ABSTRACT
We constructed, in Chapter 5, for a commutative ring R with separable closure
S, a contravariant natural equivalence between (1) the category S = Scls(R) of componentially locally strongly separable extensions of R; and (2) the cate-
goryM =Mprf (X(S⊗RS)) of (pro-relatively finite) profiniteX(S⊗RS) sets, where Π1 = Π1(R,S) = X(S ⊗R S) carries its standard groupoid structure. The correspondence is constructed as the functor S →M by T 7→ X(S⊗RT ). For convenience, we recall how these latter structures are interpreted: the
groupoidX(S⊗RS) and the setX(S⊗RT ) are both described as sets of triples, the latter as (a, b, h) where a ∈ X(S) and b ∈ X(T ) where a ∩R = b ∩R = x and h : Tb → Sa is an Rx algebra homomorphism. (The former is described the same way with T = S.) X(S ⊗R T ) lies over X(S) by (a, b, h) 7→ a and if (a, b, g) ∈ X(S ⊗R S) and (b, c, h) ∈ X(S ⊗R T ) (notice that b appears in the second entry of the first tuple and the first entry of the second tuple)
then (a, b, g) · (b, c, h) = (a, c, gh). (The same formula describes the operation in X(S ⊗R S) when T = S.) Since we have an equivalence of categories, every object Y → X(S) in M is of the form X(S ⊗R T ) for some T and in turn we can interpret the action as triples acting on triples. For example,
X(S) corresponds to T = R, so it consists of triples (a, b, h) where b ∈ X(R),
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a ∩ R = b, and h : Rb → Sa is the canonical map. (Thus the triple is determined by a alone.) The action of (c, a, g) on (a, b, h) then gives (c, b, gh) where
c ∩R = a ∩R = b and gh : Rb → Sc is the canonical map. Using the fact the triples of X(S ⊗R R) are determined by their first entry, this gives the action on X(S) as (c, a, g) · a = c.
