Schreier S Hypothesis Statement

53.62. The Artin-Schreier sequence

Let $p$ be a prime number. Let $S$ be a scheme in characteristic $p$. The Artin-Schreier sequence is the short exact sequence $$ 0 \longrightarrow \underline{\mathbf{Z}/p\mathbf{Z}}_S \longrightarrow \mathbf{G}_{a, S} \xrightarrow{F-1} \mathbf{G}_{a, S} \longrightarrow 0 $$ where $F - 1$ is the map $x \mapsto x^p - x$.

Lemma 53.62.1. Let $p$ be a prime. Let $S$ be a scheme of characteristic $p$.

  1. If $S$ is affine, then $H_{\acute{e}tale}^q(S, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for all $q \geq 2$.
  2. If $S$ is a quasi-compact and quasi-separated scheme of dimension $d$, then $H_{\acute{e}tale}^q(S, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for all $q \geq 2 + d$.

Proof. Recall that the étale cohomology of the structure sheaf is equal to its cohomology on the underlying topological space (Theorem 53.22.4). The first statement follows from the Artin-Schreier exact sequence and the vanishing of cohomology of the structure sheaf on an affine scheme (Cohomology of Schemes, Lemma 29.2.2). The second statement follows by the same argument from the vanishing of Cohomology, Proposition 20.23.4 and the fact that $S$ is a spectral space (Properties, Lemma 27.2.4). $\square$

Lemma 53.62.2. Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $V$ be a finite dimensional $k$-vector space. Let $F : V \to V$ be a frobenius linear map, i.e., an additive map such that $F(\lambda v) = \lambda^p F(v)$ for all $\lambda \in k$ and $v \in V$. Then $F - 1 : V \to V$ is surjective with kernel a finite dimensional $\mathbf{F}_p$-vector space of dimension $\leq \dim_k(V)$.

Proof. If $F = 0$, then the statement holds. If we have a filtration of $V$ by $F$-stable subvector spaces such that the statement holds for each graded piece, then it holds for $(V, F)$. Combining these two remarks we may assume the kernel of $F$ is zero.

Choose a basis $v_1, \ldots, v_n$ of $V$ and write $F(v_i) = \sum a_{ij} v_j$. Observe that $v = \sum \lambda_i v_i$ is in the kernel if and only if $\sum \lambda_i^p a_{ij} v_j = 0$. Since $k$ is algebraically closed this implies the matrix $(a_{ij})$ is invertible. Let $(b_{ij})$ be its inverse. Then to see that $F - 1$ is surjective we pick $w = \sum \mu_i v_i \in V$ and we try to solve $$ (F - 1)(\sum \lambda_iv_i) = \sum \lambda_i^p a_{ij} v_j - \sum \lambda_j v_j = \sum \mu_j v_j $$ This is equivalent to $$ \sum \lambda_j^p v_j - \sum b_{ij} \lambda_i v_j = \sum b_{ij} \mu_i v_j $$ in other words $$ \lambda_j^p - \sum b_{ij} \lambda_i = \sum b_{ij} \mu_i, \quad j = 1, \ldots, \dim(V). $$ The algebra $$ A = k[x_1, \ldots, x_n]/ (x_j^p - \sum b_{ij} x_i - \sum b_{ij} \mu_i) $$ is standard smooth over $k$ (Algebra, Definition 10.135.6) because the matrix $(b_{ij})$ is invertible and the partial derivatives of $x_j^p$ are zero. A basis of $A$ over $k$ is the set of monomials $x_1^{e_1} \ldots x_n^{e_n}$ with $e_i < p$, hence $\dim_k(A) = p^n$. Since $k$ is algebraically closed we see that $\mathop{\mathrm{Spec}}(A)$ has exactly $p^n$ points. It follows that $F - 1$ is surjective and every fibre has $p^n$ points, i.e., the kernel of $F - 1$ is a group with $p^n$ elements. $\square$

Lemma 53.62.3. Let $X$ be a separated scheme of finite type over a field $k$. Let $\mathcal{F}$ be a coherent sheaf of $\mathcal{O}_X$-modules. Then $\dim_k H^d(X, \mathcal{F}) < \infty$ where $d = \dim(X)$.

