Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras (Q1903703)

Associated to a commutative graded ring, in this note, the authors aim to associate to schematic algebras (several of which occur in the framework of quantum groups) a ``geometric'' object. Schematic algebras are characterized by the fact that they possess ``many'' Ore sets, the latter permitting the authors to introduce a suitable notion of ``noncommutative Grothendieck topology'' (based upon the free monoid over the Ore sets), as well as constructing an associated notion of ``sheaf'' over these. In general, as pointed out in the text, the structure ``sheaf'' thus associated to a schematic algebra \(R\) is not a ``sheaf of rings'', nor does the construction specialize to that of \(\text{Proj} (R)\) in the commutative case (as only covers of open sets induced by global ones are considered). However, the authors introduce a suitable notion of quasicoherent sheaf over these geometric objects and prove that the category of these is equivalent to Artin's \(\text{Proj} (R)\), thus providing an interesting step towards a full geometric understanding of the latter.