ors ▁on ▁any ▁regular ▁scheme , ▁and ▁so ▁the ▁first ▁Ch ern ▁class ▁is ▁an ▁isomorphism ▁for ▁X ▁regular . ▁ ▁Exp licit ly , ▁the ▁first ▁Ch ern ▁class ▁can ▁be ▁defined ▁as ▁follows . ▁For ▁a ▁line ▁bundle ▁L ▁on ▁an ▁integral ▁No ether ian ▁scheme ▁X , ▁let ▁s ▁be ▁a ▁non zero ▁rational ▁section ▁of ▁L ▁( that ▁is , ▁a ▁section ▁on ▁some ▁non empty ▁open ▁subset ▁of ▁L ), ▁which ▁exists ▁by ▁local ▁trivial ity ▁of ▁L . ▁Define ▁the ▁We il ▁divis or ▁( s ) ▁on ▁X ▁by ▁anal ogy ▁with ▁the ▁divis or ▁of ▁a ▁rational ▁function . ▁Then ▁the ▁first ▁Ch ern ▁class ▁of ▁L ▁can ▁be ▁defined ▁to ▁be ▁the ▁divis or ▁( s ). ▁Ch anging ▁the ▁rational ▁section ▁s ▁changes ▁this ▁divis or ▁by ▁linear ▁equivalence , ▁since ▁( fs ) ▁= ▁( f ) ▁+ ▁( s ) ▁for ▁a ▁non zero ▁rational ▁function ▁f ▁and ▁a ▁non zero ▁rational ▁section ▁s ▁of ▁L . ▁So ▁the ▁element ▁c 1 ( L ) ▁in ▁Cl ( X ) ▁is ▁well - defined . ▁ ▁For ▁a ▁complex ▁variety ▁X ▁of ▁dimension ▁n , ▁not ▁necessarily ▁smooth ▁or ▁proper ▁over ▁C , ▁there ▁is ▁a ▁natural ▁hom omorphism , ▁the ▁cycle ▁map , ▁from ▁the ▁divis or ▁class ▁group ▁to ▁Bor el – Mo ore ▁hom ology : ▁▁ ▁The ▁latter ▁group ▁is ▁defined ▁using ▁the ▁space ▁X ( C ) ▁of ▁complex ▁points ▁of ▁X , ▁with ▁its ▁classical