This time, we consider a more general case.

## Preliminaries

First, we recall some notions and facts.

Definition: Let $F\colon\mathcal{C}\to\mathcal{D}, G\colon\mathcal{D}\to\mathcal{C}$ be two functors. We call they form an adjuction if there exist two natrual transformations $\eta\colon\mathrm{id} \to GF$ (called unit) and $\epsilon\colon FG \to \mathrm{id}$ (called counit) such that the following two diagrams (called the triangle identities) hold. $$\array{ FGF & GFG\\ F\eta\nearrow\qquad\nwarrow\epsilon G & \eta G\nearrow\qquad\nwarrow F\epsilon \\ F\quad\stackrel{\mathrm{id}}{\longrightarrow}\quad F & G\quad\stackrel{\mathrm{id}}{\longrightarrow}\quad G }$$ If furthermore both $\eta$ and $\epsilon$ are isomorphisms, then $F,G$ are equivalences of categories and we say they form a pair of adjoint equivalences.

Proposition: Let $F\colon\mathcal{C}\to\mathcal{D}, G\colon\mathcal{D}\to\mathcal{C}$ be an adjunction with unit $\eta$ and counit $\epsilon$. Let $\mathcal{C}^{\eta}$ be the full subcategory of $\mathcal{C}$ consisting of those objects $X$ such that $\eta_X$ is an isomorphism. Define $\mathcal{D}^{\epsilon}$ likewise. Then $F$ and $G$ give rise to a pair of adjoint equivalences:
$$\mathcal{C}^{\eta} \cong \mathcal{D}^{\epsilon}.$$

### $G$-sets and $G$-spaces

Definition: Let $G$ be a group. A $G$-set is a set $S$ together with a group action $G\times S\to S$. A $G$-map is a map of sets between $G$-sets compatible with the group actions. A $G$-subset of a $G$-set $S$ is a subset $T$ of $S$ such that the inclusion $T\hookrightarrow S$ is a $G$-map. Let $s\in S$, then

• the $G$-subset $Gs:=\left\{gs \middle| g\in G\right\}$ is called the orbit of $s$;
• the subgroup $G_s:=\left\{ g\in G \middle| gs=s \right\}$ is called the stabilizer of $s$;
• $s$ is said to be $G$-invariant if $gs=s$ for all $g\in G$.

Let $g\in G$, then

• the $G$-subset $S^g:=\left\{s\in S\middle|gs=s\right\}$ is called the set of fixed point of $g$.

Finally

• the set of $G$-invariants is denoted by $S^G$, which equals $\cap_{g\in G} S^g$,
• the set of orbits is denoted by $S/G$ and called the quotient of the action.

Fundamental theorem of $G$-sets: Let $G$ be a group and $S$ a $G$-set. Then $S$ is a disjoint union of some orbits $Gs$, which, as $G$-set, is isomorphic to the quotient $G/G_s$.

Remark: The above stories can be transplanted to some geometric categories $\mathcal{C}$. In this case, we should consider group objects $G$ instead of groups and $G$-spaces instead of $G$-sets. In this case, we call the space of orbits $S/G$ 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 $G$ be a topological group. Then we call a $G$-space $S$ with discrete topology a $G$-set. Note that in thise case, the stabiliaer of any point is open.

## Galois Theory For $K$-Schemes

Let $K$ be a field and $G$ be its absolute Galois group.

Definition: A $K$-scheme is a locally ringed space which is locally the spectrum of a $K$-algebra. Equivalently, a $K$-scheme $X$ is a morphism of schemes $X\to \mathrm{Spec}(K)$. A morphism of $K$-schemes is a morphism of schemes over $\mathrm{Spec}(K)$.

Note that, there exists a continuous $G$-action on each set $\mathrm{Hom}_{\mathrm{Spec}(K)}(\mathrm{Spec}(K_{\mathrm{sep}}),X)$ for each $K$-scheme $X$: $$\array{ G\times\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),X\right) & \longrightarrow & \mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),X\right) \\ (g,f) & \longmapsto & f\circ\widetilde{g}, }$$ where $\widetilde{g}$ is the automoprhism of $\mathrm{Spec}(K_{\mathrm{sep}})$ induced by $g\colon K_{\mathrm{sep}}\to K_{\mathrm{sep}}$.

Those actions are compatible with the morphisms of $K$-schemes. Therefore, we get a functor
$$K\text{-}\mathrm{Sch} \longrightarrow G\text{-}\mathrm{Set}.$$

### From $G$-sets to $K$-schemes

Let $H$ be a subgroup of $G$, then $K_{\mathrm{sep}}^H$ is an $K$-algebra. Since any homomorphism of $K$-algebras $K_{\mathrm{sep}}^H\to K_{\mathrm{sep}}$ can be extended to an automorphism of $K_{\mathrm{sep}}$ over $K$, 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 $$\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),X\right) = \bigsqcup Gf$$ the orbit decomposition. Let $$Y = \bigsqcup \mathrm{Spec}(K_{\mathrm{sep}}^{G_f}).$$

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
$$\epsilon_X\colon Y\longrightarrow X.$$

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 $K$-scheme $X$, there is a canonical morphism of $K$-schemes $$\epsilon_X\colon F(\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),X\right))\longrightarrow X.$$

Moreover, one can verify that the above data satisfies the triangle identities. Thus we get an adjunction
$$F\dashv \mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),-\right).$$

As $\eta_S$ is always bijective, we have $(G\text{-}\mathrm{Set})^{\eta} = G\text{-}\mathrm{Set}$. How about $(K\text{-}\mathrm{Sch})^{\epsilon}$?

### Etale schemes

The essential iamge of the functor $F$ can be characterised as follows.
Definition: A $K$-scheme is said to be étale if it is isomorphic to a disjoint union of sepctra of finite separable extensions of $K$.
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 $K$ is of the form $K^H$ where $H$ is an open subgroup of $G$.

Let $X$ be an étale $K$-scheme and
$$X \cong \bigsqcup \mathrm{Spec}(K_{\mathrm{sep}}^{H_i}).$$

Then we have
$$\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),X\right)=\bigsqcup\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),\mathrm{Spec}(K_{\mathrm{sep}}^{H_i})\right).$$

Note that each $\mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),\mathrm{Spec}(K_{\mathrm{sep}}^{H_i})\right)$ 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 $\epsilon_X$ is an isomophism.

In this way, we see that $(K\text{-}\mathrm{Sch})^{\epsilon}$ consists precisely the étale $K$-schemes.

### Galois correspondence for $K$-schemes

Conclude above, we have
Theorem: Let $K$ be a field and $G$ 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