Free resolutions for the tangent cones of some homogeneous pseudo symmetric monomial curves

Year 2023, Volume: 72 Issue: 1, 129 - 136, 30.03.2023

Nil Şahin

https://doi.org/10.31801/cfsuasmas.1117855

Abstract

In this article, we study minimal graded free resolutions of Cohen-Macaulay tangent cones of some monomial curves associated to 4-generated pseudo symmetric numerical semigroups. We explicitly give the matrices in these minimal free resolutions.

Keywords

Minimal graded free resolution, pseudo symmetric monomial curves, homogeneous semigroup, Betti numbers

References

• Barucci, V., Fröberg, R. and Şahin, M., On free resolutions of some semigroup rings, J. Pure Appl. Algebra, 218(6) (2014),1107-1116. https://doi.org/10.1016/j.jpaa.2013.11.007
• Buchsbaum, D., Eisenbud, D., What makes a complex exact?, J. Algebra, 25 (1973), 259-268.
• Eto, K., Almost Gorenstein monomial curves in affine four space, Journal of Algebra, 488 (2017), 362-387. https://doi.org/10.1016/j.jalgebra.2017.05.044
• Greuel, G.-M., Pfister, G., and Schönemann, H., Singular 2.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de
• Gimenez, P., Sengupta, I. and Srinivasan, H., Minimal free resolution for certain affine monomial curves, Commutative Algebra and its Connections to Geometry (PASI 2009), A. Corso and C. Polini Eds, Contemp. Math., 555 (Amer. Math. Soc., 2011) 87–95. https://doi.org/10.1081/AGB-120021893
• Gimenez, P., Sengupta, I. and Srinivasan H., Minimal graded free resolution for monomial curves defined by arithmetic sequences, Journal of Algebra, 388 (2013) 294-310. https://doi.org/10.1016/j.jalgebra.2013.04.026
• Greuel, G.-M., Pfister, G., A Singular Introduction to Commutative Algebra, Springer-Verlag, (2002).
• Herzog, J., Rossi, M.E., Valla, G., On the depth of the symmetric algebra, Trans. Amer. Math. Soc., 296 (2) (1986), 577-606.
• Jafari, R., Zarzuela Armengou, S., Homogeneous numerical semigroups, Semigroup Forum, 97 (2018), 278–306. https://doi.org/10.1007/s00233-018-9941-6
• Komeda, J., On the existence of Weierstrass points with a certain semigroup, Tsukuba J. Math., 6(2) (1982), 237-270.
• Mete, P., Zengin, E.E., Minimal free resolutions of the tangent cones of Gorenstein monomial curves, Turkish Journal of Mathematics, 43 (2019), 2782-2793. https://doi.org/ 10.3906/mat-1903-15
• Mete, P., Zengin, E.E., On minimal free resolution of the associated graded rings of certain monomial curves: new proofs in A4., Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68(1) (2019), 1019–1029. https://doi.org/10.31801/cfsuasmas.501449
• Sengupta, I., A minimal free resolution for certain monomial curves in A4, Comm. Algebra, 31(6) (2003), 2791–2809. https://doi.org/10.1081/AGB-120021893
• Şahin, M., Şahin, N., On pseudo symmetric monomial curves, Communications in Algebra, 46(6) (2018), 2561-2573. https://doi.org/10.1080/00927872.2017.1392532
• Şahin, M., Şahin, N., Betti numbers for certain Cohen-Macaulay tangent cones, Bull. Aust. Math. Soc., 99(1) (2019), 68–77. doi:10.1017/S0004972718000898
• Stamate, D., Betti numbers for numerical semigroup rings, Multigraded Algebra and Applications-NSA 24,2016, Springer Proceedings in Mathematics and Statistics, 238 (eds.V. Ene and E. Miller) (2018)