  For an abelian category and a distinguished object with a graded endomorphism
ring a necessary and sufficient criterion is given so that the category is
equivalent to the abelian quotient of the category of finitely presented graded
modules modulo the Serre subcategory of finite length modules. A particular
example is the category of coherent sheaves on a projective variety, following
a theorem of Serre from 1955. The proof uses Auslander's theory of coherent
functors, and there are no noetherianess assumptions. A theorem of Lenzing for
representations of hereditary algebras is given as an application.
