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Higashitani, Akihiro; Matsushita, Koji

Conic divisorial ideals and non-commutative crepant resolutions of edge rings of complete multipartite graphs. (English) Zbl 1483.13027

J. Algebra 594, 685-711 (2022).
Authors’ abstract: The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by \(\Bbbk[K_{r_1,\dots,r_n}]\), where \(1\leq r_1\leq \ldots\leq r_n\). More concretely, we prove that the class group of \(\Bbbk[K_{r_1,\dots,r_n}]\) is isomorphic to \(\mathbb{Z}^n\) if \(n=3\) with \(r_1\geq 2\) or \(n\geq 4\), while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of \(\Bbbk[K_{r_1,\dots,r_n}]\), called conic divisorial ideals. We describe conic divisorial ideals for certain \(K_{r_1,\dots,r_n}\) including all cases where \(\Bbbk[K_{r_1,\dots,r_n}]\) is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of \(\Bbbk[K_{r_1,\dots,r_n}]\) in the case where it is Gorenstein.
Reviewer: Amir Mafi (Sanandaj and Tehran)

Cited in 4 Documents

MSC:

13C14 Cohen-Macaulay modules
13F65 Commutative rings defined by binomial ideals, toric rings, etc.
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Keywords:

class groups; conic divisorial ideals; non-commutative resolutions; non-commutative crepant resolutions; edge rings; complete multipartite graphs

Software:

Macaulay2
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Full Text: DOI arXiv

References:

[1] Auslander, M., Representation Dimension of Artin Algebras, Lecture Notes (1971), Queen Mary College: Queen Mary College London · Zbl 0331.16026
[2] Broomhead, N., Dimer Model and Calabi-Yau Algebras, vol. 215(1011) (2012), Mem. Amer. Math. Soc.: Mem. Amer. Math. Soc. Providence · Zbl 1237.14002
[3] Bruns, W., Conic divisor classes over a normal monoid algebra, (Commutative Algebra and Algebraic Geometry. Commutative Algebra and Algebraic Geometry, Contemp. Math., vol. 390 (2005), Amer. Math. Soc.), 63-71 · Zbl 1191.13021
[4] Bruns, W.; Gubeladze, J., Divisorial linear algebra of normal semigroup rings, Algebr. Represent. Theory, 6, 139-168 (2003) · Zbl 1046.13014
[5] Bruns, W.; Gubeladze, J., Polytopes, Rings and K-Theory, Springer Monographs in Mathematics (2009), Springer: Springer Dordrecht · Zbl 1168.13001
[6] Dao, H.; Iyama, O.; Takahashi, R.; Wemyss, M., Gorenstein modifications and \(\mathbb{Q} \)-Gorenstein rings, J. Algebraic Geom., 29, 729-751 (2020) · Zbl 1455.14005
[7] De Negri, E.; Hibi, T., Gorenstein algebras of Veronese type, J. Algebra, 193, 629-639 (1997) · Zbl 0884.13012
[8] Faber, E.; Muller, G.; Smith, K. E., Non-commutative resolutions of toric varieties, Adv. Math., 351, 236-274 (2019) · Zbl 1423.13075
[9] Grayson, D.; Stillman, M., Macaulay2, a software system for research in algebraic geometry, Available at
[10] 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
[11] Herzog, J.; Hibi, T.; Ohsugi, H., Binomial Ideals, Graduate Texts in Mathematics, vol. 279 (2018), Springer: Springer Cham · Zbl 1403.13004
[12] Hibi, T., Distributive lattices, affine semigroup rings and algebras with straightening laws, (Nagata, M.; Matsumura, H., Commutative Algebra and Combinatorics. Commutative Algebra and Combinatorics, Advanced Studies in Pure Mathematics, vol. 11 (1987), North-Holland: North-Holland Amsterdam), 93-109 · Zbl 0654.13015
[13] Hibi, T.; Li, N., Unimodular equivalence of order and chain polytopes, Math. Scand., 118, 1, 5-12 (2016) · Zbl 1335.52026
[14] Higashitani, A.; Nakajima, Y., Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions, Sel. Math., 25, Article 78 pp. (2019) · Zbl 1437.13022
[15] Ishii, A.; Ueda, K., Dimer models and the special McKay correspondence, Geom. Topol., 19, 3405-3466 (2015) · Zbl 1338.14019
[16] Iyama, O.; Wemyss, M., Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math., 197, 3, 521-586 (2014) · Zbl 1308.14007
[17] Nakajima, Y., Non-commutative crepant resolutions of Hibi rings with small class groups, J. Pure Appl. Algebra, 223, 3461-3484 (2019) · Zbl 1440.13051
[18] Ohsugi, H.; Hibi, T., Normal polytopes arising from finite graphs, J. Algebra, 207, 409-426 (1998) · Zbl 0926.52017
[19] Ohsugi, H.; Hibi, T., Compressed polytopes, initial ideals and complete multipartite graphs, Ill. J. Math., 44, 2, 391-406 (2000) · Zbl 0943.13016
[20] Simis, A.; Vasconcelos, W. V.; Villarreal, R. H., The integral closure of subrings associated to graphs, J. Algebra, 199, 281-289 (1998) · Zbl 0902.13004
[21] Smith, K. E.; 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
[22] 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
[23] Špenko, Š.; Van den Bergh, M., Non-commutative crepant resolutions for some toric singularities I, Int. Math. Res. Not., 21, 8120-8138 (2020) · Zbl 1457.14003
[24] Špenko, Š.; Van den Bergh, M., Non-commutative crepant resolutions for some toric singularities II, J. Noncommut. Geom., 14, 1, 73-103 (2020) · Zbl 1454.13011
[25] Stanley, R. P., Two poset polytopes, Discrete Comput. Geom., 1, 9-23 (1986) · Zbl 0595.52008
[26] Van den Bergh, M., Non-Commutative Crepant Resolutions, The Legacy of Niels Henrik Abel, 749-770 (2004), Springer: Springer Berlin · Zbl 1082.14005
[27] Villarreal, R. H., Monomial Algebras, Monographs and Research Notes in Mathematics (2015), CRC Press: CRC Press Boca Raton, FL · Zbl 1325.13004
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