Proof. We will prove this by induction on $d$. The case $d = 0$ holds because in that case $X$ is the spectrum of a finite dimensional $k$-algebra $A$ (Varieties, Lemma 32.20.2) and every coherent sheaf $\mathcal{F}$ corresponds to a finite $A$-module $M = H^0(X, \mathcal{F})$ which has $\dim_k M < \infty$.

Assume $d > 0$ and the result has been shown for separated schemes of finite type of dimension $< d$. The scheme $X$ is Noetherian. Consider the property $\mathcal{P}$ of coherent sheaves on $X$ defined by the rule $$ \mathcal{P}(\mathcal{F}) \Leftrightarrow \dim_k H^d(X, \mathcal{F}) < \infty $$ We are going to use the result of Cohomology of Schemes, Lemma 29.12.4 to prove that $\mathcal{P}$ holds for every coherent sheaf on $X$.

Let $$ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 $$ be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of cohomology $$ H^d(X, \mathcal{F}_1) \to H^d(X, \mathcal{F}) \to H^d(X, \mathcal{F}_2) $$ Thus if $\mathcal{P}$ holds for $\mathcal{F}_1$ and $\mathcal{F}_2$, then it hods for $\mathcal{F}$.

Let $Z \subset X$ be an integral closed subscheme. Let $\mathcal{I}$ be a coherent sheaf of ideals on $Z$. To finish the proof have to show that $H^d(X, i_*\mathcal{I}) = H^d(Z, \mathcal{I})$ is finite dimensional. If $\dim(Z) < d$, then the result holds because the cohomology group will be zero (Cohomology, Proposition 20.21.7). In this way we reduce to the situation discussed in the following paragraph.

Assume $X$ is a variety of dimension $d$ and $\mathcal{F} = \mathcal{I}$ is a coherent ideal sheaf. In this case we have a short exact sequence $$ 0 \to \mathcal{I} \to \mathcal{O}_X \to i_*\mathcal{O}_Z \to 0 $$ where $i : Z \to X$ is the closed subscheme defined by $\mathcal{I}$. By induction hypothesis we see that $H^{d - 1}(Z, \mathcal{O}_Z) = H^{d - 1}(X, i_*\mathcal{O}_Z)$ is finite dimensional. Thus we see that it suffices to prove the result for the structure sheaf.

We can apply Chow's lemma (Cohomology of Schemes, Lemma 29.18.1) to the morphism $X \to \mathop{\mathrm{Spec}}(k)$. Thus we get a diagram $$ \xymatrix{ X \ar[rd]_g & X' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_k \ar[dl] \\ & \mathop{\mathrm{Spec}}(k) & } $$ as in the statement of Chow's lemma. Also, let $U \subset X$ be the dense open subscheme such that $\pi^{-1}(U) \to U$ is an isomorphism. We may assume $X'$ is a variety as well, see Cohomology of Schemes, Remark 29.18.2. The morphism $i' = (i, \pi) : X' \to \mathbf{P}^n_X$ is a closed immersion (loc. cit.). Hence $$ \mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_k}(1) \cong (i')^*\mathcal{O}_{\mathbf{P}^n_X}(1) $$ is $\pi$-relatively ample (for example by Morphisms, Lemma 28.37.7). Hence by Cohomology of Schemes, Lemma 29.16.2 there exists an $n \geq 0$ such that $R^p\pi_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$. Choose any nonzero global section $s$ of $\mathcal{L}^{\otimes n}$. Since $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$, the section $s$ corresponds to section of $\mathcal{G}$, i.e., a map $\mathcal{O}_X \to \mathcal{G}$. Since $s|_U \not = 0$ as $X'$ is a variety and $\mathcal{L}$ invertible, we see that $\mathcal{O}_X|_U \to \mathcal{G}|_U$ is nonzero. As $\mathcal{G}|_U = \mathcal{KL}^{\otimes n}|_{\pi^{-1}(U)}$ is invertible we conclude that we have a short exact sequence $$ 0 \to \mathcal{O}_X \to \mathcal{G} \to \mathcal{Q} \to 0 $$ where $\mathcal{Q}$ is coherent and supported on a proper closed subscheme of $X$. Arguing as before using our induction hypothesis, we see that it suffices to prove $\dim H^d(X, \mathcal{G}) < \infty$.

