noncommutative
[email protected] Lieven Le Bruyn Universiteit Antwerpen (UIA), B-2610 Antwerp (Belgium) E-mail address:
[email protected] URL: http://win-www.uia.ac.be/u/lebruyn/ 2000 Mathematics Subject Classification. Primary 14A22,16A30 ; Secondary 13A50, 14D20, 14L24, 16G20, 53D30 Abstract. This is yet another version of the book noncommutative
[email protected] by Lieven Le Bruyn. Constructive criticism is, as always, wellcome at
[email protected] Introduction ”... La suite est trop confuse dans les notes pour ˆetre exploitable telle quelle.” Bellaiche, Dat, Marin, Racinet, Randriambololona in [33]. Rather than adding to the plethora of pet-proposals for a noncommutative geometry, we will focus in this book on some methods that are likely to prove useful in the ’final theory’. Whereas the details of this theory are unclear at the time of writing, the rough outline is slowly emerging. The starting point is that a lot of interesting (families of) moduli spaces in algebraic geometry are special cases of the isomorphism problem in suitable Abelian categories ab moduli ⊂ - iso(ab) In recent years one has come to realize that many of these naturally occurring Abelian categories are locally controlled by noncommutative algebras ab = ∪i rep Ai where rep Ai is the Abelian category of all finite dimensional representations of the affine noncommutative algebra Ai and where the covering is c