35.6 Galois descent for quasi-coherent sheaves
Galois descent for quasi-coherent sheaves is just a special case of fpqc descent for quasi-coherent sheaves. In this section we will explain how to translate from a Galois descent to an fpqc descent and then apply earlier results to conclude.
Let $k'/k$ be a field extension. Then $\{ \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)\} $ is an fpqc covering. Let $X$ be a scheme over $k$. For a $k$-algebra $A$ we set $X_ A = X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(A)$. By Topologies, Lemma 34.9.7 we see that $\{ X_{k'} \to X\} $ is an fpqc covering. Observe that
\[ X_{k'} \times _ X X_{k'} = X_{k' \otimes _ k k'} \quad \text{and}\quad X_{k'} \times _ X X_{k'} \times _ X X_{k'} = X_{k' \otimes _ k k' \otimes _ k k'} \]
Thus a descent datum for quasi-coherent sheaves with respect to $\{ X_{k'} \to X\} $ is given by a quasi-coherent sheaf $\mathcal{F}$ on $X_{k'}$, an isomorphism $\varphi : \text{pr}_0^*\mathcal{F} \to \text{pr}_1^*\mathcal{F}$ on $X_{k' \otimes _ k k'}$ which satisfies an obvious cocycle condition on $X_{k' \otimes _ k k' \otimes _ k k'}$. We will work out what this means in the case of a Galois extension below.
Let $k'/k$ be a finite Galois extension with Galois group $G = \text{Gal}(k'/k)$. Then there are $k$-algebra isomorphisms
\[ k' \otimes _ k k' \longrightarrow \prod \nolimits _{\sigma \in G} k',\quad a \otimes b \longrightarrow \prod a\sigma (b) \]
and
\[ k' \otimes _ k k' \otimes _ k k' \longrightarrow \prod \nolimits _{(\sigma , \tau ) \in G \times G} k',\quad a \otimes b \otimes c \longrightarrow \prod a\sigma (b)\sigma (\tau (c)) \]
The reason for choosing here $a\sigma (b)\sigma (\tau (c))$ and not $a\sigma (b)\tau (c)$ is that the formulas below simplify but it isn't strictly necessary. Given $\sigma \in G$ we denote
\[ f_\sigma = \text{id}_ X \times \mathop{\mathrm{Spec}}(\sigma ) : X_{k'} \longrightarrow X_{k'} \]
Please keep in mind that because $\mathop{\mathrm{Spec}}(-)$ is a contravariant functor we have $f_{\sigma \tau } = f_\tau \circ f_\sigma $ and not the other way around. Using the first isomorphism above we obtain an identification
\[ X_{k' \otimes _ k k'} = \coprod \nolimits _{\sigma \in G} X_{k'} \]
such that $\text{pr}_0$ corresponds to the map
\[ \coprod \nolimits _{\sigma \in G} X_{k'} \xrightarrow {\coprod \text{id}} X_{k'} \]
and such that $\text{pr}_1$ corresponds to the map
\[ \coprod \nolimits _{\sigma \in G} X_{k'} \xrightarrow {\coprod f_\sigma } X_{k'} \]
Thus we see that a descent datum $\varphi $ on $\mathcal{F}$ over $X_{k'}$ corresponds to a family of isomorphisms $\varphi _\sigma : \mathcal{F} \to f_\sigma ^*\mathcal{F}$. To work out the cocycle condition we use the identification
\[ X_{k' \otimes _ k k' \otimes _ k k'} = \coprod \nolimits _{(\sigma , \tau ) \in G \times G} X_{k'}. \]
we get from our isomorphism of algebras above. Via this identification the map $\text{pr}_{01}$ corresponds to the map
\[ \coprod \nolimits _{(\sigma , \tau ) \in G \times G} X_{k'} \longrightarrow \coprod \nolimits _{\sigma \in G} X_{k'} \]
which maps the summand with index $(\sigma , \tau )$ to the summand with index $\sigma $ via the identity morphism. The map $\text{pr}_{12}$ corresponds to the map
\[ \coprod \nolimits _{(\sigma , \tau ) \in G \times G} X_{k'} \longrightarrow \coprod \nolimits _{\sigma \in G} X_{k'} \]
which maps the summand with index $(\sigma , \tau )$ to the summand with index $\tau $ via the morphism $f_\sigma $. Finally, the map $\text{pr}_{02}$ corresponds to the map
\[ \coprod \nolimits _{(\sigma , \tau ) \in G \times G} X_{k'} \longrightarrow \coprod \nolimits _{\sigma \in G} X_{k'} \]
which maps the summand with index $(\sigma , \tau )$ to the summand with index $\sigma \tau $ via the identity morphism. Thus the cocycle condition
\[ \text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi \]
translates into one condition for each pair $(\sigma , \tau )$, namely
\[ \varphi _{\sigma \tau } = f_\sigma ^*\varphi _\tau \circ \varphi _\sigma \]
as maps $\mathcal{F} \to f_{\sigma \tau }^*\mathcal{F}$. (Everything works out beautifully; for example the target of $\varphi _\sigma $ is $f_\sigma ^*\mathcal{F}$ and the source of $f_\sigma ^*\varphi _\tau $ is $f_\sigma ^*\mathcal{F}$ as well.)
