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 34.9.8 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
Comments (0)