|
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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Parafactorial local ring」の詳細全文を読む スポンサード リンク
|