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Abstract
Crepant resolutions of threedimensional toric Gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models. By choosing a special stability parameter and hence a distinguished crepant resolution (Formula presented.), this derived equivalence generalises the FourierMukai transform relating the (Formula presented.)Hilbert scheme and the skew group algebra (Formula presented.) for a finite abelian subgroup of (Formula presented.). We show that this equivalence sends the vertex simples to pure sheaves, except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus. This generalises results of CautisLogvinenko (J Reine Angew Math 636:193236, 2009) and CautisCrawLogvinenko (J Reine Angew Math arXiv:1205.3110, 2014) to the dimer setting, though our approach is different in each case. We also describe some of these pure sheaves explicitly and compute the support of the remainder, providing a dimer model analogue of results from Logvinenko (J Algebra 324:20642087, 2010).
Original language  English 

Pages (fromto)  689723 
Number of pages  34 
Journal  Mathematische Annalen 
Volume  361 
Issue number  34 
Early online date  21 Aug 2014 
DOIs  
Publication status  Published  30 Apr 2015 
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 1 Finished

Alastair Craw  Bridgeland Stability and the Moveable Cone
Engineering and Physical Sciences Research Council
3/04/13 → 2/10/16
Project: Research council