## 59.67 Galois cohomology

In this section we prove a result on Galois cohomology (Proposition 59.67.4) using étale cohomology and the trick from Section 59.66. This will allow us to prove vanishing of higher étale cohomology groups over the spectrum of a field.

Lemma 59.67.1. Let $\ell $ be a prime number and $n$ an integer $> 0$. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be the limit of a directed system of $S$-schemes each $X_ i \to S$ being finite étale of constant degree relatively prime to $\ell $. The following are equivalent:

there exists an $\ell $-power torsion sheaf $\mathcal{G}$ on $S$ such that $H_{\acute{e}tale}^ n(S, \mathcal{G}) \neq 0$ and

there exists an $\ell $-power torsion sheaf $\mathcal{F}$ on $X$ such that $H_{\acute{e}tale}^ n(X, \mathcal{F}) \neq 0$.

In fact, given $\mathcal{G}$ we can take $\mathcal{F} = g^{-1}\mathcal{F}$ and given $\mathcal{F}$ we can take $\mathcal{G} = g_*\mathcal{F}$.

**Proof.**
Let $g : X \to S$ and $g_ i : X_ i \to S$ denote the structure morphisms. Fix an $\ell $-power torsion sheaf $\mathcal{G}$ on $S$ with $H^ n_{\acute{e}tale}(S, \mathcal{G}) \not= 0$. The system given by $\mathcal{G}_ i = g_ i^{-1}\mathcal{G}$ satisify the conditions of Theorem 59.51.3 with colimit sheaf given by $g^{-1}\mathcal{G}$. This tells us that:

\[ \mathop{\mathrm{colim}}\nolimits _{i\in I} H^ n_{\acute{e}tale}(X_ i, g_ i^{-1}\mathcal{G}) = H^ n_{\acute{e}tale}(X, \mathcal{G}) \]

By virtue of the $g_ i$ being finite étale morphism of degree prime to $\ell $ we can apply “la méthode de la trace” and we find the maps

\[ H^ n_{\acute{e}tale}(S, \mathcal{G}) \to H^ n_{\acute{e}tale}(X_ i, g_ i^{-1}\mathcal{G}) \]

are all injective (and compatible with the transition maps). See Section 59.66. Thus, the colimit is non-zero, i.e., $H^ n(X,g^{-1}\mathcal{G}) \neq 0$, giving us the desired result with $\mathcal{F} = g^{-1}\mathcal{G}$.

Conversely, suppose given an $\ell $-power torsion sheaf $\mathcal{F}$ on $X$ with $H^ n_{\acute{e}tale}(X, \mathcal{F}) \not= 0$. We note that since the $g_ i$ are finite morphisms the higher direct images vanish (Proposition 59.55.2). Then, by applying Lemma 59.51.7 we may also conclude the same for $g$. The vanishing of the higher direct images tells us that $H^ n_{\acute{e}tale}(X, \mathcal{F}) = H^ n(S, g_*\mathcal{F}) \neq 0$ by Leray (Proposition 59.54.2) giving us what we want with $\mathcal{G} = g_*\mathcal{F}$.
$\square$

Lemma 59.67.2. Let $\ell $ be a prime number and $n$ an integer $> 0$. Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let $H \subset G$ be a maximal pro-$\ell $ subgroup with $L/K$ being the corresponding field extension. Then $H^ n_{\acute{e}tale}(\mathop{\mathrm{Spec}}(K), \mathcal{F}) = 0$ for all $\ell $-power torsion $\mathcal{F}$ if and only if $H^ n_{\acute{e}tale}(\mathop{\mathrm{Spec}}(L), \underline{\mathbf{Z}/\ell \mathbf{Z}}) = 0$.

**Proof.**
Write $L = \bigcup L_ i$ as the union of its finite subextensions over $K$. Our choice of $H$ implies that $[L_ i : K]$ is prime to $\ell $. Thus $\mathop{\mathrm{Spec}}(L) = \mathop{\mathrm{lim}}\nolimits _{i \in I} \mathop{\mathrm{Spec}}(L_ i)$ as in Lemma 59.67.1. Thus we may replace $K$ by $L$ and assume that the absolute Galois group $G$ of $K$ is a profinite pro-$\ell $ group.

