Galois closure data for extensions of rings (Q1742049)

Let \(A\) be a locally free commutative algebra of rank \(n\) over a commutative ring \(B\) with unit. \textit{M. Bhargawa} and \textit{M. Satriano} [J. Eur. Math. Soc. (JEMS) 16, No. 9, 1881--1913 (2014; Zbl 1396.13007)] defined the \(S_n\)-closure \(\bar A\) for the pair \((A,B)\) putting \[ \bar A = A^{\otimes n}/I(A,B), \] where \(A^{\otimes n}=\bigotimes_{i=1}^nA_i\) with \(A_1=A_2=\cdots=A_n=A\), and \(I(A,B)\) is a certain \(S_n\)-invariant ideal of \(A^{\otimes n}\), the group \(S_n\) acting on \( A^{\otimes n}\) by permuting tensor factors. The author generalizes this construction to the case when the group \(S_n\) is replaced by a subgroup \(G\) of \(S_n\) and studies its properties. The case when \(A\) is an étale algebra over a connected ring with an étale fundamental group is treated in more detail (Theorem 1.2).