▁the ▁Riemann ▁sphere ▁CP 1 . ▁ ▁We il ▁divis ors ▁Let ▁X ▁be ▁an ▁integral ▁locally ▁No ether ian ▁scheme . ▁A ▁prime ▁divis or ▁or ▁ir reducible ▁divis or ▁on ▁X ▁is ▁an ▁integral ▁closed ▁sub sch eme ▁Z ▁of ▁cod im ension ▁ 1 ▁in ▁X . ▁A ▁We il ▁divis or ▁on ▁X ▁is ▁a ▁formal ▁sum ▁over ▁the ▁prime ▁divis ors ▁Z ▁of ▁X , ▁ ▁where ▁the ▁collection ▁ ▁is ▁locally ▁finite . ▁If ▁X ▁is ▁quasi - comp act , ▁local ▁fin iten ess ▁is ▁equivalent ▁to ▁ ▁being ▁finite . ▁The ▁group ▁of ▁all ▁We il ▁divis ors ▁is ▁denoted ▁. ▁A ▁We il ▁divis or ▁D ▁is ▁effective ▁if ▁all ▁the ▁coefficients ▁are ▁non - negative . ▁One ▁writes ▁ ▁if ▁the ▁difference ▁ ▁is ▁effective . ▁ ▁For ▁example , ▁a ▁divis or ▁on ▁an ▁algebraic ▁curve ▁over ▁a ▁field ▁is ▁a ▁formal ▁sum ▁of ▁finit ely ▁many ▁closed ▁points . ▁A ▁divis or ▁on ▁ ▁is ▁a ▁formal ▁sum ▁of ▁prime ▁numbers ▁with ▁integer ▁coefficients ▁and ▁therefore ▁corresponds ▁to ▁a ▁non - zero ▁fraction al ▁ideal ▁in ▁Q . ▁A ▁similar ▁character ization ▁is ▁true ▁for ▁divis ors ▁on ▁ ▁where ▁K ▁is ▁a ▁number ▁field . ▁ ▁If ▁Z ▁ ⊂ ▁X ▁is ▁a ▁prime ▁divis or , ▁then ▁the ▁local ▁ring ▁ ▁has ▁Kr ull ▁dimension ▁one . ▁If ▁ ▁is ▁non - zero , ▁then ▁the ▁order ▁of ▁van ishing ▁of ▁f ▁along ▁Z , ▁written ▁, ▁is ▁the