Subspaces of non-commutative spaces. (English) Zbl 0998.14003
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.
Reviewer: Nicolás Andruskiewitsch (Cordoba)
MSC:
| 14A22 | Noncommutative algebraic geometry |
| 16S38 | Rings arising from noncommutative algebraic geometry |
| 18E15 | Grothendieck categories (MSC2010) |
| 16P40 | Noetherian rings and modules (associative rings and algebras) |
| 18F99 | Categories in geometry and topology |
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