More generally, a schemeX 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)
Examplesedit
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.
parafactorial, local, ring, algebraic, geometry, noetherian, local, ring, called, parafactorial, depth, least, picard, group, spec, spectrum, with, closed, point, removed, trivial, more, generally, scheme, called, parafactorial, along, closed, subset, subset, . 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 editEvery 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 editGrothendieck Alexander Raynaud Michele 2005 1968 Laszlo Yves ed Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux SGA 2 Documents Mathematiques Paris vol 4 Paris Societe Mathematique de France arXiv math 0511279 Bibcode 2005math 11279G ISBN 978 2 85629 169 6 MR 2171939 Grothendieck Alexandre Dieudonne Jean 1967 Elements de geometrie algebrique IV Etude locale des schemas et des morphismes de schemas Quatrieme partie Publications Mathematiques de l IHES 32 doi 10 1007 bf02732123 MR 0238860 nbsp This commutative algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Parafactorial local ring amp oldid 1170051660, wikipedia, wiki, book, books, library,
