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

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.

Keywords

• Betti numbers,Path ideals,Rooted trees,Monomial ideals

References

1. [1] A. Alilooee, S. Faridi, On the resolution of path ideals of cycles, Commun. Algebra 43(12) (2015) 5413–5433.
2. [2] R. R. Bouchat, T. M. Brown, Multi–graded Betti numbers of path ideals of trees, to appear in J. Algebra Appl.
3. [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. [4] A. Conca, E. De Negri, M–sequences, graph ideals, and ladder ideals of linear type, J. Algebra 211(2) (1999) 599–624.
5. [5] R. Ehrenborg, G. Hetyei, The topology of the independence complex, European J. Combin. 27(6) (2006) 906–923.
6. [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. [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. [8] M. Katzman, Characteristic–independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113(3) (2006) 435–454.

9. [9] M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30(4) (2009) 429–445.
10. [10] U. Nagel, V. Reiner, Betti numbers of monomial ideals and shifted skew shapes, Electron. J. Combin. 16(2) (2009) 1–59.