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

Examples

*Every Noetherian local ring of dimension at least 2 that is factorial is parafactorial. However local rings of dimension at most 1 are not parafactorial, even if they are factorial.
*Every Noetherian complete intersection local ring of dimension at least 4 is parafactorial.
*For a locally Noetherian scheme, a closed subset is parafactorial if the local ring at every point of the subset is parafactorial. For a locally Noetherian regular scheme, the closed parafactorial subsets are those of codimension at least 2.

References

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Commutative algebra

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