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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?

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up vote 3 down vote accepted

This is proved for any noetherian separated regular schemes in SGA 6, exposé II, Corollaire 2.2.7.1 (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).

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