{F}$ is a constructible sheaf of $\mathbf{F}_\ell $-vector spaces for some prime number $\ell $.
By definition this means there is a dense open $U \subset X$ such that $\mathcal{F}|_ U$ is finite locally constant sheaf of $\mathbf{F}_\ell $-vector spaces. 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 59.83.6 we reduce to the case $\mathcal{F} = j_!\mathcal{G}$ where $\mathcal{G}$ is a finite locally constant sheaf of $\mathbf{F}_\ell $-vector spaces on $U$.
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 59.83.9. $\square$
Theorem 59.83.11. Let $X$ be a finite type, dimension $1$ scheme over an algebraically closed field $k$. Let $\mathcal{F}$ be a torsion sheaf on $X_{\acute{e}tale}$. Then If $X$ affine then also $H_{\acute{e}tale}^2(X, \mathcal{F}) = 0$.
\[ H_{\acute{e}tale}^ q(X, \mathcal{F}) = 0, \quad \forall q \geq 3. \]
Proof. If $X$ is separated, this follows immediately from the more precise Theorem 59.83.10. If $X$ is nonseparated, choose an affine open covering $X = X_1 \cup \ldots \cup X_ n$. By induction on $n$ we may assume the vanishing holds over $U = X_1 \cup \ldots \cup X_{n - 1}$. Then Mayer-Vietoris (Lemma 59.50.1) gives
\[ H^2_{\acute{e}tale}(U, \mathcal{F}) \oplus H^2_{\acute{e}tale}(X_ n, \mathcal{F}) \to H^2_{\acute{e}tale}(U \cap X_ n, \mathcal{F}) \to H^3_{\acute{e}tale}(X, \mathcal{F}) \to 0 \]
However, since $U \cap X_ n$ is an open of an affine scheme and hence affine by our dimension assumption, the group $H^2_{\acute{e}tale}(U \cap X_ n, \mathcal{F})$ vanishes by Theorem 59.83.10. $\square$
Lemma 59.83.12. Let $k'/k$ be an extension of separably closed fields. Let $X$ be a proper scheme over $k$ of dimension $\leq 1$. Let $\mathcal{F}$ be a torsion abelian sheaf on $X$. Then the map $H^ q_{\acute{e}tale}(X, \mathcal{F}) \to H^ q_{\acute{e}tale}(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for $q \geq 0$.
Proof. We have seen this for algebraically closed fields in Theorem 59.83.10. Given $k \subset k'$ as in the statement of the lemma we can choose a diagram
\[ \xymatrix{ k' \ar[r] & \overline{k}' \\ k \ar[u] \ar[r] & \overline{k} \ar[u] } \]
where $k \subset \overline{k}$ and $k' \subset \overline{k}'$ are the algebraic closures. Since $k$ and $k'$ are separably closed the field extensions $\overline{k}/k$ and $\overline{k}'/k'$ are algebraic and purely inseparable. In this case the morphisms $X_{\overline{k}} \to X$ and $X_{\overline{k}'} \to X_{k'}$ are universal homeomorphisms. Thus the cohomology of $\mathcal{F}$ may be computed on $X_{\overline{k}}$ and the cohomology of $\mathcal{F}|_{X_{k'}}$ may be computed on $X_{\overline{k}'}$, see Proposition 59.45.4. Hence we deduce the general case from the case of algebraically closed fields. $\square$
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