By the Leray spectral sequence (Cohomology, Lemma 20.14.6) we see that $H^d(X, \mathcal{G}) = H^d(X', \mathcal{L}^{\otimes n})$. Let $\overline{X}' \subset \mathbf{P}^n_k$ be the closure of $X'$. Then $\overline{X}'$ is a projective variety of dimension $d$ over $k$ and $X' \subset \overline{X}'$ is a dense open. The invertible sheaf $\mathcal{L}$ is the restriction of $\mathcal{O}_{\overline{X}'}(n)$ to $X$. By Cohomology, Proposition 20.23.4 the map $$ H^d(\overline{X}', \mathcal{O}_{\overline{X}'}(n)) \longrightarrow H^d(X', \mathcal{L}^{\otimes n}) $$ is surjective. Since the cohomology group on the left has finite dimension by Cohomology of Schemes, Lemma 29.14.1 the proof is complete. $\square$

Lemma 53.62.4. Let $X$ be separated of finite type over an algebraically closed field $k$ of characteristic $p > 0$. Then $H_{\acute{e}tale}^q(X, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for $q \geq dim(X) + 1$.

Proof. Let $d = \dim(X)$. By the vanishing established in Lemma 53.62.1 it suffices to show that $H_{\acute{e}tale}^{d + 1}(X, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$. By Lemma 53.62.3 we see that $H^d(X, \mathcal{O}_X)$ is a finite dimensional $k$-vector space. Hence the long exact cohomology sequence associated to the Artin-Schreier sequence ends with $$ H^d(X, \mathcal{O}_X) \xrightarrow{F - 1} H^d(X, \mathcal{O}_X) \to H^{d + 1}_{\acute{e}tale}(X, \mathbf{Z}/p\mathbf{Z}) \to 0 $$ By Lemma 53.62.2 the map $F - 1$ in this sequence is surjective. This proves the lemma. $\square$

Lemma 53.62.5. Let $X$ be a proper scheme over an algebraically closed field $k$ of characteristic $p > 0$. Then

  1. $H_{\acute{e}tale}^q(X, \underline{\mathbf{Z}/p\mathbf{Z}})$ is a finite $\mathbf{Z}/p\mathbf{Z}$-module for all $q$, and
  2. $H^q_{\acute{e}tale}(X, \underline{\mathbf{Z}/p\mathbf{Z}}) \to H^q_{\acute{e}tale}(X_{k'}, \underline{\mathbf{Z}/p\mathbf{Z}}))$ is an isomorphism if $k \subset k'$ is an extension of algebraically closed fields.

Proof. By Cohomology of Schemes, Lemma 29.19.2) and the comparison of cohomology of Theorem 53.22.4 the cohomology groups $H^q_{\acute{e}tale}(X, \mathbf{G}_a) = H^q(X, \mathcal{O}_X)$ are finite dimensional $k$-vector spaces. Hence by Lemma 53.62.2 the long exact cohomology sequence associated to the Artin-Schreier sequence, splits into short exact sequences $$ 0 \to H_{\acute{e}tale}^q(X, \underline{\mathbf{Z}/p\mathbf{Z}}) \to H^q(X, \mathcal{O}_X) \xrightarrow{F - 1} H^q(X, \mathcal{O}_X) \to 0 $$ and moreover the $\mathbf{F}_p$-dimension of the cohomology groups $H_{\acute{e}tale}^q(X, \underline{\mathbf{Z}/p\mathbf{Z}})$ is equal to the $k$-dimension of the vector space $H^q(X, \mathcal{O}_X)$. This proves the first statement. The second statement follows as $H^q(X, \mathcal{O}_X) \otimes_k k' \to H^q(X_{k'}, \mathcal{O}_{X_{k'}})$ is an isomorphism by flat base change (Cohomology of Schemes, Lemma 29.5.2). $\square$

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