▁be ▁the ▁project ive ▁n - space ▁with ▁the ▁hom ogeneous ▁coordinates ▁x 0 , ▁... , ▁x n . ▁Let ▁U ▁= ▁{ x 0 ▁ ≠ ▁ 0 }. ▁Then ▁U ▁is ▁is omorphic ▁to ▁the ▁aff ine ▁n - space ▁with ▁the ▁coordinates ▁y i ▁= ▁x i / x 0 . ▁Let ▁ ▁Then ▁ ω ▁is ▁a ▁rational ▁differential ▁form ▁on ▁U ; ▁thus , ▁it ▁is ▁a ▁rational ▁section ▁of ▁ ▁which ▁has ▁simple ▁pol es ▁along ▁Z i ▁= ▁{ xi ▁= ▁ 0 }, ▁i ▁= ▁ 1 , ▁... , ▁n . ▁Switch ing ▁to ▁a ▁different ▁aff ine ▁chart ▁changes ▁only ▁the ▁sign ▁of ▁ ω ▁and ▁so ▁we ▁see ▁ ω ▁has ▁a ▁simple ▁pole ▁along ▁Z 0 ▁as ▁well . ▁Thus , ▁the ▁divis or ▁of ▁ ω ▁is ▁ ▁and ▁its ▁divis or ▁class ▁is ▁ ▁where ▁[ H ] ▁= ▁[ Z i ], ▁i ▁= ▁ 0 , ▁... , ▁n . ▁( See ▁also ▁the ▁E uler ▁sequence .) ▁ ▁Cart ier ▁divis ors ▁Let ▁X ▁be ▁an ▁integral ▁No ether ian ▁scheme . ▁Then ▁X ▁has ▁a ▁she af ▁of ▁rational ▁functions ▁ ▁All ▁regular ▁functions ▁are ▁rational ▁functions , ▁which ▁leads ▁to ▁a ▁short ▁exact ▁sequence ▁ ▁A ▁Cart ier ▁divis or ▁on ▁X ▁is ▁a ▁global ▁section ▁of ▁ ▁An ▁equivalent ▁description ▁is ▁that ▁a ▁Cart ier ▁divis or ▁is ▁a ▁collection ▁ ▁where ▁ ▁is ▁an ▁open ▁cover ▁of ▁ ▁is ▁a ▁section