▁is ▁an ▁ideal ▁she af ▁I ▁which ▁is ▁invert ible ▁and ▁such ▁that ▁for ▁every ▁point ▁x ▁in ▁X , ▁the ▁st alk ▁I x ▁is ▁principal . ▁It ▁is ▁equivalent ▁to ▁require ▁that ▁around ▁each ▁x , ▁there ▁exists ▁an ▁open ▁aff ine ▁subset ▁ ▁such ▁that ▁, ▁where ▁f ▁is ▁a ▁non - zero ▁divis or ▁in ▁A . ▁The ▁sum ▁of ▁two ▁effective ▁Cart ier ▁divis ors ▁corresponds ▁to ▁multiplication ▁of ▁ideal ▁she aves . ▁ ▁There ▁is ▁a ▁good ▁theory ▁of ▁families ▁of ▁effective ▁Cart ier ▁divis ors . ▁Let ▁ ▁be ▁a ▁morph ism . ▁A ▁relative ▁effective ▁Cart ier ▁divis or ▁for ▁X ▁over ▁S ▁is ▁an ▁effective ▁Cart ier ▁divis or ▁D ▁on ▁X ▁which ▁is ▁flat ▁over ▁S . ▁Because ▁of ▁the ▁flat ness ▁assumption , ▁for ▁every ▁ ▁there ▁is ▁a ▁pull back ▁of ▁D ▁to ▁ ▁and ▁this ▁pull back ▁is ▁an ▁effective ▁Cart ier ▁divis or . ▁In ▁particular , ▁this ▁is ▁true ▁for ▁the ▁fib ers ▁of ▁ φ . ▁ ▁Fun ctor ial ity ▁Let ▁ ▁be ▁a ▁morph ism ▁of ▁integral ▁locally ▁No ether ian ▁schemes . ▁ ▁It ▁is ▁often — but ▁not ▁always — possible ▁to ▁use ▁ φ ▁to ▁transfer ▁a ▁divis or ▁D ▁from ▁one ▁scheme ▁to ▁the ▁other . ▁ ▁Whether ▁this ▁is ▁possible ▁depends ▁on ▁whether ▁the ▁divis or ▁is ▁a ▁We il ▁or ▁Cart ier ▁divis or , ▁whether ▁the ▁divis or ▁is ▁to ▁be ▁moved ▁from ▁X ▁to ▁Y ▁or ▁vice