Generalized splittings of monomial ideals

Yıl 2024, Early Access, 1 - 13

Rachelle R. Bouchat Tricia Muldoon Brown

https://doi.org/10.24330/ieja.1488479

Öz

Eliahou and Kervaire defined splittable monomial ideals and provided a relationship between the Betti numbers of the more complicated ideal in terms of the less complicated pieces. We extend the concept of splittable monomial ideals showing that an ideal which was not splittable according to the original definition is splittable in this more general definition. Further, we provide a generalized version of the result concerning the relationship between the Betti numbers.

Anahtar Kelimeler

Betti number, splitting, ideal

Kaynakça

• L. Amata, A. Ficarra and M. Crupi, A numerical characterization of the extremal Betti numbers of {$t$}-spread strongly stable ideals, J. Algebraic Combin., 55(3) (2022), 891-918.
• R. R. Bouchat and T. M. Brown, Fibonacci numbers and resolutions of domino ideals, J. Algebra Comb. Discrete Struct. Appl., 6(2) (2019), 63-74.
• S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra, 129(1) (1990), 1-25.
• G. Fatabbi, On the resolution of ideals of fat points, J. Algebra, 242(1) (2001), 92-108.
• C. A. Francisco, Resolutions of small sets of fat points, J. Pure Appl. Algebra, 203(1-3) (2005), 220-236.
• C. A. Francisco, H. T. Hà and A. Van Tuyl, Splittings of monomial ideals, Proc. Amer. Math. Soc., 137(10) (2009), 3271-3282.
• D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www2.macaulay2.com.
• S. Güntürkün, Boij-Söderberg decompositions of lexicographic ideals, J. Commut. Algebra, 13(2) (2021), 209-234.
• H. T. Hà and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin., 27(2) (2008), 215-245.