Galois Correspondences II
This time, we consider a more general case: we give the Galois correspondence for schemes over a field.
Preliminaries
First, we recall some notions and facts.
Adjunctions and adjoint equivalences
Definition: Let be two functors. We call they form an adjuction if there exist two natrual transformations (called unit) and (called counit) such that the following two diagrams (called the triangle identities) hold.
If furthermore both and are isomorphisms, then are equivalences of categories and we say they form a pair of adjoint equivalences.
Proposition: Let be an adjunction with unit and counit . Let be the full subcategory of consisting of those objects such that is an isomorphism. Define likewise. Then and give rise to a pair of adjoint equivalences:
-sets and -spaces
Definition: Let be a group. A -set is a set together with a group action . A -map is a map of sets between -sets compatible with the group actions. A -subset of a -set is a subset of such that the inclusion is a -map. Let , then
- the -subset is called the orbit of ;
- the subgroup is called the stabilizer of ;
- is said to be -invariant if for all .
Let , then
- the -subset is called the set of fixed point of .
Finally
- the set of -invariants is denoted by , which equals ,
- the set of orbits is denoted by and called the quotient of the action.
Fundamental theorem of -sets: Let be a group and a -set. Then is a disjoint union of some orbits , which, as -set, is isomorphic to the quotient .
Remark: The above stories can be transplanted to some geometric categories . In this case, we should consider group objects instead of groups and -spaces instead of -sets. In this case, we call the space of orbits the orbit space. Examples include:
- continuous actions of topological groups on topological spaces,
- smooth actions of Lie groups on smooth manifolds,
- regular actions of algebraic groups on algebraic varieties, and
- actions of group schemes on schemes.
Remark: Let be a topological group. Then we call a -space with discrete topology a -set. Note that in thise case, the stabiliaer of any point is open.
Galois Theory For -Schemes
Let be a field and be its absolute Galois group.
Definition: A -scheme is a locally ringed space which is locally the spectrum of a -algebra. Equivalently, a -scheme is a morphism of schemes . A morphism of -schemes is a morphism of schemes over .
Note that, there exists a continuous -action on each set for each -scheme :
\begin{align*} G\times\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),X\right) & \longrightarrow \mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),X\right) \\ (g,f) & \longmapsto f\circ\widetilde{g}, \end{align*}
where is the automoprhism of induced by .
Those actions are compatible with the morphisms of -schemes. Therefore, we get a functor
From -sets to -schemes
Let be a subgroup of , then is an -algebra. Since any homomorphism of -algebras can be extended to an automorphism of over , we have
\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),\mathrm{Spec}(K_{\mathrm{sep}}^H)\right) = \mathrm{Hom}_K\left(K_{\mathrm{sep}}^H,K_{\mathrm{sep}}\right) \cong G/H $$ as $G$-sets. Let $S$ be a $G$-set and
S = \bigsqcup Gs
be all the distinct orbits of $S$. Then
X = \bigsqcup \mathrm{Spec}(K_{\mathrm{sep}}^{G_s})
is a $K$-scheme. Since $\mathrm{Spec}(K_{\mathrm{sep}})$ is a single point hence connected, the $G$-map
\eta_S\colon S = \bigsqcup Gs \cong \bigsqcup\mathrm{Hom}{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K{\mathrm{sep}}),\mathrm{Spec}(K_{\mathrm{sep}}^{G_s})\right)\longrightarrow
\mathrm{Hom}{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K{\mathrm{sep}}),\bigsqcup\mathrm{Spec}(K_{\mathrm{sep}}^{G_s})\right)
is bijective. Let $f\colon S\to T$ be a $G$-map and let $X, Y$ be the $F$-algebras associated to $S, T$ respectively. Then for any point $s\in S$, the stabilizer of $s$ is a subgroup of $f(s)$. Thus $K_{\mathrm{sep}}^{G_{f(s)}}$ is a subalgebra of $K_{\mathrm{sep}}^{G_s}$. This induces a morphism of $K$-schemes $X\to Y$. In this way, we get a functor
F\colon G\text{-}\mathrm{sets} \longrightarrow K\text{-}\mathrm{Sch}.
### From $K$-schemes to $G$-sets Let $X$ be a $K$-scheme and
\mathrm{Hom}{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K{\mathrm{sep}}),X\right) = \bigsqcup Gf
Y = \bigsqcup \mathrm{Spec}(K_{\mathrm{sep}}^{G_f}).
