A coherent O X {\displaystyle {}{\mathcal {O}}_{X}} -module F {\displaystyle {}{\mathcal {F}}} on a scheme X {\displaystyle {}X} is called locally free of rank r {\displaystyle {}r} , if there exists an open covering X = ⋃ i ∈ I U i {\displaystyle {}X=\bigcup _{i\in I}U_{i}} and O U i {\displaystyle {}{\mathcal {O}}_{U_{i}}} -module-isomorphisms F | U i ≅ ( O U i ) r {\displaystyle {}{\mathcal {F}}{|}_{U_{i}}\cong {\left({\mathcal {O}}_{U_{i}}\right)}^{r}} for every i ∈ I {\displaystyle {}i\in I} .
