ors . ▁ ▁In ▁terms ▁of ▁local ▁sections , ▁the ▁pull back ▁of ▁ ▁is ▁defined ▁to ▁be ▁. ▁ ▁P ull back ▁is ▁always ▁defined ▁if ▁ φ ▁is ▁dominant , ▁but ▁it ▁cannot ▁be ▁defined ▁in ▁general . ▁ ▁For ▁example , ▁if ▁ ▁and ▁ φ ▁is ▁the ▁inclusion ▁of ▁Z ▁into ▁Y , ▁then ▁ φ * Z ▁is ▁undefined ▁because ▁the ▁corresponding ▁local ▁sections ▁would ▁be ▁everywhere ▁zero . ▁ ▁( The ▁pull back ▁of ▁the ▁corresponding ▁line ▁bundle , ▁however , ▁is ▁defined .) ▁ ▁If ▁ φ ▁is ▁flat , ▁then ▁pull back ▁of ▁We il ▁divis ors ▁is ▁defined . ▁ ▁In ▁this ▁case , ▁the ▁pull back ▁of ▁Z ▁is ▁. ▁ ▁The ▁flat ness ▁of ▁ φ ▁ens ures ▁that ▁the ▁inverse ▁image ▁of ▁Z ▁continues ▁to ▁have ▁cod im ension ▁one . ▁ ▁This ▁can ▁fail ▁for ▁morph isms ▁which ▁are ▁not ▁flat , ▁for ▁example , ▁for ▁a ▁small ▁contra ction . ▁ ▁The ▁first ▁Ch ern ▁class ▁For ▁an ▁integral ▁No ether ian ▁scheme ▁X , ▁the ▁natural ▁hom omorphism ▁from ▁the ▁group ▁of ▁Cart ier ▁divis ors ▁to ▁that ▁of ▁We il ▁divis ors ▁gives ▁a ▁hom omorphism ▁ ▁known ▁as ▁the ▁first ▁Ch ern ▁class . ▁The ▁first ▁Ch ern ▁class ▁is ▁inject ive ▁if ▁X ▁is ▁normal , ▁and ▁it ▁is ▁an ▁isomorphism ▁if ▁X ▁is ▁factor ial ▁( as ▁defined ▁above ). ▁In ▁particular , ▁Cart ier ▁divis ors ▁can ▁be ▁identified ▁with ▁We il ▁divis