Divisorial ascent in rings with the approximation property (Q1906636)

A local Noetherian ring \((R,m)\) is said to have the complete approximation property if for every ideal \(I\) of \(R\), the \(I\)-adic completion of \(R\) has the approximation property. For example an excellent Henselian ring which contains the rationals has the complete approximation property. The author gives two main theorems. The first is that if \((R,m)\) is a reduced local ring of dimension \(\geq 3\) which satisfies Serre's condition \((S_3)\) and has the complete approximation property, then \(R\) is parafactorial (if and) only if its \(m\)-adic completion \(\widehat R\) is parafactorial. For the second main theorem, let \((R,m)\) be a local normal Noetherian domain which has the complete approximation property and satisfies Serre's condition \((R_2)\). Then: (i) if the divisor class group \(\text{Cl} (R)\) of \(R\) is finite, then \(\text{Cl} (R)\cong\text{Cl} (\widehat R)\), and (ii) if the torsion subgroup \(\text{Cl}_0(R)\) of \(\text{Cl}(R)\) is finite, then \(\text{Cl}_0(R) \cong \text{Cl}_0 (\widehat R)\).