The Stacks project

Proof. Recall that the étale cohomology of the structure sheaf is equal to its cohomology on the underlying topological space (Theorem 59.22.4). 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 30.2.2). The second statement follows by the same argument from the vanishing of Cohomology, Proposition 20.22.4 and the fact that $S$ is a spectral space (Properties, Lemma 28.2.4). $\square$

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

This is equivalent to

in other words

The algebra

is standard smooth over $k$ (Algebra, Definition 10.137.6) 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. $\square$

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 33.20.2) 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

We are going to use the result of Cohomology of Schemes, Lemma 30.12.4 to prove that $\mathcal{P}$ holds for every coherent sheaf on $X$.

Let

be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of cohomology

Thus if $\mathcal{P}$ holds for $\mathcal{F}_1$ and $\mathcal{F}_2$, then it holds 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 we 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 20.20.7). 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

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 30.18.1) to the morphism $X \to \mathop{\mathrm{Spec}}(k)$. Thus we get a diagram

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 30.18.2. The morphism $i' = (i, \pi ) : X' \to \mathbf{P}^ n_ X$ is a closed immersion (loc. cit.). Hence

is $\pi $-relatively ample (for example by Morphisms, Lemma 29.39.7). Hence by Cohomology of Schemes, Lemma 30.16.2 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 $\mathcal{G}|_ U = \mathcal{L}^{\otimes n}|_{\pi ^{-1}(U)}$ is invertible we conclude that we have a short exact sequence

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 20.13.6) 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 20.22.4 the map

is surjective. Since the cohomology group on the left has finite dimension by Cohomology of Schemes, Lemma 30.14.1 the proof is complete. $\square$

Proof. Let $d = \dim (X)$. By the vanishing established in Lemma 59.63.1 it suffices to show that $H_{\acute{e}tale}^{d + 1}(X, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$. By Lemma 59.63.3 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

By Lemma 59.63.2 the map $F - 1$ in this sequence is surjective. This proves the lemma. $\square$

Proof. By Cohomology of Schemes, Lemma 30.19.2) and the comparison of cohomology of Theorem 59.22.4 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 59.63.2 the long exact cohomology sequence associated to the Artin-Schreier sequence, splits into short exact sequences

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 30.5.2). $\square$