Araştırma Makalesi
BibTex RIS Kaynak Göster

A constructive approach to minimal free resolutions of path ideals of trees

Yıl 2017, , 23 - 35, 11.01.2017
https://doi.org/10.13069/jacodesmath.63088

Öz

For a rooted tree $\Gamma ,$ we consider path ideals of $\Gamma$, which are ideals that are generated by all directed paths of a fixed length in $\Gamma$. In this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. In particular, we provide a class of subforests of $\Gamma$ that are in one-to-one correspondence with the multi-graded Betti numbers of the path ideal as well as providing a method for determining the projective dimension and the Castelnuovo-Mumford regularity of a given path ideal.

Kaynakça

  • [1] A. Alilooee, S. Faridi, On the resolution of path ideals of cycles, Commun. Algebra 43(12) (2015) 5413–5433.
  • [2] R. R. Bouchat, T. M. Brown, Multi–graded Betti numbers of path ideals of trees, to appear in J. Algebra Appl.
  • [3] R. Bouchat, A. O’Keefe, H. Tài Hà, Path ideals of rooted trees and their graded Betti numbers, J. Combin. Theory Ser. A 118(8) (2011) 2411–2425.
  • [4] A. Conca, E. De Negri, M–sequences, graph ideals, and ladder ideals of linear type, J. Algebra 211(2) (1999) 599–624.
  • [5] R. Ehrenborg, G. Hetyei, The topology of the independence complex, European J. Combin. 27(6) (2006) 906–923.
  • [6] D. Grayson, M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.
  • [7] H. Tài Hà, A. Van Tuyl, Monomial ideals, edge ideals of hyper graphs, and their graded Betti numbers, J. Algebraic Combin. 27(2) (2008) 215–245.
  • [8] M. Katzman, Characteristic–independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113(3) (2006) 435–454.
  • [9] M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30(4) (2009) 429–445.
  • [10] U. Nagel, V. Reiner, Betti numbers of monomial ideals and shifted skew shapes, Electron. J. Combin. 16(2) (2009) 1–59.
Yıl 2017, , 23 - 35, 11.01.2017
https://doi.org/10.13069/jacodesmath.63088

Öz

Kaynakça

  • [1] A. Alilooee, S. Faridi, On the resolution of path ideals of cycles, Commun. Algebra 43(12) (2015) 5413–5433.
  • [2] R. R. Bouchat, T. M. Brown, Multi–graded Betti numbers of path ideals of trees, to appear in J. Algebra Appl.
  • [3] R. Bouchat, A. O’Keefe, H. Tài Hà, Path ideals of rooted trees and their graded Betti numbers, J. Combin. Theory Ser. A 118(8) (2011) 2411–2425.
  • [4] A. Conca, E. De Negri, M–sequences, graph ideals, and ladder ideals of linear type, J. Algebra 211(2) (1999) 599–624.
  • [5] R. Ehrenborg, G. Hetyei, The topology of the independence complex, European J. Combin. 27(6) (2006) 906–923.
  • [6] D. Grayson, M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.
  • [7] H. Tài Hà, A. Van Tuyl, Monomial ideals, edge ideals of hyper graphs, and their graded Betti numbers, J. Algebraic Combin. 27(2) (2008) 215–245.
  • [8] M. Katzman, Characteristic–independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113(3) (2006) 435–454.
  • [9] M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30(4) (2009) 429–445.
  • [10] U. Nagel, V. Reiner, Betti numbers of monomial ideals and shifted skew shapes, Electron. J. Combin. 16(2) (2009) 1–59.
Toplam 10 adet kaynakça vardır.

Ayrıntılar

Konular Mühendislik
Bölüm Makaleler
Yazarlar

Rachelle R. Bouchat Bu kişi benim

Tricia Muldoon Brown Bu kişi benim

Yayımlanma Tarihi 11 Ocak 2017
Yayımlandığı Sayı Yıl 2017

Kaynak Göster

APA Bouchat, R. R., & Brown, T. M. (2017). A constructive approach to minimal free resolutions of path ideals of trees. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(1), 23-35. https://doi.org/10.13069/jacodesmath.63088
AMA Bouchat RR, Brown TM. A constructive approach to minimal free resolutions of path ideals of trees. Journal of Algebra Combinatorics Discrete Structures and Applications. Ocak 2017;4(1):23-35. doi:10.13069/jacodesmath.63088
Chicago Bouchat, Rachelle R., ve Tricia Muldoon Brown. “A Constructive Approach to Minimal Free Resolutions of Path Ideals of Trees”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, sy. 1 (Ocak 2017): 23-35. https://doi.org/10.13069/jacodesmath.63088.
EndNote Bouchat RR, Brown TM (01 Ocak 2017) A constructive approach to minimal free resolutions of path ideals of trees. Journal of Algebra Combinatorics Discrete Structures and Applications 4 1 23–35.
IEEE R. R. Bouchat ve T. M. Brown, “A constructive approach to minimal free resolutions of path ideals of trees”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 4, sy. 1, ss. 23–35, 2017, doi: 10.13069/jacodesmath.63088.
ISNAD Bouchat, Rachelle R. - Brown, Tricia Muldoon. “A Constructive Approach to Minimal Free Resolutions of Path Ideals of Trees”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/1 (Ocak 2017), 23-35. https://doi.org/10.13069/jacodesmath.63088.
JAMA Bouchat RR, Brown TM. A constructive approach to minimal free resolutions of path ideals of trees. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:23–35.
MLA Bouchat, Rachelle R. ve Tricia Muldoon Brown. “A Constructive Approach to Minimal Free Resolutions of Path Ideals of Trees”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 4, sy. 1, 2017, ss. 23-35, doi:10.13069/jacodesmath.63088.
Vancouver Bouchat RR, Brown TM. A constructive approach to minimal free resolutions of path ideals of trees. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(1):23-35.