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The depth of the Rees algebra of three general binary forms. (English) Zbl 1423.13007
In the paper under review, the authors prove that the Rees algebra of an ideal generated by three general binary forms of same degree \(d \geq 5\) has depth one. This result is in sharp contrast to recent akin statements regarding the depth of the Rees algebra of an ideal \(I\) when \(I\) is an almost complete intersection, (see [J. Hong et al., J. Symb. Comput. 43, No. 4, 275–292 (2008; Zbl 1139.13013); J. Commut. Algebra 5, No. 2, 231–267 (2013; Zbl 1274.13015); M. E. Rossi and I. Swanson Contemp. Math. 331, 313–328 (2003; Zbl 1089.13501); A. Simis and S. O. Tohǎneanu, Collect. Math. 66, No. 1, 1–31 (2015; Zbl 1329.13008)], where it has been proved that the Rees algebra is almost Cohen-Macaulay.
The main tools that are used in the proof are: the Ratliff-Rush filtration and the Huckaba-Marley test. The authors also give a conjecture that implies the main theorem of the paper.
13A02 Graded rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Full Text: DOI arXiv
[1] 1994
[2] Heinzer, W.; Lantz, D.; Shah, K., The Ratliff-Rush ideals in a noetherian ring, Comm. Algebra, 20, 2, 591-622, (1992) · Zbl 0747.13002
[3] Hong, J.; Simis, A.; Vasconcelos, W. V., On the homology of two-dimensional elimination, J. Symb. Comp, 43, 4, 275-292, (2008) · Zbl 1139.13013
[4] Hong, J.; Simis, A.; Vasconcelos, W. V., The equations of almost complete intersections, Bull. Braz. Math. Soc, 43, 2, 171-199, (2012) · Zbl 1260.13021
[5] Hong, J.; Simis, A.; Vasconcelos, W. V., Extremal Rees algebras, J. Comm. Algebra, 5, 2, 231-267, (2013) · Zbl 1274.13015
[6] Huckaba, S.; Marley, T., Hilbert coefficients and the depths of associated graded rings, J. London Math. Soc, 56, 1, 64-76, (1997) · Zbl 0910.13008
[7] Kustin, A.; Polini, C.; Ulrich, B., Rational normal scrolls and the defining equations of Rees algebras, J. Reine Angew. Math, 650, 23-65, (2011) · Zbl 1211.13005
[8] Rossi, M. E.; Swanson, I., Contemporary Mathematics, 331, Notes on the behavior of the Ratliff-Rush filtration, (2003), Providence, RI: American Mathematical Society, Providence, RI
[9] Simis, A.; Tohǎneanu, S., The ubiquity of Sylvester forms in almost complete intersections, Collect. Math, 66, 1, 1-31, (2015) · Zbl 1329.13008
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