Grothendieck's Galois theory: fundamental theorem

Solution 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.