A Grothendieck topology on a subcategory of opposite category of non commutative algebras. (English) Zbl 0876.18006
Let \(\mathcal C\) be the category whose objects consist of \(R\)-algebras (possibly non-commutative) over a possibly non-commutative ring and whose opposite morphisms consist of direct sums of bi-flat epimorphisms \(A\to B_{i}\). A map \(A\to B\) of \(R\)-algebras is a bi-flat epimorphism if every \(A\)-algebra map \(g: B\to C\) is unique. Bi-flat epimorphisms are viewed dually as localizations. An \(R\)-algebra \(A\) in \(\mathcal C\), denoted \(Geo(A)\) as an element of \(\mathcal C\), acquires via such localizations a Grothendieck topology and the representation functor is a sheaf for this Grothendieck topology. An analogue of affine scheme is thus introduced in the non-commutative case.
Reviewer: P.Cherenack (Rondebosch)
MSC:
18F10 | Grothendieck topologies and Grothendieck topoi |
14F20 | Étale and other Grothendieck topologies and (co)homologies |
18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
14A22 | Noncommutative algebraic geometry |