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Tag 04JM

Chapter 53: Étale Cohomology > Section 53.55: Galois action on stalks

Lemma 53.55.5. Assumptions and notations as in Theorem 53.55.3. There is a functorial bijection $$ \Gamma(S, \mathcal{F}) = (\mathcal{F}_{\overline{s}})^G $$

Proof. We can prove this using formal arguments and the result of Theorem 53.55.3 as follows. Given a sheaf $\mathcal{F}$ corresponding to the $G$-set $M = \mathcal{F}_{\overline{s}}$ we have \begin{eqnarray*} \Gamma(S, \mathcal{F}) & = & \mathop{Mor}\nolimits_{\mathop{\mathit{Sh}}\nolimits(S_{\acute{e}tale})}(h_{\mathop{\mathrm{Spec}}(K)}, \mathcal{F}) \\ & = & \mathop{Mor}\nolimits_{G\textit{-Sets})}(\{*\}, M) \\ & = & M^G \end{eqnarray*} Here the first identification is explained in Sites, Sections 7.2 and 7.12, the second results from Theorem 53.55.3 and the third is clear. We will also give a direct proof1.

Suppose that $t \in \Gamma(S, \mathcal{F})$ is a global section. Then the triple $(S, \overline{s}, t)$ defines an element of $\mathcal{F}_{\overline{s}}$ which is clearly invariant under the action of $G$. Conversely, suppose that $(U, \overline{u}, t)$ defines an element of $\mathcal{F}_{\overline{s}}$ which is invariant. Then we may shrink $U$ and assume $U = \mathop{\mathrm{Spec}}(L)$ for some finite separable field extension of $K$, see Proposition 53.26.2. In this case the map $\mathcal{F}(U) \to \mathcal{F}_{\overline{s}}$ is injective, because for any morphism of étale neighbourhoods $(U', \overline{u}') \to (U, \overline{u})$ the restriction map $\mathcal{F}(U) \to \mathcal{F}(U')$ is injective since $U' \to U$ is a covering of $S_{\acute{e}tale}$. After enlarging $L$ a bit we may assume $K \subset L$ is a finite Galois extension. At this point we use that $$ \mathop{\mathrm{Spec}}(L) \times_{\mathop{\mathrm{Spec}}(K)} \mathop{\mathrm{Spec}}(L) = \coprod\nolimits_{\sigma \in \text{Gal}(L/K)} \mathop{\mathrm{Spec}}(L) $$ where the maps $\mathop{\mathrm{Spec}}(L) \to \mathop{\mathrm{Spec}}(L \otimes_K L)$ come from the ring maps $a \otimes b \mapsto a\sigma(b)$. Hence we see that the condition that $(U, \overline{u}, t)$ is invariant under all of $G$ implies that $t \in \mathcal{F}(\mathop{\mathrm{Spec}}(L))$ maps to the same element of $\mathcal{F}(\mathop{\mathrm{Spec}}(L) \times_{\mathop{\mathrm{Spec}}(K)} \mathop{\mathrm{Spec}}(L))$ via restriction by either projection (this uses the injectivity mentioned above; details omitted). Hence the sheaf condition of $\mathcal{F}$ for the étale covering $\{\mathop{\mathrm{Spec}}(L) \to \mathop{\mathrm{Spec}}(K)\}$ kicks in and we conclude that $t$ comes from a unique section of $\mathcal{F}$ over $\mathop{\mathrm{Spec}}(K)$. $\square$

  1. For the doubting Thomases out there.

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 7903–7911 (see updates for more information).

\begin{lemma}
\label{lemma-global-sections-point}
Assumptions and notations as in
Theorem \ref{theorem-equivalence-sheaves-point}.
There is a functorial bijection
$$
\Gamma(S, \mathcal{F}) = (\mathcal{F}_{\overline{s}})^G
$$
\end{lemma}

\begin{proof}
We can prove this using formal arguments and the result of
Theorem \ref{theorem-equivalence-sheaves-point}
as follows. Given a sheaf $\mathcal{F}$ corresponding to
the $G$-set $M = \mathcal{F}_{\overline{s}}$ we have
\begin{eqnarray*}
\Gamma(S, \mathcal{F}) & = &
\Mor_{\Sh(S_\etale)}(h_{\Spec(K)},  \mathcal{F})
\\
& = & \Mor_{G\textit{-Sets})}(\{*\}, M) \\
& = & M^G
\end{eqnarray*}
Here the first identification is explained in
Sites, Sections \ref{sites-section-presheaves} and
\ref{sites-section-representable-sheaves},
the second results from
Theorem \ref{theorem-equivalence-sheaves-point}
and the third is clear. We will also give a direct proof\footnote{For
the doubting Thomases out there.}.

\medskip\noindent
Suppose that $t \in \Gamma(S, \mathcal{F})$ is a global section.
Then the triple $(S, \overline{s}, t)$ defines an element of
$\mathcal{F}_{\overline{s}}$ which is clearly invariant under the
action of $G$. Conversely, suppose that $(U, \overline{u}, t)$
defines an element of $\mathcal{F}_{\overline{s}}$ which is invariant.
Then we may shrink $U$ and assume $U = \Spec(L)$ for some
finite separable field extension of $K$, see
Proposition \ref{proposition-etale-morphisms}.
In this case the map $\mathcal{F}(U) \to \mathcal{F}_{\overline{s}}$
is injective, because for any morphism of \'etale neighbourhoods
$(U', \overline{u}') \to (U, \overline{u})$ the restriction map
$\mathcal{F}(U) \to \mathcal{F}(U')$ is injective since $U' \to U$
is a covering of $S_\etale$.
After enlarging $L$ a bit we may assume $K \subset L$ is a finite
Galois extension. At this point we use that
$$
\Spec(L) \times_{\Spec(K)} \Spec(L)
=
\coprod\nolimits_{\sigma \in \text{Gal}(L/K)} \Spec(L)
$$
where the maps $\Spec(L) \to \Spec(L \otimes_K L)$
come from the ring maps $a \otimes b \mapsto a\sigma(b)$. Hence we
see that the condition that $(U, \overline{u}, t)$ is invariant
under all of $G$ implies that $t \in \mathcal{F}(\Spec(L))$
maps to the same element of
$\mathcal{F}(\Spec(L) \times_{\Spec(K)} \Spec(L))$
via restriction by either projection (this uses the injectivity mentioned
above; details omitted). Hence the sheaf condition of $\mathcal{F}$
for the \'etale covering $\{\Spec(L) \to \Spec(K)\}$ kicks
in and we conclude that $t$ comes from a unique section of $\mathcal{F}$
over $\Spec(K)$.
\end{proof}

Comments (1)

Comment #3201 by Dario WeiƟmann on February 14, 2018 a 2:13 am UTC

Typo in the second equality of the short proof: '$\text{Mor}_{G-Sets)}$' should have no bracket.

There are also 2 comments on Section 53.55: Étale Cohomology.

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