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Generalized mixed product ideals. (English) Zbl 1303.13012

Mixed product ideals were introduced in [G. Restuccia and R. H. Villarreal, Commun. Algebra 29, No. 8, 3571–3580 (2001; Zbl 1080.13509)]. Afterwards, algebraic invariants of this class of ideals were studied in [C. Ionescu and G. Rinaldo, Arch. Math. 91, No. 1, 20–30 (2008; Zbl 1229.13025); G. Rinaldo, Arch. Math. 91, No. 5, 416–426 (2008; Zbl 1168.13020); L. T. Hoa and N. D. Tam, Arch. Math. 94, No. 4, 327–337 (2010; Zbl 1191.13032)]. In the present paper, the authors are generalizing this construction introducing the generalized mixed product ideal \(L\) induced by a monomial ideal \(I.\) This also generalizes the expansion construction in [S. Bayati and J. Herzog, Rocky Mountain J. Math., to appear]. The authors are computing the minimal graded free resolution of \(L\) and show that \(I\) and \(L\) have the same regularity.

MSC:

13C13 Other special types of modules and ideals in commutative rings
13D02 Syzygies, resolutions, complexes and commutative rings
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References:

[1] S. Bayati and J. Herzog, Expansion of monomial ideals and multigraded modules, Rocky Mountain J. Math. (2014, to appear). arXiv:1205.3599v1. · Zbl 1327.13042
[2] Conca A., De Negri E.: M-sequences, graph ideals, and ladder ideals of linear type. J. Algebra 211, 599-624 (1999) · Zbl 0924.13012 · doi:10.1006/jabr.1998.7740
[3] Hoa T., Tam N.: On some invariants of a mixed product idals. Arch. Math. 94, 327-337 (2010) · Zbl 1191.13032 · doi:10.1007/s00013-010-0112-6
[4] C. Ionescu and G. Rinaldo, Some algebraic invariants related to mixed product ideals, Arch. Math. 91 (2008), 20-30. · Zbl 1229.13025
[5] Restuccia G., Villarreal R.: On the normality of monomial ideals of mixed products. Commun. Algebra 29, 3571-3580 (2001) · Zbl 1080.13509 · doi:10.1081/AGB-100105039
[6] Rinaldo G.: Betti numbers of mixed product ideals. Arch. Math. 91, 416-426 (2008) · Zbl 1168.13020 · doi:10.1007/s00013-008-2781-y
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