Grothendieck's Galois theory: fundamental theorem

Question

I've been learning about Grothendieck's Galois theory, and I just haven't been able to understand the fundamental theorem properly. Let's phrase the fundamental theorem in the case of fields:

Let $k$ be a field, and $k_s$ its separable closure. There is an anti-equivalence between the category of finite separable $k$-algebras and the category of finite sets equipped with a continuous Gal($k_s/k$)-action. Under this equivalence, separable field extensions of $k$ correspond to sets with a transitive action, and Galois extensions of $k$ correspond to finite quotients of Gal($k_s/k$).

Everywhere I read that the above theorem generalizes the fundamental theorem of Galois theory, but I don't quite understand this. For example, given a Galois extension, why do subfields correspond to subgroups of its Galois group (which is the finite quotient of Gal($k_s/k$))? All I can tell is that a subfield $K$ of a Galois extension $L$ corresponds to a surjective map of Gal$(k_s/k)$-sets $$\text{Hom}_k(L,k_s) \to \text{Hom}_k(K,k_s),$$ and the left-hand side can be identified with a group. How does that realise the right-hand side as a subgroup?

Moreover, how do we obtain that the normal subgroups of the Galois group of a finite extension correspond to normal subfields of that extension?

Many thanks.

Answer 1

Let $L$ be a Galois extension of $k$ embedded in $k_s$. Then $L$ is the union of its finite Galois subextensions $L'$,and $\hom(L,k_s)$ is the projective limit of $\hom(L',k_s)$ along Galois subextensions $L'$. Grothendieck's theorem gives you a structure of group on $\hom(L',k_s)$ for each finite subextension and these are compatible with the limit, hence you get a structure of profinite group on $\hom(L,k_s)=Gal(L/k)$.

Now take a subextension $K$ of $L$, as you say there is a natural map $\hom(L,k_s)\to\hom(K,k_s)$. But now you say something wrong: $\hom(K,k_s)$ is not a subgroup, the subgroup is the inverse image in $\hom(L,k_s)=Gal(L/k)$ of the fixed embedding $K\subset L\subset k_s$ which is an element of $\hom(K,k_s)$. It is a subgroup since you can see it as a stabilizer of the action of $\hom(L,k_s)$ on $\hom(K,k_s)$.

In this construction, if $K$ is a normal subextension of $L$ then Grothendieck's theorem gives you a group structure on $\hom(K,k_s)$ compatible with the group structure on $\hom(L,k_s)$, hence the inverse image in $\hom(L,k_s)$ of the fixed embedding $K\subset L\subset k_s$ is a kernel of an homomorphism, and hence it is normal.

On the other hand, if $H\subset Gal(L/k)=\hom(L,k_s)$ is a closed subgroup, $\hom(L,k_s)/H$ has a natural structure of profinite set with continuos action of $Gal(k_s/k)$ induced by the projection $Gal(k_s/k)\to \hom(L,k_s)=Gal(L/k)$ (this projection comes with the construction, we are not using classical Galois theory). Then, since the projection $Gal(k_s/k)\to \hom(L,k_s)/H$ is surjective (because $Gal(k_s/k)\to \hom(L,k_s)$ is surjective) and hence the action is transitive, Grothendieck's theorem gives you a separable extension $K/k$ with a natural identification $Gal(L/k)/H=\hom(K,k_s)$, hence with a natural map $\hom(L,k_s)\to\hom(K,k_s)$ which in turn gives you an embedding $K\subset L$.

If moreover $H$ is normal, then $Gal(L/k)/H$ is a group and hence Grothendieck's theorem tells you that $K$ is Galois.