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Semi-steady non-commutative crepant resolutions via regular dimer models. (English) Zbl 1419.13019
Non-commutative crepant resolutions (NCCRs) were introduced in [M. Van den Bergh, 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)]. Given a Cohen-Macaulay normal domain \(R\), an NCCR is the endomorphism ring \(\Lambda = \mathrm{End}_R(M)\) of a reflexive \(R\)-module \(M\), with \(\Lambda\) satisfying some non-singularity conditions. Many NCCRs can be realized as path algebras of quivers. The paper under review investigates NCCRs where \(M\) is called semi-steady, showing an equivalence to path algebras from square dimer models.
This paper is a sequel to [O. Iyama and Y. Nakajima, J. Noncommut. Geom. 12, No. 2, 457–471 (2018; Zbl 1419.16012)]. That previous work studied steady modules. A reflexive module \(M\) is steady if \(M\) is a generator (that is, \(R \in \mathrm{add}_R(M)\)) and \(\mathrm{End}_R(M) \in \mathrm{add}_R(M)\). An NCCR is steady if \(M\) is. Here \(\mathrm{add}_R(M)\) is the full subcategory consisting of direct summands of finite direct sums of some copies of \(M\). The authors found that a dimer model \(\Gamma\) was homotopy equivalent to a dimer model of regular hexagons if and only if \(\Gamma\) gives a steady NCCR.
The paper under review generalizes to semi-steady \(M\). That is, when \(M\) is again a generator, and given a decomposition \(M = \oplus M_i\) into non-isomorphic summands, \(\mathrm{Hom}_R(M_i, M) \in \mathrm{add}_R(M)\) or \(\mathrm{Hom}_R(M_i, M) \in \mathrm{add}_R(M^*)\). It is shown that \(M\) is steady if and only if \(M\) is semi-steady and \(\mathrm{add}_R(M) = \mathrm{add}_R(M^*)\).
The main result is that a dimer model \(\Gamma\) is homotopy equivalent to a dimer model of squares if and only if \(\Gamma\) gives a semi-steady, but not steady, NCCR. A tiling of equilateral triangles cannot be realized as a dimer model. This leads to a satisfying conclusion, that a dimer model of regular polygons is in some sense equivalent to semi-steady NCCRs, with regular hexagons and squares corresponding to steady and non-steady NCCRs, respectively.
MSC:
13C14 Cohen-Macaulay modules
05B45 Combinatorial aspects of tessellation and tiling problems
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
16S38 Rings arising from noncommutative algebraic geometry
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References:
[1] Auslander, M., Rational singularities and almost split sequences, Trans. Am. Math. Soc., 293, 2, 511-531, (1986) · Zbl 0594.20030
[2] Bocklandt, R., Consistency conditions for dimer models, Glasg. Math. J., 54, 2, 429-447, (2012) · Zbl 1244.14042
[3] Bocklandt, R., Generating toric noncommutative crepant resolutions, J. Algebra, 364, 119-147, (2012) · Zbl 1263.14006
[4] Bocklandt, R., Toric systems and mirror symmetry, Compos. Math., 149, 11, 1839-1855, (2013) · Zbl 1396.14045
[5] Bocklandt, R., A dimer ABC, Bull. Lond. Math. Soc., 48, 3, 387-451, (2016) · Zbl 1345.82006
[6] Bondal, A.; Orlov, D., Proceedings of the International Congress of Mathematicians (Beijing, 2002). Vol.D II: Invited lectures, Derived categories of coherent sheaves, 47-56, (2002), Higher Education Press · Zbl 0996.18007
[7] Bridgeland, T., Flops and derived categories, Invent. Math., 147, 3, 613-632, (2002) · Zbl 1085.14017
[8] Bridgeland, T.; King, A.; Reid, M., The McKay correspondence as an equivalence of derived categories, J. Am. Math. Soc., 14, 3, 535-554, (2001) · Zbl 0966.14028
[9] Broomhead, N., Dimer model and Calabi-Yau algebras, Mem. Am. Math. Soc., 215, 1011, (2012) · Zbl 1237.14002
[10] Bruns, W.; Gubeladze, J., Polytopes, rings and K-theory, (2009), Springer · Zbl 1073.14065
[11] Buchweitz, R.-O.; Leuschke, G. J.; Bergh, M. Van den, Non-commutative desingularization of determinantal varieties I, Invent. Math., 182, 1, 47-115, (2010) · Zbl 1204.14003
[12] Burban, I.; Iyama, O.; Keller, B.; Reiten, I., Cluster tilting for one-dimensional hypersurface singularities, Adv. Math., 217, 6, 2443-2484, (2008) · Zbl 1143.13014
[13] Cox, D. A.; Little, J. B.; Schenck, H. K., Toric varieties, 124, (2011), American Mathematical Society · Zbl 1223.14001
