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Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions. (English) Zbl 1437.13022
Summary: In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.
##### MSC:
 13C14 Cohen-Macaulay modules 06A11 Algebraic aspects of posets 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 16S38 Rings arising from noncommutative algebraic geometry
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##### References:
 [1] Auslander, M., Representation Dimension of Artin Algebras (1971), London: Queen Mary College, London [2] Auslander, M.; Reiten, I.; Smalo, So, Representation Theory of Artin Algebras (1995), Cambridge: Cambridge University Press, Cambridge [3] Baeţica, C., Cohen-Macaulay classes which are not conic, Commun. Algebra, 32, 1183-1188 (2004) · Zbl 1080.13005 [4] Bocklandt, R., Generating toric noncommutative crepant resolutions, J. Algebra, 364, 119-147 (2012) · Zbl 1263.14006 [5] 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 [6] Broomhead, N., Dimer Model and Calabi-Yau Algebras (2012), Providence: Mem. Amer. Math. Soc., Providence · Zbl 1237.14002 [7] Bruns, W., Conic Divisor Classes Over a Normal Monoid Algebra. Commutative Algebra and Algebraic Geometry, 63-71 (2005), Providence: Amer. Math. Soc., Providence · Zbl 1191.13021 [8] Bruns, W.; Gubeladze, J., Divisorial linear algebra of normal semigroup rings, Algebra Represent. Theory, 6, 139-168 (2003) · Zbl 1046.13014 [9] Bruns, W.; Gubeladze, J., Polytopes, Rings and K-Theory (2009), Dordrecht: Springer, Dordrecht [10] Buchweitz, R-O; Leuschke, Gj; Van Den Bergh, M., Non-commutative desingularization of determinantal varieties II: arbitrary minors, Int. Math. Res. Not. IMRN, 9, 2748-2812 (2016) · Zbl 1434.14001 [11] 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 [12] Dao, H., Iyama, O., Takahashi, R., Wemyss, M.: Gorenstein modifications and $${\mathbb{Q}}$$-Gorenstein rings. arXiv:1611.04137 [13] Dong, X., Canonical modules of semigroup rings and a conjecture of Reiner, Discrete Comput. Geom., 27, 85-97 (2002) · Zbl 1001.20053 [14] Faber, Eleonore; Muller, Greg; Smith, Karen E., Non-commutative resolutions of toric varieties, Advances in Mathematics, 351, 236-274 (2019) · Zbl 1423.13075 [15] Hashimoto, M.: Equivariant class group. III. Almost principal fiber bundles. arXiv:1503.02133 [16] Hashimoto, M.; Hibi, T.; Noma, A., Divisor class groups of affine semigroup rings associated with distributive lattices, J. Algebra, 149, 2, 352-357 (1992) · Zbl 0759.13009 [17] Hibi, T.; Nagata, M.; Matsumura, H., Distributive lattices, affine semigroup rings and algebras with straightening laws, Commutative Algebra and Combinatorics, 93-109 (1987), Amsterdam: North-Holland, Amsterdam [18] Ishii, A.; Ueda, K., Dimer models and the special McKay correspondence, Geom. Topol., 19, 3405-3466 (2015) · Zbl 1338.14019 [19] Iyama, O., Auslander correspondence, Adv. Math., 210, 1, 51-82 (2007) · Zbl 1115.16006 [20] Iyama, O.; Nakajima, Y., On steady non-commutative crepant resolutions, J. Noncommut. Geom., 12, 2, 457-471 (2018) · Zbl 1419.16012 [21] 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 [22] Iyama, O.; Wemyss, M., Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math., 197, 3, 521-586 (2014) · Zbl 1308.14007 [23] Kuznetsov, A., Lefschetz decompositions and categorical resolutions of singularities, Sel. Math. (N.S.), 13, 4, 661-696 (2008) · Zbl 1156.18006 [24] Nakajima, Y., Mutations of splitting maximal modifying modules: the case of reflexive polygons, Int. Math. Res. Not. IMRN, 23, 2, 470-550 (2019) · Zbl 07130846 [25] Nakajima, Y., Non-commutative crepant resolutions of Hibi rings with small class group, J. Pure Appl. Algebra, 223, 8, 3461-3484 (2019) · Zbl 1440.13051 [26] Rouquier, R., Dimensions of triangulated categories, J. K Theory, 1, 2, 193-256 (2008) · Zbl 1165.18008 [27] Smith, Ke; Van Den Bergh, M., Simplicity of rings of differential operators in prime characteristic, Proc. Lond. Math. Soc. (3), 75, 1, 32-62 (1997) · Zbl 0948.16019 [28] 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 [29] Špenko, Š., Van den Bergh, M.: Non-commutative crepant resolutions for some toric singularities I. arXiv:1701.05255 [30] Špenko, Š., Van den Bergh, M.: Non-commutative crepant resolutions for some toric singularities II. J. Noncommut. Geom. (to appear). arXiv:1707.08245 [31] Stanley, R.P.: Combinatorics and invariant theory. Relations Between Combinatorics and Other Parts of Mathematics. Proceedings of Symposia in Pure Mathematics, vol. 34, pp. 345-355. Amer. Math. Soc., Providence (1979) [32] Stanley, Rp, Two poset polytopes, Discrete Comput. Geom., 1, 9-23 (1986) · Zbl 0595.52008 [33] Van Den Bergh, M., Cohen-Macaulayness of semi-invariants for tori, Trans. Am. Math. Soc., 336, 2, 557-580 (1993) · Zbl 0791.13003 [34] Van Den Bergh, M., Three-dimensional flops and noncommutative rings, Duke Math. J., 122, 3, 423-455 (2004) · Zbl 1074.14013 [35] Van Den Bergh, M., Non-Commutative Crepant Resolutions, 749-770 (2004), Berlin: Springer, Berlin · Zbl 1082.14005 [36] Wemyss, M., Flops and clusters in the homological minimal model program, Invent. Math., 211, 2, 435-521 (2018) · Zbl 1390.14012
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