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LIFTINGS OF A MONOMIAL CURVE

Part of: Theory of modules and ideals Rings and algebras arising under various constructions General commutative ring theory Commutative algebra: Homological methods Computational aspects and applications

Published online by Cambridge University Press:  05 July 2018

MESUT ŞAHİN Open the ORCID record for MESUT ŞAHİN [Opens in a new window]
MESUT ŞAHİN*
Affiliation:
Department of Mathematics, Hacettepe University, Beytepe, 06800, Ankara, Turkey email [email protected]
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Abstract

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We study an operation, that we call lifting, creating nonisomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen–Macaulay tangent cones even if the tangent cone of the original curve is not Cohen–Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen–Macaulay tangent cones when the original monomial curve has a Cohen–Macaulay tangent cone. In this case, all the Betti sequences are just the Betti sequence of the original curve.

Keywords

numerical semigroup ringsmonomial curvestangent conesBetti numbers

MSC classification

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Secondary: 13C13: Other special types 13D02: Syzygies, resolutions, complexes 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 16S36: Ordinary and skew polynomial rings and semigroup rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 230 - 238
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Arslan, F., ‘Cohen–Macaulayness of tangent cones’, Proc. Amer. Math. Soc. 128 (2000), 2243–2251.Google Scholar
Arslan, F., Katsabekis, A. and Nalbandiyan, M., ‘On the Cohen–Macaulayness of tangent cones of monomial curves in $\mathbb{A}^{4}(K)$ ’, Preprint, 2015, arXiv:1512.04204.Google Scholar
Arslan, F. and Mete, P., ‘Hilbert functions of Gorenstein monomial curves’, Proc. Amer. Math. Soc. 135 (2007), 1993–2002.Google Scholar
Barucci, V., Fröberg, R. and Şahin, M., ‘On free resolutions of some semigroup rings’, J. Pure Appl. Algebra 218(6) (2014), 1107–1116.Google Scholar
Charalambous, H., Katsabekis, A. and Thoma, A., ‘Minimal systems of binomial generators and the indispensable complex of a toric ideal’, Proc. Amer. Math. Soc. 135 (2007), 3443–3451.Google Scholar
Charalambous, H. and Thoma, A., ‘On the generalized Scarf complex of lattice ideals’, J. Algebra 323 (2010), 1197–1211.Google Scholar
Cortadellas, T. and Zarzuela, S., ‘Apéry and micro-invariants of a one-dimensional Cohen–Macaulay local ring and invariants of its tangent cone’, J. Algebra 328 (2011), 94–113.Google Scholar
Garcia-Sanchez, P. A. and Ojeda, I., ‘Uniquely presented finitely generated commutative monoids’, Pacific J. Math. 248 (2010), 91–105.Google Scholar
Herzog, J., ‘When is a regular sequence super regular?’, Nagoya Math. J. 83 (1981), 183–195.Google Scholar
Herzog, J. and Stamate, D. I., ‘On the defining equations of the tangent cone of a numerical semigroup ring’, J. Algebra 418 (2014), 8–28.Google Scholar
Huang, I. C., ‘Extensions of tangent cones of monomial curves’, J. Pure Appl. Algebra 220(10) (2016), 3437–3449.Google Scholar
Jafari, R. and Zarzuela, S., ‘On monomial curves obtained by gluing’, Semigroup Forum 88(2) (2014), 397–416.Google Scholar
Morales, M., ‘Noetherian symbolic blow-ups’, J. Algebra 140(1) (1991), 12–25.Google Scholar
Numata, T., ‘A variation of gluing of numerical semigroups’, Semigroup Forum 93(1) (2016), 152–160.Google Scholar
O’Neill, C. and Pelayo, R., ‘Realisable sets of catenary degrees of numerical monoids’, Bull. Aust. Math. Soc. 97 (2018), 240–245.Google Scholar
Şahin, M. and Şahin, N., ‘On pseudo symmetric monomial curves’, Comm. Algebra 46 (2018), 2561–2573.Google Scholar
Shen, Y.-H., ‘Tangent cone of numerical semigroup rings of embedding dimension three’, Comm. Algebra 39(5) (2011), 1922–1940.Google Scholar
Shibuta, T., ‘Cohen–Macaulayness of almost complete intersection tangent cones’, J. Algebra 319 (2008), 3222–3243.Google Scholar
Takemura, A. and Aoki, S., ‘Some characterizations of minimal Markov basis for sampling from discrete conditional distributions’, Ann. Inst. Statist. Math. 56(1) (2004), 1–17.Google Scholar
Thoma, A., ‘On the set-theoretic complete intersection problem for monomial curves in A n and ℙ n ’, J. Pure Appl. Algebra 104(3) (1995), 333–344.Google Scholar
Watanabe, K., ‘Some examples of one dimensional Gorenstein domains’, Nagoya Math. J. 49 (1973), 101–109.Google Scholar