Journal of Algebraic Combinatorics

, Volume 27, Issue 2, pp 215–245

# Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

• Huy Tài Hà
Article

## Abstract

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph ℋ appears within the resolution of its edge ideal ℐ(ℋ). We discuss when recursive formulas to compute the graded Betti numbers of ℐ(ℋ) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405–425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are “well behaved.” For such a hypergraph ℋ (and thus, for any simple graph), we give a lower bound for the regularity of ℐ(ℋ) via combinatorial information describing ℋ and an upper bound for the regularity when ℋ=G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When ℋ is a triangulated hypergraph, we explicitly compute the regularity of ℐ(ℋ) and show that the graded Betti numbers of ℐ(ℋ) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs.

## Keywords

Hypergraphs Chordal graphs Monomial ideals Graded resolutions Regularity

## References

1. 1.
Barile, M. (2006). A note on the edge ideals of Ferrers graphs. Preprint. math.AC/0606353. Google Scholar
2. 2.
Berge, C. (1989). Hypergraphs: combinatorics of finite sets. New York: North-Holland.
3. 3.
Caboara, M., Faridi, S., & Selinger, P. (2007). Simplicial cycles and the computation of simplicial trees. Journal of Symbolic Computation, 42, 74–88.
4. 4.
CoCoATeam. CoCoA: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it.
5. 5.
Corso, A., & Nagel, U. (2006). Monomial and toric ideals associated to Ferrers graphs. Preprint. math.AC/0609371. Google Scholar
6. 6.
Dirac, G. A. (1961). On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 25, 71–76.
7. 7.
Eisenbud, D., Green, M., Hulek, K., & Popescu, S. (2005). Restricting linear syzygies: algebra and geometry. Compositio Mathematica, 141, 1460–1478.
8. 8.
Eliahou, S., & Kervaire, M. (1990). Minimal resolutions of some monomial ideals. Journal of Algebra, 129, 1–25.
9. 9.
Eliahou, S., & Villarreal, R. H. (1999). The second Betti number of an edge ideal. In Aportaciones matematicas, comunicaciones : Vol. 25. XXXI national congress of the Mexican mathematical society (pp. 115–119), Hermosillo, 1998. México: Soc. Mat. Mexicana. Google Scholar
10. 10.
Fatabbi, G. (2001). On the resolution of ideals of fat points. Journal of Algebra, 242, 92–108.
11. 11.
Faridi, S. (2002). The facet ideal of a simplicial complex. Manuscripta Mathematica, 109, 159–174.
12. 12.
Francisco, C., & Hà, H. T. (2006, in press). Whiskers and sequentially Cohen–Macaulay graphs. Journal of Combinatorial Theory, Series A. Preprint. math.AC/0605487. Google Scholar
13. 13.
Francisco, C., & Van Tuyl, A. (2007). Sequentially Cohen–Macaulay edge ideals. Proceedings of the American Mathematical Society, 135, 2327–2337.
14. 14.
Fröberg, R. (1990). On Stanley–Reisner rings. Topics in algebra, 26(2) 57–70. Google Scholar
15. 15.
Herzog, J., Hibi, T., Trung, N. V., & Zheng, X. (2006). Standard graded vertex cover algebras, cycles, and leaves. Preprint. math.AC/0606357. Google Scholar
16. 16.
Herzog, J., Hibi, T., & Zheng, X. (2004). Monomial ideals whose powers have a linear resolution. Mathematica Scandinavica, 95, 23–32.
17. 17.
Herzog, J., Hibi, T., & Zheng, X. (2006). Cohen–Macaulay chordal graphs. Journal of Combinatorial Theory Series A, 113, 911–916.
18. 18.
Herzog, J., Hibi, T., & Zheng, X. (2004). Dirac’s theorem on chordal graphs and Alexander duality. European Journal of Combinatorics, 25, 949–960.
19. 19.
Jacques, S. (2004). Betti numbers of graph ideals. University of Sheffield: PhD thesis. math.AC/0410107. Google Scholar
20. 20.
Jacques, S., & Katzman, M. (2005). The Betti numbers of forests. Preprint. math.AC/0401226. Google Scholar
21. 21.
Katzman, M. (2006). Characteristic-independence of Betti numbers of graph ideals. Journal of Combinatorial Theory Series A, 113, 435–454.
22. 22.
Hà, H. T., & Van Tuyl, A. (2007). Splittable ideals and the resolutions of monomial ideals. Journal of Algebra, 309, 405–425.
23. 23.
Hà, H. T., & Van Tuyl, A. (2006, in press). Resolutions of squarefree monomial ideals via facet ideals: a survey. Contemporary Mathematics (AMS). Preprint. math.AC/0604301. Google Scholar
24. 24.
Miller, E., & Sturmfels, B. (2004). In Springer GTM : Vol. 227. Combinatorial commutative algebra. Berlin: Springer. Google Scholar
25. 25.
Pelsmajer, M. J., Tokaz, J., & West, D. (2004). New proofs for strongly chordal graphs and chordal bipartite graphs. Preprint. Google Scholar
26. 26.
Roth, M., & Van Tuyl, A. (2007) On the linear strand of edge ideals. Communications in Algebra, 35, 821–832.
27. 27.
Simis, A. (1998). On the Jacobian module associated to a graph. Proceedings of the American Mathematical Society, 126, 989–997.
28. 28.
Simis, A., Vasconcelos, W. V., & Villarreal, R. H. (1994). On the ideal theory of graphs. Journal of Algebra, 167, 389–416.
29. 29.
Sturmfels, B., & Sullivant, S. (2006). Combinatorial secant varieties. Quarterly Journal of Pure and Applied Mathematics, 2, 285–309.
30. 30.
Villarreal, R. H. (1995). Rees algebras of edge ideals. Communications in Algebra, 23, 3513–3524.
31. 31.
Villarreal, R. H. (1990). Cohen–Macaulay graphs. Manuscripta Mathematica, 66, 277–293.
32. 32.
Villarreal, R. H. (2001). Monographs and textbooks in pure and applied mathematics : Vol. 238. Monomial algebras. New York: Dekker.
33. 33.
Zheng, X. (2004). Resolutions of facet ideals. Communications in Algebra, 32, 2301–2324.