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We show that for a quasicompact quasiseparated scheme   X  , the following assertions are equivalent: (1) the category    QCoh  (  X  )    of all quasicoherent sheaves on   X   has a flat generator; (2) for every injective object   E   of    QCoh  (  X  )   , the internal hom functor into   E   is exact; (3) the scheme   X   is semiseparated.

On flat generators and Matlis duality for quasicoherent sheaves