▁We il ▁divis or ▁D ▁determ ines ▁a ▁coh er ent ▁she af ▁ ▁on ▁X . ▁Con cret ely ▁it ▁may ▁be ▁defined ▁as ▁sub she af ▁of ▁the ▁she af ▁of ▁rational ▁functions ▁ ▁That ▁is , ▁a ▁non zero ▁rational ▁function ▁f ▁is ▁a ▁section ▁of ▁ ▁over ▁U ▁if ▁and ▁only ▁if ▁for ▁any ▁prime ▁divis or ▁Z ▁intersect ing ▁U , ▁ ▁where ▁n Z ▁is ▁the ▁coefficient ▁of ▁Z ▁in ▁D . ▁If ▁D ▁is ▁principal , ▁so ▁D ▁is ▁the ▁divis or ▁of ▁a ▁rational ▁function ▁g , ▁then ▁there ▁is ▁an ▁isomorphism ▁▁▁▁▁ ▁since ▁ ▁is ▁an ▁effective ▁divis or ▁and ▁so ▁ ▁is ▁regular ▁thanks ▁to ▁the ▁normal ity ▁of ▁X . ▁Con vers ely , ▁if ▁ ▁is ▁is omorphic ▁to ▁ ▁as ▁an ▁- module , ▁then ▁D ▁is ▁principal . ▁It ▁follows ▁that ▁D ▁is ▁locally ▁principal ▁if ▁and ▁only ▁if ▁ ▁is ▁invert ible ; ▁that ▁is , ▁a ▁line ▁bundle . ▁ ▁If ▁D ▁is ▁an ▁effective ▁divis or ▁that ▁corresponds ▁to ▁a ▁sub sch eme ▁of ▁X ▁( for ▁example ▁D ▁can ▁be ▁a ▁reduced ▁divis or ▁or ▁a ▁prime ▁divis or ), ▁then ▁the ▁ideal ▁she af ▁of ▁the ▁sub sch eme ▁D ▁is ▁equal ▁to ▁This ▁leads ▁to ▁an ▁often ▁used ▁short ▁exact ▁sequence , ▁ ▁The ▁she af ▁coh om ology ▁of ▁this ▁sequence ▁shows ▁that ▁ ▁contains ▁information ▁on ▁whether ▁regular ▁functions ▁on ▁D ▁are ▁the ▁restrictions ▁of ▁regular ▁functions ▁on ▁X . ▁