×

Specializations of Ferrers ideals. (English) Zbl 1179.13008

A Ferrers graph is a bipartite graph on two distinct vertex sets \(\{x_1,\dots, x_n\}\) and \(\{y_1, \dots , y_m\}\) with the property that if \((x_i, y_j)\) is an edge of \(G\), then so is \((x_p, y_q)\) for \(1\leq p \leq i\) and \(1 \leq q \leq j\). In addition, \((x_1, y_m)\) and \((x_n, y_1)\) are required to be edges of \(G.\) A Ferrers ideals is the edge ideal associated with a Ferrers graph. The central idea of this paper is to obtain other monomial ideals (in general non square-free) from Ferrers ideals, by means of a specialization process which, roughly speaking, consists in identifying each \(y\)-vertex with an \(x\)-vertex. The authors extend this specialization to the polyhedral cell complex that resolves the Ferrers ideals and obtain, under suitable hypotheses, a cellular minimal free resolution of the specialized Ferrers ideals. Various classes of ideals and graphs are given, which can be obtained as a specializations of Ferrers ideals; in particular, all threshold graphs and all strongly stable ideals generated in degree two can be obtained as such a specialization.

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aramova, A., Herzog, J., Hibi, T.: Squarefree lexsegment ideals. Math. Z. 228, 353-378 (1998) · Zbl 0914.13007 · doi:10.1007/PL00004621
[2] Bayer, D., Sturmfels, B.: Cellular resolutions of monomial modules. J. Reine Angew. Math. 502, 123-140 (1998) · Zbl 0909.13011
[3] Bayer, D., Peeva, I., Sturmfels, B.: Monomial resolutions. Math. Res. Lett. 5, 31-46 (1998) · Zbl 0909.13010
[4] Corso, A., Nagel, U.: Monomial and toric ideals associated to Ferrers graphs. Trans. Am. Math. Soc. (2007, to appear) · Zbl 1228.05068
[5] Eisenbud, D., Green, M., Hulek, K., Popescu, S.: Small schemes and varieties of minimal degree. Am. J. Math. 128, 1363-1389 (2006) · Zbl 1108.14042 · doi:10.1353/ajm.2006.0043
[6] Eliahou, S., Kervaire, M.: Minimal resolutions of some monomial ideals. J. Algebra 129, 1-25 (1990) · Zbl 0701.13006 · doi:10.1016/0021-8693(90)90237-I
[7] Horwitz, N.: Linear resolutions of quadratic monomial ideals. J. Algebra 318, 981-1001 (2007) · Zbl 1142.13011 · doi:10.1016/j.jalgebra.2007.06.006
[8] Klivans, C., Reiner, V.: Shifted set families, degree sequences, and plethysm. Electron. J. Comb. (2007, to appear) · Zbl 1180.05033
[9] Mahadev, N.V.R., Peled, U.N.: Threshold Graphs and Related Topics. Annals Discrete Mathematics, vol. 56. North-Holland, Amsterdam (1995) · Zbl 0852.05001
[10] Migliore, J., Nagel, U.: Lifting monomial ideals. Commun. Algebra 28, 5679-5701 (2000). Special volume in honor of R. Hartshorne · Zbl 1003.13005 · doi:10.1080/00927870008827182
[11] Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005) · Zbl 1090.13001
[12] Villarreal, R.: Monomial Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 238. Marcel Dekker, New York (2001) · Zbl 1002.13010
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.