Non-commutative crepant resolutions: scenes from categorical geometry. (English) Zbl 1254.13001

Leuschke, Graham J.

Francisco, Christopher (ed.) et al., Progress in commutative algebra 1. Combinatorics and homology. Berlin: Walter de Gruyter (ISBN 978-3-11-025034-3/hbk; 978-3-11-025040-4/ebook). De Gruyter Proceedings in Mathematics, 293-361 (2012).

In [Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 47–56 (2002; Zbl 0996.18007)], A. Bondal and D. Orlov suggested the possibility of the existence of a categorical resolution of an algebraic variety \(X\). Such a resolution is a pair \((C,K)\) consisting of an abelian category \(C\) of finite homological dimension and of a thick subcategory \(K\subset D^b(C)\) such that the bounded derived category of coherent sheaves on \(X\) is \(D^b(C)/K\). In [The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer. 749–770 (2004; Zbl 1082.14005)], M. Van den Bergh defined a noncommutative crepant resolution, realizing the Bondal–Orlov categorical resolution, and in some cases proved the existence of such a resolution. The paper under review is an expository paper describing recent advances related to this topic.
Very recently, A. Kuznetsov and V. A. Lunts [“Categorical resolutions of irrational singularities”, arXiv:1212.6170] constructed a categorical resolution of any singularity over a field of characteristic zero. It is slightly weaker than the one conjectured by Bondal and Orlov.
For the entire collection see [Zbl 1237.13005].

Reviewer: Adrian Langer (Warszawa)

MSC:

13-02	Research exposition (monographs, survey articles) pertaining to commutative algebra
14E15	Global theory and resolution of singularities (algebro-geometric aspects)
16E35	Derived categories and associative algebras
16S38	Rings arising from noncommutative algebraic geometry
13C14	Cohen-Macaulay modules
14A22	Noncommutative algebraic geometry

Keywords:

non-commutative algebraic geometry; categorical resolution; derived categories; non-commutative crepant resolution; McKay correspondence; minimal model program; tilting

Software:

Macaulay2