Grothendieck wrote in his letter to Serre(Nov. 12,1957) that every coherent algebraic sheaf on a non-singular algebraic variety(not necessarily quasi-projective) is a quotient of a direct sum of sheaves defined by divisors. I think "sheaves defined by divisors" means locally free sheaves of rank one(i.e. invertible sheaves). How do you prove this?
This is proved for any noetherian separated regular schemes in SGA 6, exposé II, Corollaire 126.96.36.199 (I learn this result from a comment here: such schemes are "divisorial".) To see that this answers your question, look at op. cit. Définition 2.2.3(ii).