Subspaces of non-commutative spaces (Q2781401)

Non-commutative algebraic geometry is an active area of research, impulsed by the work of Artin, Rosenberg, van den Bergh and others. Following Rosenberg and van den Bergh, a non-commutative space \(X\) ``is'' a Grothendieck category mod \(X\). The paper under review is a contribution to the theoretical foundations of this subject. The author discusses the notions of weakly open, weakly closed, open and closed subspaces of a non-commutative space \(X\). Closed points of \(X\) are also defined; it is shown that point modules of a graded connected algebra \(A\) give rise to closed points of the projective non-commutative space Proj \(A\). Finally, relations between effective divisors as defined by van den Berg, and weakly closed subspaces are also established.