Specializations of cofinite subalgebras of a polynomial ring (Q1820210)

Let \(k\) be an algebraically closed field, \(T\), \(X\), \(Y\) some variables, \(S=k[[ T]]\), \(R:=S[X,Y]\), \(K:=Fr(S)\) \((= quotient\) field of \(S\)), \(\bar K\) an algebraic closure of \(K\) and \(A\) a normal sub-\(S\)-algebra of \(R\) such that: (i) \(A\) is a cofinite \(S\)- algebra of \(R\), i.e. \(R\) is a finite \(A\)-module; (ii) \(Fr(R)\) is a quasi- Galois extension of \(Fr(A)\) over \(K\). - Then: (1) the Galois group \(G\) of the extension \(Fr(R)\otimes_ K\bar K/Fr(A)\otimes_ K\bar K\) acts effectively on \(R\) and \(A=R^ G;\) (2) \(R_ k:=R/TR\) is a Galois extension of \(A_ k:=A/TA\) with group \(G\); (3) If \(A_{\bar K}:=A\otimes_ K\bar K\) is a polynomial ring in two variables over \(\bar K\) so is \(A_ k\) over \(k\).