Non-Koszul quadratic Gorenstein toric rings

  • Kazunori Matsuda

Abstract

Koszulness of Gorenstein quadratic algebras of small socle degree is studied. In this paper, we construct non-Koszul Gorenstein quadratic toric ring such that its socle degree is more than $3$ by using stable set polytopes.

References

Aramova, A., Herzog, J., and Hibi, T., Shellability of semigroup rings, Nagoya Math. J. 168 (2002), 65–84. https://doi.org/10.1017/S0027763000008357

>

Blum, S., Initially Koszul algebras, Beiträge Algebra Geom. 41 (2000), no. 2, 455–467.

Chappell, T., Friedl, T., and Sanyal, R., Two double poset polytopes, SIAM J. Discrete Math. 31 (2017), no. 4, 2378–2413. https://doi.org/10.1137/16M1091800

>

Chudnovsky, M., Robertson, N., Seymour, P., and Thomas, R., The strong perfect graph theorem, Ann. of Math. (2) 164 (2006), no. 1, 51–229. https://doi.org/10.4007/annals.2006.164.51

>

Conca, A., Universally Koszul algebras, Math. Ann. 317 (2000), no. 2, 329–346. https://doi.org/10.1007/s002080000100

>

Conca, A., De Negri, E., and Rossi, M. E., Koszul algebras and regularity, in “Commutative algebra”, Springer, New York, 2013, pp. 285--315. https://doi.org/10.1007/978-1-4614-5292-8_8

>

Conca, A., Rossi, M. E., and Valla, G., Gröbner flags and Gorenstein algebras, Compositio Math. 129 (2001), no. 1, 95–121. https://doi.org/10.1023/A:1013160203998

>

Diestel, R., Graph theory, fourth ed., Graduate Texts in Mathematics, vol. 173, Springer, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-14279-6

>

Ene, V., Herzog, J., and Hibi, T., Koszul binomial edge ideals, in “Bridging algebra, geometry, and topology”, Springer Proc. Math. Stat., vol. 96, Springer, Cham, 2014, pp. 125--136. https://doi.org/10.1007/978-3-319-09186-0_8

>

Engström, A. and Norén, P., Ideals of graph homomorphisms, Ann. Comb. 17 (2013), no. 1, 71–103. https://doi.org/10.1007/s00026-012-0169-y

>

Fröberg, R., Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29–39. https://doi.org/10.7146/math.scand.a-11585

>

Fulkerson, D. R., Hoffman, A. J., and McAndrew, M. H., Some properties of graphs with multiple edges, Canad. J. Math. 17 (1965), 166–177. https://doi.org/10.4153/CJM-1965-016-2

>

Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.

>

Herzog, J., Hibi, T., and Restuccia, G., Strongly Koszul algebras, Math. Scand. 86 (2000), no. 2, 161–178. https://doi.org/10.7146/math.scand.a-14287

>

Hibi, T., Distributive lattices, affine semigroup rings and algebras with straightening laws, in “Commutative algebra and combinatorics (Kyoto, 1985)'', Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 93--109.

Hibi, T. and Li, N., Chain polytopes and algebras with straightening laws, Acta Math. Vietnam. 40 (2015), no. 3, 447–452. https://doi.org/10.1007/s40306-015-0115-2

>

Hibi, T. and Matsuda, K., Quadratic Gröbner bases of twinned order polytopes, European J. Combin. 54 (2016), 187–192. https://doi.org/10.1016/j.ejc.2015.12.014

>

Hibi, T., Matsuda, K., and Ohsugi, H., Strongly Koszul edge rings, Acta Math. Vietnam. 41 (2016), no. 1, 69–76. https://doi.org/10.1007/s40306-014-0097-5

>

Hibi, T., Matsuda, K., Ohsugi, H., and Shibata, K., Centrally symmetric configurations of order polytopes, J. Algebra 443 (2015), 469–478. https://doi.org/10.1016/j.jalgebra.2015.06.010

>

Hibi, T., Matsuda, K., and Tsuchiya, A., Gorenstein Fano polytopes arising from order polytopes and chain polytopes, preprint arXiv:1507.03221 [math.CO], 2015.

Hibi, T., Matsuda, K., and Tsuchiya, A., Quadratic Gröbner bases arising from partially ordered sets, Math. Scand. 121 (2017), no. 1, 19–25. https://doi.org/10.7146/math.scand.a-26246

>

Hibi, T., Nishiyama, K., Ohsugi, H., and Shikama, A., Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases, J. Algebra 408 (2014), 138–146. https://doi.org/10.1016/j.jalgebra.2013.09.039

>

Hochster, M., Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337. https://doi.org/10.2307/1970791

>

Ishida, M.-N., Torus embeddings and dualizing complexes, Tôhoku Math. J. (2) 32 (1980), no. 1, 111–146. https://doi.org/10.2748/tmj/1178229687

>

Mancini, F., Graph modification problems related to graph classes, Ph.D. thesis, University of Bergen, 2008.

Matsuda, K., Strong Koszulness of toric rings associated with stable set polytopes of trivially perfect graphs, J. Algebra Appl. 13 (2014), no. 4, 1350138, 11 pp. https://doi.org/10.1142/S0219498813501387

>

Matsuda, K. and Ohsugi, H., Reverse lexicographic Gröbner bases and strongly Koszul toric rings, Math. Scand. 119 (2016), no. 2, 161–168. https://doi.org/10.7146/math.scand.a-24741

>

Matsuda, K., Ohsugi, H., and Shibata, K., Toric rings and ideals of stable set polytopes, preprint arXiv:1603.01850 [math.AC], 2016.

Ohsugi, H., Herzog, J., and Hibi, T., Combinatorial pure subrings, Osaka J. Math. 37 (2000), no. 3, 745–757.

Ohsugi, H. and Hibi, T., Toric ideals generated by quadratic binomials, J. Algebra 218 (1999), no. 2, 509–527. https://doi.org/10.1006/jabr.1999.7918

>

Ohsugi, H. and Hibi, T., Special simplices and Gorenstein toric rings, J. Combin. Theory Ser. A 113 (2006), no. 4, 718–725. https://doi.org/10.1016/j.jcta.2005.06.002

>

Priddy, S. B., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. https://doi.org/10.2307/1995637

>

Roos, J.-E. and Sturmfels, B., A toric ring with irrational Poincaré-Betti series, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 2, 141–146. https://doi.org/10.1016/S0764-4442(97)89459-1

>

Shibata, K., Strong Koszulness of the toric ring associated to a cut ideal, Comment. Math. Univ. St. Pauli 64 (2015), no. 1, 71–80.

Stanley, R. P., Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57–83. https://doi.org/10.1016/0001-8708(78)90045-2

>

Sturmfels, B., Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996.

Sturmfels, B., Four counterexamples in combinatorial algebraic geometry, J. Algebra 230 (2000), no. 1, 282–294. https://doi.org/10.1006/jabr.1999.7950

>

Tate, J., Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–27.

Published
2018-09-05
How to Cite
Matsuda, K. (2018). Non-Koszul quadratic Gorenstein toric rings. MATHEMATICA SCANDINAVICA, 123(2), 161-173. https://doi.org/10.7146/math.scand.a-105278
Section
Articles