Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras

Abstract

We define schematic algebras to be algebras which have “enough” Ore-sets. Many graded algebras studied nowadays are schematic. We construct a generalised Grothendieck topology for the free monoid on all Ore-sets of a schematic algebra R. This allows us to develop a sheaf theory which is similar to the scheme theory for commutative algebras. In particular, we obtain an equivalence between the category of all coherent sheaves and the category Proj R as it is defined in (Artin; 1992).