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 Grothendieck (1967, 21.13, 1968, XI 3.1,3.2)

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

• Grothendieck, Alexander; Raynaud, Michele (2005) [1968], Laszlo, Yves, ed., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents Mathématiques (Paris), 4, Paris: Société Mathématique de France, arXiv:math/0511279, ISBN 978-2-85629-169-6, MR 2171939
• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32. doi:10.1007/bf02732123. MR 0238860.