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

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

Abstract

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.

References

  • [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.
Year 2017, , 23 - 35, 11.01.2017
https://doi.org/10.13069/jacodesmath.63088

Abstract

References

  • [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.
There are 10 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Rachelle R. Bouchat This is me

Tricia Muldoon Brown This is me

Publication Date January 11, 2017
Published in Issue Year 2017

Cite

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. January 2017;4(1):23-35. doi:10.13069/jacodesmath.63088
Chicago Bouchat, Rachelle R., and Tricia Muldoon Brown. “A Constructive Approach to Minimal Free Resolutions of Path Ideals of Trees”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, no. 1 (January 2017): 23-35. https://doi.org/10.13069/jacodesmath.63088.
EndNote Bouchat RR, Brown TM (January 1, 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 and T. M. Brown, “A constructive approach to minimal free resolutions of path ideals of trees”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 1, pp. 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 (January 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. and Tricia Muldoon Brown. “A Constructive Approach to Minimal Free Resolutions of Path Ideals of Trees”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 1, 2017, pp. 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.