Noncommutative resolutions of singularities
Overview
The broad idea would be to learn something about noncommutative analogues of resolutions of singularities (in particular crepant resolutions). This was originally inspired by the McKay correspondence; we have seen in the case of Kleinian singularities the McKay correspondence gives rise to a noncommutative resolution, namely the skew group algebra.We could begin with this example and the definition of a noncommutative crepant resolution as introduced in [VdB1] and [VdB2], as well as the connection between the existence of commutative and noncommutative crepant resolutions and derived equivalences (together with some background on crepant resolutions). It would then be interesting to understand further examples, such as determinantal varieties [BLVdB] as well as restrictions on the existence of noncommutative crepant resolutions in terms of rationality [SVdB] and K-theory [DITV], [D]. In the very unlikely event that time permitted (or there was sufficient interest at the expense of some of the other topics) we could also consider other versions of noncommutative or categorical resolutions which exist in greater generality.
Talks
There will be one session on Friday, December 13, 2013, in room V2-210/216 at the Universität Bielefeld.- 10:15 – 11:30
Andreas Hochenegger: (Crepant) resolutions of singularities in the geometric sense
Some overview of what happens in the geometric setting to give a context for the noncommutative generalisations. - 11:45 – 13:00
David Ploog: On Van den Bergh's Noncommutative crepant resolutions paper
Gives the basic definition and discusses the equivalence between the existence of commutative and noncommutative crepant resolutions for 3-folds as well as the derived equivalence conjectured generally by Bondal and Orlov (and links up with seminar of last term). - 14:15 – 15:30
Julia Sauter: Obstructions to existence of (noncommutative) crepant resolutions
Follows the work of Dao and Dao-Iyama-Takahashi-Vial. It will motivate the many more general definitions of noncommutative resolution that exist. - 15:45 – 17:00
Greg Stevenson: On more general notions of noncommutative or categorical resolutions
For instance the approach of Kuznetzsov and Lunts. Gives a (very small) feeling for the various generalisations of NCR around.
Literature
[B] Noncommutative counterparts of the Springer resolution by R. Bezrukavnikov.[BLVdB] Non-commutative desingularization of determinantal varieties I by R. Buchweitz, G.J. Leuschke and M. Van den Bergh.
[D] Remarks on non-commutative crepant resolutions of complete intersections by H. Dao.
[DITV] Non-commutative resolutions and Grothendieck groups by H. Dao, O. Iyama, R. Takahashi and C. Vial.
[KL] Categorical resolutions of irrational singularities by A. Kuznetsov and V.A. Lunts.
[L] Non-commutative crepant resolutions: scenes from categorical geometry by G.J. Leuschke.
[SVdB] Noncommutative resolutions and rational singularities by J.T. Stafford and M. Van den Berg.
[VdB1] Three-dimensional flops and non-commutative rings by M. Van den Bergh.
[VdB2] Non-commutative crepant resolutions by M. Van den Bergh.