Assume $H^ n(\mathop{\mathrm{Spec}}(K), \underline{\mathbf{Z}/\ell \mathbf{Z}}) = 0$. Let $\mathcal{F}$ be an $\ell $-power torsion sheaf on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$. We will show that $H^ n_{\acute{e}tale}(\mathop{\mathrm{Spec}}(K), \mathcal{F}) = 0$. By the correspondence specified in Lemma 59.59.1 our sheaf $\mathcal{F}$ corresponds to an $\ell $-power torsion $G$-module $M$. Any finite set of elements $x_1, \ldots , x_ m \in M$ must be fixed by an open subgroup $U$ by continuity. Let $M'$ be the module spanned by the orbits of $x_1, \ldots , x_ m$. This is a finite abelian $\ell $-group as each $x_ i$ is killed by a power of $\ell $ and the orbits are finite. Since $M$ is the filtered colimit of these submodules $M'$, we see that $\mathcal{F}$ is the filtered colimit of the corresponding subsheaves $\mathcal{F}' \subset \mathcal{F}$. Applying Theorem 59.51.3 to this colimit, we reduce to the case where $\mathcal{F}$ is a finite locally constant sheaf.

Let $M$ be a finite abelian $\ell $-group with a continuous action of the profinite pro-$\ell $ group $G$. Then there is a $G$-invariant filtration

\[ 0 = M_0 \subset M_1 \subset \ldots \subset M_ r = M \]

such that $M_{i + 1}/M_ i \cong \mathbf{Z}/\ell \mathbf{Z}$ with trivial $G$-action (this is a simple lemma on representation theory of finite groups; insert future reference here). Thus the corresponding sheaf $\mathcal{F}$ has a filtration

\[ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ r = \mathcal{F} \]

with successive quotients isomorphic to $\underline{\mathbf{Z}/\ell \mathbf{Z}}$. Thus by induction and the long exact cohomology sequence we conclude.
$\square$

Lemma 59.67.3. Let $\ell $ be a prime number and $n$ an integer $> 0$. Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let $H \subset G$ be a maximal pro-$\ell $ subgroup with $L/K$ being the corresponding field extension. Then $H^ q_{\acute{e}tale}(\mathop{\mathrm{Spec}}(K),\mathcal{F}) = 0$ for $q \geq n$ and all $\ell $-torsion sheaves $\mathcal{F}$ if and only if $H^ n_{\acute{e}tale}(\mathop{\mathrm{Spec}}(L), \underline{\mathbf{Z}/\ell \mathbf{Z}}) = 0$.

**Proof.**
The forward direction is trivial, so we need only prove the reverse direction. We proceed by induction on $q$. The case of $q = n$ is Lemma 59.67.2. Now let $\mathcal{F}$ be an $\ell $-power torsion sheaf on $\mathop{\mathrm{Spec}}(K)$. Let $f : \mathop{\mathrm{Spec}}(K^{sep}) \rightarrow \mathop{\mathrm{Spec}}(K)$ be the inclusion of a geometric point. Then consider the exact sequence:

\[ 0 \rightarrow \mathcal{F} \xrightarrow {res} f_* f^{-1} \mathcal{F} \rightarrow f_* f^{-1} \mathcal{F}/\mathcal{F} \rightarrow 0 \]

Note that $K^{sep}$ may be written as the filtered colimit of finite separable extensions. Thus $f$ is the limit of a directed system of finite étale morphisms. We may, as was seen in the proof of Lemma 59.67.1, conclude that $f$ has vanishing higher direct images. Thus, we may express the higher cohomology of $f_* f^{-1} \mathcal{F}$ as the higher cohomology on the geometric point which clearly vanishes. Hence, as everything here is still $\ell $-torsion, we may use the inductive hypothesis in conjunction with the long-exact cohomology sequence to conclude the result for $q + 1$.
$\square$

reference
Proposition 59.67.4. Let $K$ be a field with separable algebraic closure $K^{sep}$. Assume that for any finite extension $K'$ of $K$ we have $\text{Br}(K') = 0$. Then

$H^ q(\text{Gal}(K^{sep}/K), (K^{sep})^*) = 0$ for all $q \geq 1$, and

$H^ q(\text{Gal}(K^{sep}/K), M) = 0$ for any torsion $\text{Gal}(K^{sep}/K)$-module $M$ and any $q \geq 2$,

**Proof.**
Set $p = \text{char}(K)$. By Lemma 59.59.2, Theorem 59.61.6, and Example 59.59.3 the proposition is equivalent to showing that if $H^2(\mathop{\mathrm{Spec}}(K'),\mathbf{G}_ m|_{\mathop{\mathrm{Spec}}(K')_{\acute{e}tale}}) = 0$ for all finite extensions $K'/K$ then:

$H^ q(\mathop{\mathrm{Spec}}(K),\mathbf{G}_ m|_{\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}}) = 0$ for all $q \geq 1$, and

$H^ q(\mathop{\mathrm{Spec}}(K),\mathcal{F}) = 0$ for any torsion sheaf $\mathcal{F}$ and any $q \geq 2$.