Lemma 35.6.1. Let $k'/k$ be a (finite) Galois extension with Galois group $G$. Let $X$ be a scheme over $k$. The category of quasi-coherent $\mathcal{O}_ X$-modules is equivalent to the category of systems $(\mathcal{F}, (\varphi _\sigma )_{\sigma \in G})$ where
$\mathcal{F}$ is a quasi-coherent module on $X_{k'}$,
$\varphi _\sigma : \mathcal{F} \to f_\sigma ^*\mathcal{F}$ is an isomorphism of modules,
$\varphi _{\sigma \tau } = f_\sigma ^*\varphi _\tau \circ \varphi _\sigma $ for all $\sigma , \tau \in G$.
Here $f_\sigma = \text{id}_ X \times \mathop{\mathrm{Spec}}(\sigma ) : X_{k'} \to X_{k'}$.
Proof.
As seen above a datum $(\mathcal{F}, (\varphi _\sigma )_{\sigma \in G})$ as in the lemma is the same thing as a descent datum for the fpqc covering $\{ X_{k'} \to X\} $. Thus the lemma follows from Proposition 35.5.2.
$\square$
A slightly more general case of the above is the following. Suppose we have a surjective finite étale morphism $X \to Y$ and a finite group $G$ together with a group homomorphism $G^{opp} \to \text{Aut}_ Y(X), \sigma \mapsto f_\sigma $ such that the map
\[ G \times X \longrightarrow X \times _ Y X,\quad (\sigma , x) \longmapsto (x, f_\sigma (x)) \]
is an isomorphism. Then the same result as above holds.
Lemma 35.6.2. Let $X \to Y$, $G$, and $f_\sigma : X \to X$ be as above. The category of quasi-coherent $\mathcal{O}_ Y$-modules is equivalent to the category of systems $(\mathcal{F}, (\varphi _\sigma )_{\sigma \in G})$ where
$\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module,
$\varphi _\sigma : \mathcal{F} \to f_\sigma ^*\mathcal{F}$ is an isomorphism of modules,
$\varphi _{\sigma \tau } = f_\sigma ^*\varphi _\tau \circ \varphi _\sigma $ for all $\sigma , \tau \in G$.
Proof.
Since $X \to Y$ is surjective finite étale $\{ X \to Y\} $ is an fpqc covering. Since $G \times X \to X \times _ Y X$, $(\sigma , x) \mapsto (x, f_\sigma (x))$ is an isomorphism, we see that $G \times G \times X \to X \times _ Y X \times _ Y X$, $(\sigma , \tau , x) \mapsto (x, f_\sigma (x), f_{\sigma \tau }(x))$ is an isomorphism too. Using these identifications, the category of data as in the lemma is the same as the category of descent data for quasi-coherent sheaves for the covering $\{ x \to Y\} $. Thus the lemma follows from Proposition 35.5.2.
$\square$
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