Serre’s theorem for coherent sheaves via Auslander’s techniques

Abstract

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 noetherian assumptions. A theorem of Lenzing for representations of hereditary algebras is given as an application.