Topics:
Etale fundamental groups
Section conjecture
This is meant to be a generalization of Galois theory and covering spaces.
An etale algebra \(A\) over a field \(k\) is a finite dimensional \(k\)-algebra isomorphic to a finite direct product of separable extensions of \(k\).
Theorem. \(A\mapsto \mathrm{Hom}_k(A,k^s)\) is an anti-equivalence between the category of finite etale \(k\)-algebras and the category of finite \(G_k\)-sets (with continuous \(G_k\)-action). Here \(\mathrm{Hom}_k\) is for \(k\)-algebra maps, not \(k\)-linear maps.
\[\{\text{Finite etale $k$-algebras}\} \longleftrightarrow \{\text{Finite continuous $G_k$-sets}\}\] \[A \mapsto \mathrm{Hom}_k(A,k^s)\mathrel{\circlearrowright}\mathrm{Gal}(k^s/k) \]
(the functor depends on the choice of \(k^s\))
Let \(x\in X\), the functor
\[\mathrm{Fib}_x:\{\text{Covering }p:Y\to X\} \to \{\pi_1(X,x)\text{-sets}\}: \quad p\mapsto p^{-1}(x)\]
is an equivalence of categories, depends on the choice of basepoint \(x\). It is representable as
\[\mathrm{Fib}_x \cong \mathrm{Hom}_{\mathrm{Cov}(X)}(\tilde{X}_x,-)\]
where \(\pi:\tilde{X}_x\to X\) is the universal cover of \(X\) based at \(x\), the space of homotopy (homotopy with fixed end points) classes of paths starting from \(x\).
By Yoneda’s Lemma we have
\[\mathrm{Hom}_{\mathrm{Hom}(\mathrm{Cov}(X),\pi_1-{\bf Sets})}(\mathrm{Fib}_x,\mathrm{Fib}_x) \cong \mathrm{Hom}(h^{\tilde{X}_x}, h^{\tilde{X}_x}) \cong^{op} \mathrm{Hom}(\tilde{X}_x, \tilde{X}_x) \]
i.e.
\[\mathrm{Aut}(\tilde{X}_x)^{op} \cong \mathrm{Aut}(\mathrm{Fib}_x) \cong \pi_1(X,x)\]
The last isomorphism is called the monodromy action.
For a profinite group \(G\) acting continuously on a finite set \(S\), the action factor through a finite quotient of the group since the finite intersection of stablizers \(\cap_{s\in S} G_{s}\) are open.
The profinite completion of a group \(G\) is the inverse limit of the finite quotients of \(G\). For Galois groups or fundamental groups these corresponds to finite covers or finite extensions. You can think of profinite completion as a group that remembers all the finite actions.
The functor (restricted to subcategory of finite coverings)
\[\mathrm{Fib}_x: \mathrm{FinCov}(X)\to \widehat{\pi_1(X,x)}\text{-}{\bf Sets}\]
induces an equivalence between the category of finite covers of \(X\) and the category of finite continuous \(\widehat{\pi_1(X,x)}\)-sets.
Let \(S\) be connected scheme, a finite etale map \(X\to S\) is also called a finite etale cover of \(S\). Its fibre at each \(s\in S\) is a finite etale \(k(s)\)-algebra.
For a geometric point \(\bar{s}=\mathrm{Spec}(\Omega)\to S\), the geometric fibre \(X\times_S \bar{s}\) is a finite etale \(\Omega\)-algebra whose underlying set is \(\mathrm{Fib}_{\bar{s}}(X)\), this is a set valued functor on \(\mathrm{FinEt}(S)\).
The algebraic fundamental group is defined as
\[\pi_1(S,\bar{s}) := \mathrm{Aut}_{\mathrm{FinEt}(S)\to{\bf Set}}(\mathrm{Fib}_{\bar{s}})\]
There is a natural action of \(\pi_1(S,\bar{s})\) on \(\mathrm{Fib}_{\bar{s}}(X)\) for any finite etale cover \(X\to S\).
Theorem (Grothendieck)
The group \(\pi_1(S,\bar{s})\) is profinite and acts continuously on each fibre \(\mathrm{Fib}_{\bar{s}}(X)\).
This fibre functor induces an equivalence
\[\mathrm{FinEt}(S) \cong \widehat{\pi_1(S,\bar{s})}\text{-}{\bf Sets}\]
between the category of finite etale covers of \(S\) and the category of finite continuous \(\widehat{\pi_1(S,\bar{s})}\)-sets.
The functor \(\mathrm{Fib}_{\bar{s}}\) is pro-representable, for some filtered inverse system of finite etale covers \(P_i\to S\) we have
\[\mathrm{Fib}_{\bar{s}}(X) \cong \mathrm{colim}\, \mathrm{Hom}(P_i, X)\]
A path is an isomorphism of fibre functors
\[\gamma: \mathrm{Fib}_{\bar{s}} \cong \mathrm{Fib}_{\bar{s}'}\]
and path induce a map of fundamental groups
\[\gamma(\cdot)\gamma^{-1} : \pi_1(S,\bar{s}) \to \pi_1(S,\bar{s}')\]