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 F ⁣:CD,G ⁣:DCF\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 η ⁣:idGF\eta\colon\mathrm{id} \to GF (called unit) and ϵ ⁣:FGid\epsilon\colon FG \to \mathrm{id} (called counit) such that the following two diagrams (called the triangle identities) hold.

FGFFηϵGFidFGFGηGFϵGidG\begin{gathered} FGF \\ F\eta\nearrow\qquad\nwarrow\epsilon G \\ F\quad\stackrel{\mathrm{id}}{\longrightarrow}\quad F \end{gathered} \qquad\qquad \begin{gathered} GFG\\ \eta G\nearrow\qquad\nwarrow F\epsilon \\ G\quad\stackrel{\mathrm{id}}{\longrightarrow}\quad G \end{gathered}

If furthermore both η\eta and ϵ\epsilon are isomorphisms, then F,GF,G are equivalences of categories and we say they form a pair of adjoint equivalences.

Proposition: Let F ⁣:CD,G ⁣:DCF\colon\mathcal{C}\to\mathcal{D}, G\colon\mathcal{D}\to\mathcal{C} be an adjunction with unit η\eta and counit ϵ\epsilon. Let Cη\mathcal{C}^{\eta} be the full subcategory of C\mathcal{C} consisting of those objects XX such that ηX\eta_X is an isomorphism. Define Dϵ\mathcal{D}^{\epsilon} likewise. Then FF and GG give rise to a pair of adjoint equivalences:

CηDϵ.\mathcal{C}^{\eta} \cong \mathcal{D}^{\epsilon}.

GG-sets and GG-spaces

Definition: Let GG be a group. A GG-set is a set SS together with a group action G×SSG\times S\to S. A GG-map is a map of sets between GG-sets compatible with the group actions. A GG-subset of a GG-set SS is a subset TT of SS such that the inclusion TST\hookrightarrow S is a GG-map. Let sSs\in S, then

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

Let gGg\in G, then

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

Finally

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

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

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

Galois Theory For KK-Schemes

Let KK be a field and GG be its absolute Galois group.

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

Note that, there exists a continuous GG-action on each set HomSpec(K)(Spec(Ksep),X)\mathrm{Hom}_{\mathrm{Spec}(K)}(\mathrm{Spec}(K_{\mathrm{sep}}),X) for each KK-scheme XX:

\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 g~\widetilde{g} is the automoprhism of Spec(Ksep)\mathrm{Spec}(K_{\mathrm{sep}}) induced by g ⁣:KsepKsepg\colon K_{\mathrm{sep}}\to K_{\mathrm{sep}}.

Those actions are compatible with the morphisms of KK-schemes. Therefore, we get a functor

K-SchG-Set.K\text{-}\mathrm{Sch} \longrightarrow G\text{-}\mathrm{Set}.

From GG-sets to KK-schemes

Let HH be a subgroup of GG, then KsepHK_{\mathrm{sep}}^H is an KK-algebra. Since any homomorphism of KK-algebras KsepHKsepK_{\mathrm{sep}}^H\to K_{\mathrm{sep}} can be extended to an automorphism of KsepK_{\mathrm{sep}} over KK, 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

theorbitdecomposition.Let 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.

### 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 KK-scheme XX, there is a canonical morphism of KK-schemes

    ϵX ⁣:F(HomSpec(K)(Spec(Ksep),X))X.\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

FHomSpec(K)(Spec(Ksep),).F\dashv \mathrm{Hom}_{\mathrm{Spec}(K)}\left(\mathrm{Spec}(K_{\mathrm{sep}}),-\right).

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

Etale schemes

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

Let XX be an étale KK-scheme and

XSpec(KsepHi).X \cong \bigsqcup \mathrm{Spec}(K_{\mathrm{sep}}^{H_i}).

Then we have

HomSpec(K)(Spec(Ksep),X)=HomSpec(K)(Spec(Ksep),Spec(KsepHi)).\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 HomSpec(K)(Spec(Ksep),Spec(KsepHi))\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 ϵX\epsilon_X is an isomophism.

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

Galois correspondence for KK-schemes

Conclude above, we have
Theorem: Let KK be a field and GG 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).