ABSTRACT
The classical theory of Tannaka duality, which is an extension of Pontryagin
duality to non-commutative groups, shows that a compact topological group
is determined by its category of (continuous, finite dimensional) representa-
tions. Grothendieck extended this notion to algebraic groups (and beyond).
Somewhat earlier, Grothendieck had used a related idea to define the (e´tale)
fundamental group of a scheme. The idea was that the category of finite e´tale
covering spaces of the (connected) scheme X was equivalent to the category
of finite sets on which a certain profinite group Π1(X) acts continuously, in
fact the group was defined to make this statement true. More precisely, a
geometric point x0 of X is selected, and to the finite e´tale cover (surjective
morphism) p : Y → X is associated with the fiber p−1(x0), which is a finite set. The category of such finite sets (there is a product condition related to
the fact that if Yi → X , i = 1, 2 are e´tale covers so is Y1×X Y2) turns out to be one of those categories which are the sets on which a profinite group acts, and
the group is recoverable from the category with the product condition. When
X = Spec(R) is affine, a geometric point corresponds to a homomorphism
R → L where L is an algebraically closed field. When R itself is a field (for convenience we will assume that the separable and algebraic closures of R are
the same field L) then finite e´tale covers of Spec(R) are of the form Spec(S)