We prove the second part first. Since $\mathcal{F}$ is a torsion sheaf, we may use the $\ell $-primary decomposition as well as the compatibility of cohomology with colimits (i.e, direct sums, see Theorem 59.51.3) to reduce to showing $H^ q(\mathop{\mathrm{Spec}}(K),\mathcal{F}) = 0$, $q \geq 2$ for all $\ell $-power torsion sheaves for every prime $\ell $. This allows us to analyze each prime individually.

Suppose that $\ell \neq p$. For any extension $K'/K$ consider the Kummer sequence (Lemma 59.28.1)

\[ 0 \to \mu _{\ell , \mathop{\mathrm{Spec}}{K'}} \to \mathbf{G}_{m, \mathop{\mathrm{Spec}}{K'}} \xrightarrow {(\cdot )^{\ell }} \mathbf{G}_{m, \mathop{\mathrm{Spec}}{K'}} \to 0 \]

Since $H^ q(\mathop{\mathrm{Spec}}{K'},\mathbf{G}_ m|_{\mathop{\mathrm{Spec}}(K')_{\acute{e}tale}}) = 0$ for $q = 2$ by assumption and for $q = 1$ by Theorem 59.24.1 combined with $\mathop{\mathrm{Pic}}\nolimits (K) = (0)$. Thus, by the long-exact cohomology sequence we may conclude that $H^2(\mathop{\mathrm{Spec}}{K'}, \mu _\ell ) = 0$ for any separable $K'/K$. Now let $H$ be a maximal pro-$\ell $ subgroup of the absolute Galois group of $K$ and let $L$ be the corresponding extension. We can write $L$ as the colimit of finite extensions, applying Theorem 59.51.3 to this colimit we see that $H^2(\mathop{\mathrm{Spec}}(L), \mu _\ell ) = 0$. Now $\mu _\ell $ must be the constant sheaf. If it weren't, that would imply there exists a Galois extension of degree relatively prime to $\ell $ of $L$ which is not true by definition of $L$ (namely, the extension one gets by adjoining the $\ell $th roots of unity to $L$). Hence, via Lemma 59.67.3, we conclude the result for $\ell \neq p$.

Now suppose that $\ell = p$. We consider the Artin-Schrier exact sequence (Section 59.63)

\[ 0 \longrightarrow \underline{\mathbf{Z}/p\mathbf{Z}}_{\mathop{\mathrm{Spec}}{K}} \longrightarrow \mathbf{G}_{a, \mathop{\mathrm{Spec}}{K}} \xrightarrow {F-1} \mathbf{G}_{a, \mathop{\mathrm{Spec}}{K}} \longrightarrow 0 \]

where $F - 1$ is the map $x \mapsto x^ p - x$. Then note that the higher Cohomology of $\mathbf{G}_{a, \mathop{\mathrm{Spec}}{K}}$ vanishes, by Remark 59.23.4 and the vanishing of the higher cohomology of the structure sheaf of an affine scheme (Cohomology of Schemes, Lemma 30.2.2). Note this can be applied to any field of characteristic $p$. In particular, we can apply it to the field extension $L$ defined by a maximal pro-$p$ subgroup $H$. This allows us to conclude $H^ n(\mathop{\mathrm{Spec}}{L}, \underline{\mathbf{Z}/p\mathbf{Z}}_{\mathop{\mathrm{Spec}}{L}}) = 0$ for $n \geq 2$, from which the result follows for $\ell = p$, by Lemma 59.67.3.

To finish the proof we still have to show that $H^ q(\text{Gal}(K^{sep}/K), (K^{sep})^*) = 0$ for all $q \geq 1$. Set $G = \text{Gal}(K^{sep}/K)$ and set $M = (K^{sep})^*$ viewed as a $G$-module. We have already shown (above) that $H^1(G, M) = 0$ and $H^2(G, M) = 0$. Consider the exact sequence

\[ 0 \to A \to M \to M \otimes \mathbf{Q} \to B \to 0 \]

of $G$-modules. By the above we have $H^ i(G, A) = 0$ and $H^ i(G, B) = 0$ for $i > 1$ since $A$ and $B$ are torsion $G$-modules. By Lemma 59.57.6 we have $H^ i(G, M \otimes \mathbf{Q}) = 0$ for $i > 0$. It is a pleasant exercise to see that this implies that $H^ i(G, M) = 0$ also for $i \geq 3$.
$\square$

Definition 59.67.5. A field $K$ is called *$C_ r$* if for every $0 < d^ r < n$ and every $f \in K[T_1, \ldots , T_ n]$ homogeneous of degree $d$, there exist $\alpha = (\alpha _1, \ldots , \alpha _ n)$, $\alpha _ i \in K$ not all zero, such that $f(\alpha ) = 0$. Such an $\alpha $ is called a *nontrivial solution* of $f$.

