Class groups and Picard groups of normal schemes (Q1060254)

Let \(\phi\) : \(X\to Y\) be a dominant morphism of irreducible noetherian schemes with Y reduced and normal. Let \(T=\phi^{-1}(Sing(Y))\). Assume that T has codimension \(\geq 2\) in X and contains Sing(X) and that \(\phi\) \(| T: T\to Sing(Y)\) is an isomorphism. Under these conditions it is proved that the map \(\ker (Pic(Y)\to Pic(X))\to \ker (Cl(Y)\to Cl(X))\) is an isomorphism. Some standard results on the relation between the divisor class groups of a normal domain A and those of certain localizations (resp. completions) of A are obtained as immediate corollaries.