▁of ▁ ▁on ▁ ▁and ▁ ▁on ▁ ▁up ▁to ▁multiplication ▁by ▁a ▁section ▁of ▁▁ ▁Cart ier ▁divis ors ▁also ▁have ▁a ▁she af - the or etic ▁description . ▁A ▁fraction al ▁ideal ▁she af ▁is ▁a ▁sub -- module ▁of ▁ ▁A ▁fraction al ▁ideal ▁she af ▁J ▁is ▁invert ible ▁if , ▁for ▁each ▁x ▁in ▁X , ▁there ▁exists ▁an ▁open ▁neighborhood ▁U ▁of ▁x ▁on ▁which ▁the ▁restriction ▁of ▁J ▁to ▁U ▁is ▁equal ▁to ▁ ▁where ▁ ▁and ▁the ▁product ▁is ▁taken ▁in ▁ ▁Each ▁Cart ier ▁divis or ▁defines ▁an ▁invert ible ▁fraction al ▁ideal ▁she af ▁using ▁the ▁description ▁of ▁the ▁Cart ier ▁divis or ▁as ▁a ▁collection ▁ ▁and ▁convers ely , ▁invert ible ▁fraction al ▁ideal ▁she aves ▁define ▁Cart ier ▁divis ors . ▁If ▁the ▁Cart ier ▁divis or ▁is ▁not ated ▁D , ▁then ▁the ▁corresponding ▁fraction al ▁ideal ▁she af ▁is ▁not ated ▁O ( D ) ▁or ▁L ( D ). ▁ ▁By ▁the ▁exact ▁sequence ▁above , ▁there ▁is ▁an ▁exact ▁sequence ▁of ▁she af ▁coh om ology ▁groups : ▁ ▁A ▁Cart ier ▁divis or ▁is ▁said ▁to ▁be ▁principal ▁if ▁it ▁is ▁in ▁the ▁image ▁of ▁the ▁hom omorphism ▁ ▁that ▁is , ▁if ▁it ▁is ▁the ▁divis or ▁of ▁a ▁rational ▁function ▁on ▁X . ▁Two ▁Cart ier ▁divis ors ▁are ▁linear ly ▁equivalent ▁if ▁their ▁difference ▁is ▁principal . ▁Every ▁line ▁bundle ▁L ▁on ▁X ▁on ▁an ▁integral ▁No ether ian ▁scheme ▁is