On the conjecture of Vasconcelos for Artinian almost complete intersection monomial ideals. (English) Zbl 1467.13007
Summary: In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen-Macaulay.
MSC:
| 13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
Software:
Macaulay2References:
| [1] | Burity, R., Simis, A. and Tohǎneanu, S. O., On a conjecture of Vasconcelos via Sylvester forms, J. Symbolic Comput.77 (2016), 39-62. · Zbl 1356.13001 |
| [2] | Cortadellas Benítez, T. and D’Andrea, C., The Rees algebra of a monomial plane parametrization, J. Symbolic Comput.70 (2015), 71-105. · Zbl 1327.13018 |
| [3] | Cox, D. A., Lin, K.-N. and Sosa, G., Multi-Rees Algebras and Toric Dynamical Systems, Proc. Amer. Math. Soc.147 (2019), 4605-4616. · Zbl 1429.13006 |
| [4] | Cox, D. A., The moving curve ideal and the Rees algebra, Theoret. Comput. Sci.392 (2008), 23-36. · Zbl 1170.13004 |
| [5] | Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, 2018, available at http://www.math.uiuc.edu/Macaulay2/. |
| [6] | Herzog, J. and Hibi, T., Monomial Ideals, , Springer, London, 2011. · Zbl 1206.13001 |
| [7] | Herzog, J., Simis, A. and Vasconcelos, W. V., “Koszul homology and blowing-up rings”, in Commutative Algebra (Trento, 1981), , Dekker, New York, 1983, 79-169. · Zbl 0499.13002 |
| [8] | Hong, J., Simis, A. and Vasconcelos, W. V., The equations of almost complete intersections, Bull. Braz. Math. Soc. (N.S.)43 (2012), 171-199. · Zbl 1260.13021 |
| [9] | Hong, J., Simis, A. and Vasconcelos, W. V., Extremal Rees algebras, J. Commut. Algebra5 (2013), 231-267. · Zbl 1274.13015 |
| [10] | Huneke, C., On the symmetric and Rees algebra of an ideal generated by a d-sequence, J. Algebra62 (1980), 268-275. · Zbl 0439.13001 |
| [11] | Kustin, A. R., Polini, C. and Ulrich, B., Rational normal scrolls and the defining equations of Rees algebras, J. Reine Angew. Math.650 (2011), 23-65. · Zbl 1211.13005 |
| [12] | Lin, K.-N. and Shen, Y.-H., Koszul blowup algebras associated to three-dimensional Ferrers diagrams, J. Algebra514 (2018), 219-253. · Zbl 1433.13019 |
| [13] | Lin, K.-N. and Shen, Y.-H., Regularity and multiplicity of toric rings of three-dimensional Ferrers diagrams, available at arXiv:1809.08351. |
| [14] | Rossi, M. E. and Swanson, I., “Notes on the behavior of the Ratliff-Rush filtration”, in Commutative Algebra, Grenoble/Lyon, 2001, , American Mathematical Society, Providence, RI, 2003, 313-328. · Zbl 1089.13501 |
| [15] | Rossi, M. E., Trung, D. T. and Trung, N. V., Castelnuovo-Mumford regularity and Ratliff-Rush closure, J. Algebra504 (2018), 568-586. · Zbl 1412.13015 |
| [16] | Simis, A. and Tohǎneanu, S. O., The ubiquity of Sylvester forms in almost complete intersections, Collect. Math.66 (2015), 1-31. · Zbl 1329.13008 |
| [17] | Sturmfels, B., Gröbner Bases and Convex Polytopes, , American Mathematical Society, Providence, RI, 1996. · Zbl 0856.13020 |
| [18] | Taylor, D. K., Ideals generated by monomials in an R-sequence, Ph.D. thesis, The University of Chicago. ProQuest LLC, Ann Arbor, MI, 1966. |
| [19] | Vasconcelos, W. V., Integral Closure, , Springer, Berlin, 2005. · Zbl 1082.13006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
