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ON THE CONJECTURE OF VASCONCELOS FOR ARTINIAN ALMOST COMPLETE INTERSECTION MONOMIAL IDEALS

Part of: Local rings and semilocal rings General commutative ring theory Computational aspects and applications Commutative algebra: Homological methods

Published online by Cambridge University Press:  10 December 2019

KUEI-NUAN LIN Open the ORCID record for KUEI-NUAN LIN [Opens in a new window]  and
YI-HUANG SHEN Open the ORCID record for YI-HUANG SHEN [Opens in a new window]
KUEI-NUAN LIN
Affiliation:
Department of Mathematics, The Penn State University, Greater Allegheny Campus, McKeesport, PA, 15132, USA email [email protected]
YI-HUANG SHEN
Affiliation:
Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China email [email protected]
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Abstract

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 classification

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13D02: Syzygies, resolutions, complexes 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Type
Article
Information
Nagoya Mathematical Journal , Volume 243 , September 2021 , pp. 263 - 277
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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References

Burity, R., Simis, A. and Tohǎneanu, S. O., On a conjecture of Vasconcelos via Sylvester forms, J. Symbolic Comput. 77 (2016), 39–62.CrossRefGoogle Scholar
Cortadellas Benítez, T. and D’Andrea, C., The Rees algebra of a monomial plane parametrization, J. Symbolic Comput. 70 (2015), 71–105.CrossRefGoogle Scholar
Cox, D. A., Lin, K.-N. and Sosa, G., Multi-Rees Algebras and Toric Dynamical Systems, Proc. Amer. Math. Soc. 147 (2019), 4605–4616.CrossRefGoogle Scholar
Cox, D. A., The moving curve ideal and the Rees algebra, Theoret. Comput. Sci. 392 (2008), 23–36.CrossRefGoogle Scholar
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/.Google Scholar
Herzog, J. and Hibi, T., Monomial Ideals, Graduate Texts in Mathematics 260, Springer, London, 2011.CrossRefGoogle Scholar
Herzog, J., Simis, A. and Vasconcelos, W. V., “Koszul homology and blowing-up rings”, in Commutative Algebra (Trento, 1981), Lecture Notes in Pure and Applied Mathematics 84, Dekker, New York, 1983, 79–169.Google Scholar
Hong, J., Simis, A. and Vasconcelos, W. V., The equations of almost complete intersections, Bull. Braz. Math. Soc. (N.S.) 43 (2012), 171–199.CrossRefGoogle Scholar
Hong, J., Simis, A. and Vasconcelos, W. V., Extremal Rees algebras, J. Commut. Algebra 5 (2013), 231–267.10.1216/JCA-2013-5-2-231CrossRefGoogle Scholar
Huneke, C., On the symmetric and Rees algebra of an ideal generated by a d-sequence, J. Algebra 62 (1980), 268–275.CrossRefGoogle Scholar
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.Google Scholar
Lin, K.-N. and Shen, Y.-H., Koszul blowup algebras associated to three-dimensional Ferrers diagrams, J. Algebra 514 (2018), 219–253.CrossRefGoogle Scholar
Lin, K.-N. and Shen, Y.-H., Regularity and multiplicity of toric rings of three-dimensional Ferrers diagrams, available at arXiv:1809.08351.Google Scholar
Rossi, M. E. and Swanson, I., “Notes on the behavior of the Ratliff-Rush filtration”, in Commutative Algebra, Grenoble/Lyon, 2001, Contemporary Mathematics, 331, American Mathematical Society, Providence, RI, 2003, 313–328.Google Scholar
Rossi, M. E., Trung, D. T. and Trung, N. V., Castelnuovo-Mumford regularity and Ratliff-Rush closure, J.  Algebra 504 (2018), 568–586.CrossRefGoogle Scholar
Simis, A. and Tohǎneanu, S. O., The ubiquity of Sylvester forms in almost complete intersections, Collect. Math. 66 (2015), 1–31.CrossRefGoogle Scholar
Sturmfels, B., Gröbner Bases and Convex Polytopes, University Lecture Series 8, American Mathematical Society, Providence, RI, 1996.Google Scholar
Taylor, D. K., Ideals generated by monomials in an R-sequence, Ph.D. thesis, The University of Chicago. ProQuest LLC, Ann Arbor, MI, 1966.Google Scholar
Vasconcelos, W. V., Integral Closure, Springer Monographs in Mathematics, Springer, Berlin, 2005.Google Scholar