The functor $Proj$ of a gradedcommutative algebra is a projective variety. Projective schemes? are particular cases of schemes, so they have nontrivial covers by open affine subschemes. In the noncommutative case, Ore localizations (even on one-sided Ore sets) on Ore sets which are multiplicatively generated by finite subsets behave very much like Zariski-open subsets. F. van Oystaeyen produced an appropriate notion of a cover in terms of Ore localizations at two-sided Ore subsets in a graded algebra to define a notion of schematic algebra.

Definition

…to be written

Literature

F. van Oystaeyen, Algebraic geometry for associative algebras, 287 p. Dekker, 2000.

F. van Oystaeyen, L. Willaert, Cohomology of schematic algebras, J. algebra 185:1 (1996), p. 74–84

F. van Oystaeyen, L. Willaert, Grothendieck topology, coherent sheaves and Serre’s theorem for schematic algebras, J. Pure and Appl. Algebra, 104 (1995), p. 109–122

F. van Oystaeyen, L. Willaert, The quantum site of a schematic algebra, Comm. Alg. 24:1(1996), p. 209–222