[14] Dao, H., Remarks on non-commutative crepant resolutions of complete intersections, Adv. Math., 224, 3, 1021-1030, (2010) · Zbl 1192.13011
[15] Dao, H.; Faber, E.; Ingalls, C., Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings, Algebr. Represent. Theory, 18, 3, 633-664, (2015) · Zbl 1327.14020
[16] Dao, H.; Huneke, C., Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings, Am. J. Math., 135, 2, 561-578, (2013) · Zbl 1274.13031
[17] Dao, H.; Iyama, O.; Takahashi, R.; Vial, C., Non-commutative resolutions and Grothendieck groups, J. Noncommut. Geom., 9, 1, 21-34, (2015) · Zbl 1327.13053
[18] Dao, H.; Iyama, O.; Takahashi, R.; Wemyss, M., Gorenstein modifications and \(\mathbb{Q}\)-Gorenstein rings, (2016)
[19] Duffin, R. J., Potential theory on a rhombic lattice, J. Comb. Theory, 5, 258-272, (1968) · Zbl 0247.31003
[20] Grünbaum, B.; Shephard, G. C., Tilings and patterns, xii+700 p. pp., (1987), W. H. Freeman and Company · Zbl 0601.05001
[21] Gulotta, D. R., Properly ordered dimers, \({R}\)-charges, and an efficient inverse algorithm, J. High Energy Phys., 10, 31 p. pp., (2008) · Zbl 1245.81091
[22] Hanany, A.; Vegh, D., Quivers, tilings, branes and rhombi, J. High Energy Phys., 10, 35 p. pp., (2007)
[23] Higashitani, A.; Nakajima, Y., Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions, (2017)
[24] Ishii, A.; Ueda, K., A note on consistency conditions on dimer models, RIMS Kôkyûroku Bessatsu, B24, 143-164, (2011) · Zbl 1270.16012
[25] Ishii, A.; Ueda, K., Dimer models and the special McKay correspondence, Geom. Topol., 19, 3405-3466, (2015) · Zbl 1338.14019
[26] Iyama, O., Auslander correspondence, Adv. Math., 210, 1, 51-82, (2007) · Zbl 1115.16006
[27] Iyama, O.; Nakajima, Y., On steady non-commutative crepant resolutions, J. Noncommut. Geom., 12, 2, 457-471, (2018) · Zbl 1419.16012
[28] Iyama, O.; Reiten, I., Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, Am. J. Math., 130, 4, 1087-1149, (2008) · Zbl 1162.16007
[29] Iyama, O.; Wemyss, M., On the Noncommutative Bondal-Orlov Conjecture, J. Reine Angew. Math., 683, 119-128, (2013) · Zbl 1325.14007
[30] Iyama, O.; Wemyss, M., Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math., 197, 3, 521-586, (2014) · Zbl 1308.14007
[31] Iyama, O.; Wemyss, M., Reduction of triangulated categories and maximal modification algebras for \(c{A}_n\) singularities, J. Reine Angew. Math., 738, 149-202, (2018) · Zbl 1428.18012
[32] Kapranov, M.; Vasserot, E., Kleinian singularities, derived categories and Hall algebras, Math. Ann., 316, 3, 565-576, (2000) · Zbl 0997.14001
[33] Kenyon, R.; Schlenker, J. M., Rhombic embeddings of planar quadgraphs, Trans. Am. Math. Soc., 357, 9, 3443-3458, (2005) · Zbl 1062.05045
[34] Leuschke, G. J., Combinatorics and Homology, 1, Non-commutative crepant resolutions: scenes from categorical geometry, 293-361, (2012), de Gruyter · Zbl 1254.13001
[35] Leuschke, G. J.; Wiegand, R., Cohen-Macaulay Representations, 181, (2012), American Mathematical Society · Zbl 1252.13001
[36] Mercat, C., Discrete Riemann surfaces and the Ising model, Commun. Math. Phys., 218, 1, 177-216, (2001) · Zbl 1043.82005
[37] Nakajima, Y., Mutations of splitting maximal modifying modules: The case of reflexive polygons, Int. Math. Res. Not., (2017)
[38] Špenko, Š.; Van den Bergh, M., Non-commutative resolutions of quotient singularities for reductive groups, Invent. Math., 210, 1, 3-67, (2017) · Zbl 1375.13007
[39] Stafford, J. T.; Bergh, M. Van den, Noncommutative resolutions and rational singularities, Mich. Math. J., 57, 659-674, (2008) · Zbl 1177.14026
[40] Ueda, K.; Yamazaki, M., A note on dimer models and McKay quivers, Commun. Math. Phys., 301, 3, 723-747, (2011) · Zbl 1211.81090
[41] Van den Bergh, M., The Legacy of Niels Henrik Abel, Non-Commutative Crepant Resolutions, 749-770, (2004), Springer · Zbl 1082.14005
[42] Van den Bergh, M., Three-dimensional flops and noncommutative rings, Duke Math. J., 122, 3, 423-455, (2004) · Zbl 1074.14013
[43] Wemyss, M., Flops and Clusters in the Homological Minimal Model Program, Invent. Math., 211, 2, 435-521, (2018) · Zbl 1390.14012
[44] Yoshino, Y., Cohen-Macaulay modules over Cohen-Macaulay rings, 146, (1990), Cambridge University Press · Zbl 0745.13003
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