▁the ▁class ▁of ▁some ▁Cart ier ▁divis or . ▁As ▁a ▁result , ▁the ▁exact ▁sequence ▁above ▁ident ifies ▁the ▁Pic ard ▁group ▁of ▁line ▁bund les ▁on ▁an ▁integral ▁No ether ian ▁scheme ▁X ▁with ▁the ▁group ▁of ▁Cart ier ▁divis ors ▁mod ulo ▁linear ▁equivalence . ▁This ▁holds ▁more ▁generally ▁for ▁reduced ▁No ether ian ▁schemes , ▁or ▁for ▁quasi - project ive ▁schemes ▁over ▁a ▁No ether ian ▁ring , ▁but ▁it ▁can ▁fail ▁in ▁general ▁( even ▁for ▁proper ▁schemes ▁over ▁C ), ▁which ▁less ens ▁the ▁interest ▁of ▁Cart ier ▁divis ors ▁in ▁full ▁general ity . ▁ ▁Assume ▁D ▁is ▁an ▁effective ▁Cart ier ▁divis or . ▁Then ▁there ▁is ▁a ▁short ▁exact ▁sequence ▁ ▁This ▁sequence ▁is ▁derived ▁from ▁the ▁short ▁exact ▁sequence ▁rel ating ▁the ▁structure ▁she aves ▁of ▁X ▁and ▁D ▁and ▁the ▁ideal ▁she af ▁of ▁D . ▁Because ▁D ▁is ▁a ▁Cart ier ▁divis or , ▁O ( D ) ▁is ▁locally ▁free , ▁and ▁hence ▁tensor ing ▁that ▁sequence ▁by ▁O ( D ) ▁yields ▁another ▁short ▁exact ▁sequence , ▁the ▁one ▁above . ▁When ▁D ▁is ▁smooth , ▁O D ( D ) ▁is ▁the ▁normal ▁bundle ▁of ▁D ▁in ▁X . ▁ ▁Com parison ▁of ▁We il ▁divis ors ▁and ▁Cart ier ▁divis ors ▁A ▁We il ▁divis or ▁D ▁is ▁said ▁to ▁be ▁Cart ier ▁if ▁and ▁only ▁if ▁the ▁she af ▁O ( D ) ▁is ▁invert ible . ▁When ▁this ▁happens , ▁O ( D )