Parafactorial local ring

In algebraic geometry, a Noetherian local ring "R" is called parafactorial if it has depth at least 2 and the Picard group Pic(Spec("R") − "m") of its spectrum with the closed point "m" removed is trivial.
More generally, a scheme "X" is called parafactorial along a closed subset "Z" if the subset "Z" is "too small" for invertible sheaves to detect; more precisely if for every open set "V" the map from "P"("V") to "P"("V" ∩ "U") is an equivalence of categories, where "U" = "X" – "Z" and "P"("V") is the category of invertible sheaves on "V". A Noetherian local ring is parafactorial if and only if its spectrum is parafactorial along its closed point.
Parafactorial local rings were introduced by 