ALGEBRAIC COMBINATORICS

Semi-steady non-commutative crepant resolutions via regular dimer models
Algebraic Combinatorics, Volume 2 (2019) no. 2, p. 173-195
A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a 3-dimensional Gorenstein toric singularity. In particular, it is known that a consistent dimer model gives a class of NCCRs called steady if and only if it is homotopy equivalent to a regular hexagonal dimer model. Inspired by this result, we detect another nice property on NCCRs that characterizes square dimer models. We call such NCCRs semi-steady NCCRs, and study their properties.
Received : 2017-08-17
Revised : 2018-06-05
Accepted : 2018-08-06
Published online : 2019-03-05
DOI : https://doi.org/10.5802/alco.39
Classification:  13C14,  05B45,  14E15,  16S38
Keywords: Non-commutative crepant resolutions, Dimer models, Regular tilings, Toric singularities 
@article{ALCO_2019__2_2_173_0,
     author = {Nakajima, Yusuke},
     title = {Semi-steady non-commutative crepant resolutions via regular dimer models},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {2},
     year = {2019},
     pages = {173-195},
     doi = {10.5802/alco.39},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_2_173_0}
}

Nakajima, Yusuke. Semi-steady non-commutative crepant resolutions via regular dimer models. Algebraic Combinatorics, Volume 2 (2019) no. 2, pp. 173-195. doi : 10.5802/alco.39. https://alco.centre-mersenne.org/item/ALCO_2019__2_2_173_0/

[1] Auslander, M. Rational singularities and almost split sequences, Trans. Am. Math. Soc., Volume 293 (1986) no. 2, pp. 511-531 | Article | MR 816307 | Zbl 0594.20030

[2] Bocklandt, R. Consistency conditions for dimer models, Glasg. Math. J., Volume 54 (2012) no. 2, pp. 429-447 | Article | MR 2911380 | Zbl 1244.14042

[3] Bocklandt, R. Generating toric noncommutative crepant resolutions, J. Algebra, Volume 364 (2012), pp. 119-147 | Article | MR 2927051 | Zbl 1263.14006

[4] Bocklandt, R. Toric systems and mirror symmetry, Compos. Math., Volume 149 (2013) no. 11, pp. 1839-1855 | Article | MR 3133295 | Zbl 1396.14045

[5] Bocklandt, R. A dimer ABC, Bull. Lond. Math. Soc., Volume 48 (2016) no. 3, pp. 387-451 | Article | MR 3509904 | Zbl 1345.82006

[6] Bondal, A.; Orlov, D. Derived categories of coherent sheaves, Proceedings of the International Congress of Mathematicians (Beijing, 2002). Vol.D II: Invited lectures, Higher Education Press (2002), pp. 47-56 | Zbl 0996.18007

[7] Bridgeland, T. Flops and derived categories, Invent. Math., Volume 147 (2002) no. 3, pp. 613-632 | Article | MR 1893007 | Zbl 1085.14017

[8] Bridgeland, T.; King, A.; Reid, M. The McKay correspondence as an equivalence of derived categories, J. Am. Math. Soc., Volume 14 (2001) no. 3, pp. 535-554 | Article | MR 1824990 | Zbl 0966.14028

[9] Broomhead, N. Dimer model and Calabi-Yau algebras, Mem. Am. Math. Soc., Volume 215 (2012) no. 1011 | MR 2908565 | Zbl 1237.14002

[10] Bruns, W.; Gubeladze, J. Polytopes, rings and K-theory, Springer, Springer Monographs in Mathematics (2009) | Zbl 1073.14065

[11] Buchweitz, R.-O.; Leuschke, G. J.; Bergh, M. Van Den Non-commutative desingularization of determinantal varieties I, Invent. Math., Volume 182 (2010) no. 1, pp. 47-115 | MR 2672281 | Zbl 1204.14003

[12] Burban, I.; Iyama, O.; Keller, B.; Reiten, I. Cluster tilting for one-dimensional hypersurface singularities, Adv. Math., Volume 217 (2008) no. 6, pp. 2443-2484 | Article | MR 2397457 | Zbl 1143.13014

[13] Cox, D. A.; Little, J. B.; Schenck, H. K. Toric varieties, American Mathematical Society, Graduate Studies in Mathematics, Volume 124 (2011) | MR 2810322 | Zbl 1223.14001

[14] Dao, H. Remarks on non-commutative crepant resolutions of complete intersections, Adv. Math., Volume 224 (2010) no. 3, pp. 1021-1030 | MR 2628801 | Zbl 1192.13011

[15] Dao, H.; Faber, E.; Ingalls, C. Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings, Algebr. Represent. Theory, Volume 18 (2015) no. 3, pp. 633-664 | MR 3357942 | Zbl 1327.14020

[16] Dao, H.; Huneke, C. Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings, Am. J. Math., Volume 135 (2013) no. 2, pp. 561-578 | Article | MR 3038721 | Zbl 1274.13031

[17] Dao, H.; Iyama, O.; Takahashi, R.; Vial, C. Non-commutative resolutions and Grothendieck groups, J. Noncommut. Geom., Volume 9 (2015) no. 1, pp. 21-34 | MR 3337953 | Zbl 1327.13053

[18] Dao, H.; Iyama, O.; Takahashi, R.; Wemyss, M. Gorenstein modifications and ℚ-Gorenstein rings (2016) https://arxiv.org/abs/1611.04137