Example 59.67.6. An algebraically closed field is $C_ r$.

In fact, we have the following simple lemma.

Lemma 59.67.7. Let $k$ be an algebraically closed field. Let $f_1, \ldots , f_ s \in k[T_1, \ldots , T_ n]$ be homogeneous polynomials of degree $d_1, \ldots , d_ s$ with $d_ i > 0$. If $s < n$, then $f_1 = \ldots = f_ s = 0$ have a common nontrivial solution.

**Proof.**
This follows from dimension theory, for example in the form of Varieties, Lemma 33.34.2 applied $s - 1$ times.
$\square$

The following result computes the Brauer group of $C_1$ fields.

Theorem 59.67.8. Let $K$ be a $C_1$ field. Then $\text{Br}(K) = 0$.

**Proof.**
Let $D$ be a finite dimensional division algebra over $K$ with center $K$. We have seen that

\[ D \otimes _ K K^{sep} \cong \text{Mat}_ d(K^{sep}) \]

uniquely up to inner isomorphism. Hence the determinant $\det : \text{Mat}_ d(K^{sep}) \to K^{sep}$ is Galois invariant and descends to a homogeneous degree $d$ map

\[ \det = N_\text {red} : D \longrightarrow K \]

called the *reduced norm*. Since $K$ is $C_1$, if $d > 1$, then there exists a nonzero $x \in D$ with $N_\text {red}(x) = 0$. This clearly implies that $x$ is not invertible, which is a contradiction. Hence $\text{Br}(K) = 0$.
$\square$

Definition 59.67.9. Let $k$ be a field. A *variety* is separated, integral scheme of finite type over $k$. A *curve* is a variety of dimension $1$.

Theorem 59.67.10 (Tsen's theorem). The function field of a variety of dimension $r$ over an algebraically closed field $k$ is $C_ r$.

**Proof.**
For projective space one can show directly that the field $k(x_1, \ldots , x_ r)$ is $C_ r$ (exercise).

General case. Without loss of generality, we may assume $X$ to be projective. Let $f \in k(X)[T_1, \ldots , T_ n]_ d$ with $0 < d^ r < n$. Say the coefficients of $f$ are in $\Gamma (X, \mathcal{O}_ X(H))$ for some ample $H \subset X$. Let $\mathbf{\alpha } = (\alpha _1, \ldots , \alpha _ n)$ with $\alpha _ i \in \Gamma (X, \mathcal{O}_ X(eH))$. Then $f(\mathbf{\alpha }) \in \Gamma (X, \mathcal{O}_ X((de + 1)H))$. Consider the system of equations $f(\mathbf{\alpha }) =0$. Then by asymptotic Riemann-Roch (Varieties, Proposition 33.45.13) there exists a $c > 0$ such that

the number of variables is $n\dim _ k \Gamma (X, \mathcal{O}_ X(eH)) \sim n e^ r c$, and

the number of equations is $\dim _ k \Gamma (X, \mathcal{O}_ X((de + 1)H)) \sim (de + 1)^ r c$.

Since $n > d^ r$, there are more variables than equations. The equations are homogeneous hence there is a solution by Lemma 59.67.7.
$\square$

Lemma 59.67.11. Let $C$ be a curve over an algebraically closed field $k$. Then the Brauer group of the function field of $C$ is zero: $\text{Br}(k(C)) = 0$.

**Proof.**
This is clear from Tsen's theorem, Theorem 59.67.10 and Theorem 59.67.8.
$\square$

Lemma 59.67.12. Let $k$ be an algebraically closed field and $K/k$ a field extension of transcendence degree 1. Then for all $q \geq 1$, $H_{\acute{e}tale}^ q(\mathop{\mathrm{Spec}}(K), \mathbf{G}_ m) = 0$.

**Proof.**
Recall that $H_{\acute{e}tale}^ q(\mathop{\mathrm{Spec}}(K), \mathbf{G}_ m) = H^ q(\text{Gal}(K^{sep}/K), (K^{sep})^*)$ by Lemma 59.59.2. Thus by Proposition 59.67.4 it suffices to show that if $K'/K$ is a finite field extension, then $\text{Br}(K') = 0$. Now observe that $K' = \mathop{\mathrm{colim}}\nolimits K''$, where $K''$ runs over the finitely generated subextensions of $k$ contained in $K'$ of transcendence degree $1$. Note that $\text{Br}(K') = \mathop{\mathrm{colim}}\nolimits \text{Br}(K'')$ which reduces us to a finitely generated field extension $K''/k$ of transcendence degree $1$. Such a field is the function field of a curve over $k$, hence has trivial Brauer group by Lemma 59.67.11.
$\square$

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