Let $f\colon\mathrm{Spec}(K_{\mathrm{sep}})\to X$ be a morphism of $K$-schemes. Then, for any open set $U$ of $X$ containing the image of $f$, the $K$-homomorphism $\mathcal{O}_X(U)\to K_{\mathrm{sep}}$ factors through $K_{\mathrm{sep}}^{G_f}$. Thus $f$ factors through $\mathrm{Spec}(K_{\mathrm{sep}}^{G_f})$. Therefore we get a canonical morphism of $K$-schemes
\epsilon_X\colon Y\longrightarrow X.
### The adjunction From the above discussion, we have seen: - given a $G$-set $S$, there is a canonical bijective $G$-map $$\eta_S\colon S\longrightarrow\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),F(S)\right);
- given a -scheme , there is a canonical morphism of -schemes
Moreover, one can verify that the above data satisfies the triangle identities. Thus we get an adjunction
As is always bijective, we have . How about ?
Etale schemes
The essential iamge of the functor can be characterised as follows.
Definition: A -scheme is said to be étale if it is isomorphic to a disjoint union of sepctra of finite separable extensions of .
This is of course not the original definition. One can image this as a lemma.
By the first Galois correspondence, any finite separable extension of is of the form where is an open subgroup of .
Let be an étale -scheme and
Then we have
Note that each must be connected. Therefore $$F(\mathrm{Hom}{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K{\mathrm{sep}}),X\right)) = \bigsqcup \mathrm{Spec}(K_{\mathrm{sep}}^{H_i})$$ and hence the canonical morphism is an isomophism.
In this way, we see that consists precisely the étale -schemes.
Galois correspondence for -schemes
Conclude above, we have
Theorem: Let be a field and be its absolute Galois group. Then there is a pair of adjoint equivalences of categories
F\dashv \mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),-\right). $$ between the category of $G$-sets and the category of étale $K$-schemes. Let $L/K$ be a Galois extension, then the above pair of adjoint equivalences induces a pair of adjoint of equivalences between the category of $\mathrm{Gal}(L/K)$-sets and the category of $K$-schemes which are isomorphic to disjoint unions of sepctra of finite subseparable extensions of $L/K$. ### Galois correspondence for $K$-algebras Note that finite disjoint union of affine schemes is the same as coproduct of them in the category of affine schemes, hence we have
\bigsqcup_{i=1}^{n}\mathrm{Spec}(L_i) = \mathrm{Spec}(\prod_{i=1}^{n}L_i).
Recall that a $K$-algebra is said to be **étale** if it is a finite product of finite separable extensions of $K$ and **splited** by $L$ is all those separable extensions are subextensions of the Galois extension $L/K$. Then one may expect a Galois correspondence for such kind of $K$-algebras. However, a finite disjoint union of $G$-sets may not be finite unless all the components are finite. So we have **Theorem:** Let $L/K$ be a finite Galois extension. Then the functor
\mathrm{Hom}_{K}\left(-,L\right)
is an equivalence between the category of finite $\mathrm{Gal}(L/K)$-sets and the category of finite $K$-algebras splited by $L$. Note that - any Galois extension is a filtered colimit of finite Galois extensions; - any $K$-algebra is a filtered colimit of finite subalgebras; - a $G$-space is said to be **profinite** if it is a filtered colimit of finite $G$-sets. We thus get **Theorem:** Let $L/K$ be a Galois extension. Then the functor
\mathrm{Hom}_{K}\left(-,L\right)
is an equivalence between the category of profinite $\mathrm{Gal}(L/K)$-spaces and the category of $K$-algebras splited by $L$. ### Etale coverings Recall that **Definition:** Let $X$ be a scheme. A family of morphisms $\\{f_i\colon Y_i\to X\\}$ is called a **étale covering** if all $f_i$ are étale and the images of $Y_i$ cover $X$. Equipped such coverings, $\mathrm{Sch}$ becomes a sites, call the **(big) étale site**, and $K\text{-}\mathrm{Sch}$ becomes a sites $\mathrm{Spec}(K)_{et}$, call the **(small) étale site** of $\mathrm{Spec}(K)$. One can verify that under the Galois correspondence for $K$-schemes, the étale coverings in the étale site $\mathrm{Spec}(K)_{et}$ correspond to surjective families of $G$-maps in $G\text{-}\mathrm{Set}$. Let's stop this post here. ## Reference For the notion of schemes, one can refer any textbooks such as - Robin Hartshorne, *Algebraic Geometry*, Graduate Texts in Mathematics 52, New York: Springer-Verlag, 1977. - Ravi Vakil, [*Math 216: Foundations of algebraic geometry*](https://math216.wordpress.com). - Johan de Jong, et al., [*Stack project*](http://stacks.math.columbia.edu).