Section 61.22 (09B3): Cohomology of a point

61.22 Cohomology of a point

Let $\Lambda $ be a Noetherian ring complete with respect to an ideal $I \subset \Lambda $. Let $k$ be a field. In this section we “compute”

\[ H^ i(\mathop{\mathrm{Spec}}(k)_{pro\text{-}\acute{e}tale}, \underline{\Lambda }^\wedge ) \]

where $\underline{\Lambda }^\wedge = \mathop{\mathrm{lim}}\nolimits _ m \underline{\Lambda /I^ m}$ as before. Let $k^{sep}$ be a separable algebraic closure of $k$. Then

\[ \mathcal{U} = \{ \mathop{\mathrm{Spec}}(k^{sep}) \to \mathop{\mathrm{Spec}}(k)\} \]

is a pro-étale covering of $\mathop{\mathrm{Spec}}(k)$. We will use the Čech to cohomology spectral sequence with respect to this covering. Set $U_0 = \mathop{\mathrm{Spec}}(k^{sep})$ and

\begin{align*} U_ n & = \mathop{\mathrm{Spec}}(k^{sep}) \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k^{sep}) \times _{\mathop{\mathrm{Spec}}(k)} \ldots \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k^{sep}) \\ & = \mathop{\mathrm{Spec}}(k^{sep} \otimes _ k k^{sep} \otimes _ k \ldots \otimes _ k k^{sep}) \end{align*}

($n + 1$ factors). Note that the underlying topological space $|U_0|$ of $U_0$ is a singleton and for $n \geq 1$ we have

\[ |U_ n| = G \times \ldots \times G\quad (n\text{ factors}) \]

as profinite spaces where $G = \text{Gal}(k^{sep}/k)$. Namely, every point of $U_ n$ has residue field $k^{sep}$ and we identify $(\sigma _1, \ldots , \sigma _ n)$ with the point corresponding to the surjection

\[ k^{sep} \otimes _ k k^{sep} \otimes _ k \ldots \otimes _ k k^{sep} \longrightarrow k^{sep}, \quad \lambda _0 \otimes \lambda _1 \otimes \ldots \lambda _ n \longmapsto \lambda _0 \sigma _1(\lambda _1) \ldots \sigma _ n(\lambda _ n) \]

Then we compute

\begin{align*} R\Gamma ((U_ n)_{pro\text{-}\acute{e}tale}, \underline{\Lambda }^\wedge ) & = R\mathop{\mathrm{lim}}\nolimits _ m R\Gamma ((U_ n)_{pro\text{-}\acute{e}tale}, \underline{\Lambda /I^ m}) \\ & = R\mathop{\mathrm{lim}}\nolimits _ m R\Gamma ((U_ n)_{\acute{e}tale}, \underline{\Lambda /I^ m}) \\ & = \mathop{\mathrm{lim}}\nolimits _ m H^0(U_ n, \underline{\Lambda /I^ m}) \\ & = \text{Maps}_{cont}(G \times \ldots \times G, \Lambda ) \end{align*}

The first equality because $R\Gamma $ commutes with derived limits and as $\Lambda ^\wedge $ is the derived limit of the sheaves $\underline{\Lambda /I^ m}$ by Proposition 61.21.1. The second equality by Lemma 61.19.6. The third equality by Étale Cohomology, Lemma 59.80.3. The fourth equality uses Étale Cohomology, Remark 59.23.2 to identify sections of the constant sheaf $\underline{\Lambda /I^ m}$. Then it uses the fact that $\Lambda $ is complete with respect to $I$ and hence equal to $\mathop{\mathrm{lim}}\nolimits _ m \Lambda /I^ m$ as a topological space, to see that $\mathop{\mathrm{lim}}\nolimits _ m \text{Map}_{cont}(G, \Lambda /I^ m) = \text{Map}_{cont}(G, \Lambda )$ and similarly for higher powers of $G$. At this point Cohomology on Sites, Lemmas 21.10.3 and 21.10.7 tell us that

\[ \Lambda \to \text{Maps}_{cont}(G, \Lambda ) \to \text{Maps}_{cont}(G \times G, \Lambda ) \to \ldots \]

computes the pro-étale cohomology. In other words, we see that

\[ H^ i(\mathop{\mathrm{Spec}}(k)_{pro\text{-}\acute{e}tale}, \underline{\Lambda }^\wedge ) = H^ i_{cont}(G, \Lambda ) \]

where the right hand side is Tate's continuous cohomology, see Étale Cohomology, Section 59.58. Of course, this is as it should be.

Lemma 61.22.1. Let $k$ be a field. Let $G = \text{Gal}(k^{sep}/k)$ be its absolute Galois group. Further,

let $M$ be a profinite abelian group with a continuous $G$-action, or
let $\Lambda $ be a Noetherian ring and $I \subset \Lambda $ an ideal an let $M$ be an $I$-adically complete $\Lambda $-module with continuous $G$-action.

Then there is a canonical sheaf $\underline{M}^\wedge $ on $\mathop{\mathrm{Spec}}(k)_{pro\text{-}\acute{e}tale}$ associated to $M$ such that

\[ H^ i(\mathop{\mathrm{Spec}}(k), \underline{M}^\wedge ) = H^ i_{cont}(G, M) \]

as abelian groups or $\Lambda $-modules.

Proof. Proof in case (2). Set $M_ n = M/I^ nM$. Then $M = \mathop{\mathrm{lim}}\nolimits M_ n$ as $M$ is assumed $I$-adically complete. Since the action of $G$ is continuous we get continuous actions of $G$ on $M_ n$. By Étale Cohomology, Theorem 59.56.3 this action corresponds to a (locally constant) sheaf $\underline{M_ n}$ of $\Lambda /I^ n$-modules on $\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}$. Pull back to $\mathop{\mathrm{Spec}}(k)_{pro\text{-}\acute{e}tale}$ by the comparison morphism $\epsilon $ and take the limit

\[ \underline{M}^\wedge = \mathop{\mathrm{lim}}\nolimits \epsilon ^{-1}\underline{M_ n} \]

to get the sheaf promised in the lemma. Exactly the same argument as given in the introduction of this section gives the comparison with Tate's continuous Galois cohomology. $\square$

The Stacks project

61.22 Cohomology of a point

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