GALOIS CLOSURE DATA FOR EXTENSIONS OF RINGS

Abstract

To generalize the notion of Galois closure for separable field extensions, we devise a notion of G-closure for algebras of commutative rings R → A, where A is locally free of rank n as an R-module and G is a subgroup of S _n . A G-closure datum for A over R is an R-algebra homomorphism φ : (A ^⨂n)^G → R satisfying certain properties, and we associate to a closure datum φ a closure algebra A ^⨂n ${\otimes}_{{\left({A}^{\otimes n}\right)}^G}$ R. This construction reproduces the normal closure of a finite separable field extension if G is the corresponding Galois group. We describe G-closure data and algebras of finite étale algebras over a general connected ring R in terms of the corresponding finite sets with continuous actions by the étale fundamental group of R. We show that if 2 is invertible, then A _n -closure data for free extensions correspond to square roots of the discriminant, and that D₄-closure data for quartic monogenic extensions correspond to roots of the cubic resolvent. This is an updated and revised version of the author's PhD thesis.