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- W2021645488 abstract "In [3, 1.4, p. 112] we attempted to prove that every closed subgroupoid of a profinite groupoid is an intersection of open-closed subgroupoids. There is, however, an error in the proof: It is asserted that if G is a groupoid, H a subgroupoid and f: G G' a groupoid homomorphism, then f1(/(H)) is a subgroupoid of G. This will not happen, however, if f(H) is not a subgroupoid of G', and such things can occur, as pointed out in [1, p. 7]. The intersection theorem is, however, an interesting part of the Galois theory of commutative rings [2, 1.10, p. 961 and also plays a role in some subsequent developments. Thus we give a correction here. We begin by recalling the relevant definitions. A groupoid is a category in which all morphisms are isomorphisms. A subgroupoid is a subcategory closed under inversion and containing all identities. A homomorphism of groupoids is a functor. A profinite groupoid is a (topological) inverse limit of finite groupoids, over a directed index set, where all transition maps are surjective (so two elements of a profinite groupoid are equal if and only if their homomorphic images in all finite groupoids are equal). A groupoid is connected if there is a map between any two of its objects, and every groupoid is a co-product of its maximal connected subgroupoids, which are called its (connected) components. If G is a group and X a set, then X x X x G, with composition (a, b, g) (b, c, h) = (a, c, gh) is a groupoid. As a first step towards the intersection theorem, we will show that every profinite groupoid is a subgroupoid of a groupoid of this type. Let 9 be a profinite groupoid. We will find a profinite group F(9) and a continuous functor f: g F(q) universal with respect to continuous functors from 9 to profinite groups: First, take the free group on the set 9. Let F0 be the (topological) inverse limit of all the (discrete) quotients of the free group by normal subgroups of finite index each of whose cosets meets 9 in an open-closed set. Let h map 9 to Fo and let F(q) be the quotient of F0 by the closed normal subgroup generated by all h(s) h(t)h(st)where s, t are composable maps in t. (This construction is given in detail in [4, Definition 3].)" @default.
- W2021645488 created "2016-06-24" @default.
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- W2021645488 date "1974-04-01" @default.
- W2021645488 modified "2023-09-26" @default.
- W2021645488 title "Correction to The Separable Closure of Some Commutative Rings" @default.
- W2021645488 doi "https://doi.org/10.2307/1997007" @default.
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