[19] Duffin, R. J. Potential theory on a rhombic lattice, J. Comb. Theory, Volume 5 (1968), pp. 258-272 | Article | MR 232005 | Zbl 0247.31003

[20] Grünbaum, B.; Shephard, G. C. Tilings and patterns, W. H. Freeman and Company (1987), xii+700 pages | Zbl 0601.05001

[21] Gulotta, D. R. Properly ordered dimers, R-charges, and an efficient inverse algorithm, J. High Energy Phys. (2008) no. 10, 014, 31 pages | Article | MR 2453031 | Zbl 1245.81.091

[22] Hanany, A.; Vegh, D. Quivers, tilings, branes and rhombi, J. High Energy Phys. (2007) no. 10, 029, 35 pages | Article | MR 2357949

[23] Higashitani, A.; Nakajima, Y. Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions (2017) https://arxiv.org/abs/1702.07058

[24] Ishii, A.; Ueda, K. A note on consistency conditions on dimer models, RIMS Kôkyûroku Bessatsu, Volume B24 (2011), pp. 143-164 | Zbl 1270.16012

[25] Ishii, A.; Ueda, K. Dimer models and the special McKay correspondence, Geom. Topol., Volume 19 (2015), pp. 3405-3466 | Article | MR 3447107 | Zbl 1338.14019

[26] Iyama, O. Auslander correspondence, Adv. Math., Volume 210 (2007) no. 1, pp. 51-82 | MR 2298820 | Zbl 1115.16006

[27] Iyama, O.; Nakajima, Y. On steady non-commutative crepant resolutions, J. Noncommut. Geom., Volume 12 (2018) no. 2, pp. 457-471 | Article | MR 3825193 | Zbl 06915978

[28] Iyama, O.; Reiten, I. Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, Am. J. Math., Volume 130 (2008) no. 4, pp. 1087-1149 | Article | MR 2427009 | Zbl 1162.16007

[29] Iyama, O.; Wemyss, M. On the Noncommutative Bondal-Orlov Conjecture, J. Reine Angew. Math., Volume 683 (2013), pp. 119-128 | MR 3181550 | Zbl 1325.14007

[30] Iyama, O.; Wemyss, M. Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math., Volume 197 (2014) no. 3, pp. 521-586 | MR 3251829 | Zbl 1308.14007

[31] Iyama, O.; Wemyss, M. Reduction of triangulated categories and maximal modification algebras for cA n singularities, J. Reine Angew. Math., Volume 738 (2018), pp. 149-202 | Article | Zbl 06866511

[32] Kapranov, M.; Vasserot, E. Kleinian singularities, derived categories and Hall algebras, Math. Ann., Volume 316 (2000) no. 3, pp. 565-576 | Article | MR 1752785 | Zbl 0997.14001

[33] Kenyon, R.; Schlenker, J. M. Rhombic embeddings of planar quadgraphs, Trans. Am. Math. Soc., Volume 357 (2005) no. 9, pp. 3443-3458 | Article

[34] Leuschke, G. J. Non-commutative crepant resolutions: scenes from categorical geometry, Combinatorics and Homology, de Gruyter (Progress in Commutative Algebra) Volume 1 (2012), pp. 293-361 | MR 2932589 | Zbl 1254.13001

[35] Leuschke, G. J.; Wiegand, R. Cohen-Macaulay Representations, American Mathematical Society, Mathematical Surveys and Monographs, Volume 181 (2012) | MR 2919145 | Zbl 1252.13001

[36] Mercat, C. Discrete Riemann surfaces and the Ising model, Commun. Math. Phys., Volume 218 (2001) no. 1, pp. 177-216 | MR 1824204 | Zbl 1043.82005

[37] Nakajima, Y. Mutations of splitting maximal modifying modules: The case of reflexive polygons, Int. Math. Res. Not. (2017) | Article

[38] Špenko, Š.; Van Den Bergh, M. Non-commutative resolutions of quotient singularities for reductive groups, Invent. Math., Volume 210 (2017) no. 1, pp. 3-67 | MR 3698338 | Zbl 1375.13007

[39] Stafford, J. T.; Bergh, M. Van Den Noncommutative resolutions and rational singularities, Mich. Math. J., Volume 57 (2008), pp. 659-674 | Article | MR 2492474 | Zbl 1177.14026

[40] Ueda, K.; Yamazaki, M. A note on dimer models and McKay quivers, Commun. Math. Phys., Volume 301 (2011) no. 3, pp. 723-747 | Article | MR 2784278 | Zbl 1211.81090

[41] Van Den Bergh, M. Non-Commutative Crepant Resolutions, The Legacy of Niels Henrik Abel, Springer (2004), pp. 749-770 | Article | Zbl 1082.14005

[42] Van Den Bergh, M. Three-dimensional flops and noncommutative rings, Duke Math. J., Volume 122 (2004) no. 3, pp. 423-455 | Article | MR 2057015 | Zbl 1074.14013

[43] Wemyss, M. Flops and Clusters in the Homological Minimal Model Program, Invent. Math., Volume 211 (2018) no. 2, pp. 435-521 | Article | MR 3748312 | Zbl 1390.14012

[44] Yoshino, Y. Cohen-Macaulay modules over Cohen-Macaulay rings, Cambridge University Press, London Mathematical Society Lecture Note Series, Volume 146 (1990) | MR 1079937 | Zbl 0745.13003