Betti numbers of toric ideals of graphs: A case study
Abstract
We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and -vector for all the toric ideals of graphs in this family.
Communicated by T. H. Ha
References
- 1. S. Beyarslan, H. T. Hà and A. O’Keefe, Algebraic properties of toric rings of graphs, Preprint 2017, arXiv:1703.08270. Google Scholar
- 2. , Bounds on the regularity of toric ideals of graphs, Adv. Appl. Math. 85 (2017) 84–102. Web of Science, Google Scholar
- 3. ,
Unimodality problems in Ehrhart theory , in Recent Trends in Combinatorics,IMA Mathematical Applications , Vol. 159 (Springer, Cham, 2016), pp. 687–711. Google Scholar - 4. , Ehrhart series, unimodality, and integrally closed reflexive polytopes, Ann. Comb. 20(4) (2016) 705–717. Web of Science, Google Scholar
- 5. W. Bruns and J. Herzog, Cohen–Macaulay Rings Revised Edition (Cambridge University Press, 1998). Google Scholar
- 6. A. Conca and M. Varbaro, Square-free Groebner degenerations, Preprint 2018, arXiv:1805.11923. Google Scholar
- 7. , Monomial and toric ideals associated to Ferrers graphs, Trans. Amer. Math. Soc. 361 (2009) 1371–1395. Web of Science, Google Scholar
- 8. , Toric ideals associated with gap-free graphs, J. Pure Appl. Algebra 219 (2015) 3862–3872. Web of Science, Google Scholar
- 9. , Gröbner Bases in Commutative Algebra,
Graduate Studies in Mathematics , Vol. 130 (American Mathematical Society, 2012). Google Scholar - 10. , Multiplicities of edge subrings, Discrete Math. 302 (2005) 107–123. Web of Science, Google Scholar
- 11. D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/. Google Scholar
- 12. , Resolutions by mapping cones, The Roos Festschrift volume 2, Homol. Homot. Appl. 4(part 2) (2002) 277–294. Google Scholar
- 13. , Depth of edge rings arising from finite graphs, Proc. Amer. Math. Soc. 139 (2011) 3807–3813. Web of Science, Google Scholar
- 14. , Depth of initial ideals of normal edge rings, Comm. Algebra 42 (2014) 2908–2922. Web of Science, Google Scholar
- 15. , Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes, Ann. of Math. 96 (1972) 318–337. Web of Science, Google Scholar
- 16. , Bipartite graphs whose edge algebras are complete intersections, J. Algebra 220 (1999) 519–530. Web of Science, Google Scholar
- 17. M. Michałek, Normal and very ample polytopes — old and new open problems, to appear in Oberwolfach Reports (2017), doi:10.4171/OWR/2017/44. Google Scholar
- 18. , Koszul bipartite graphs, Adv. Appl. Math. 22 (1999) 25–28. Web of Science, Google Scholar
- 19. ,
Graded syzygies , in Algebra and Applications, Vol. 14 (Springer-Verlag London Ltd., London, 2011). Google Scholar - 20. , Minimal generators of toric ideals of graphs, Adv. Appl. Math. 48 (2012) 64–78. Web of Science, Google Scholar
- 21. , Graded Betti numbers of ideals with linear quotients, Matematiche (Catania) 63 (2008) 257–265. Google Scholar
- 22. ,
Log-concave and unimodal sequences in algebra, combinatorics, and geometry , in Graph Theory and its Applications: East and West (Jinan, 1986),Annals New York Academic Science , Vol. 576 (New York Academic Science, New York, 1989), pp. 500–535. Google Scholar - 23. , Gröbner Bases and Convex Polytopes,
University Lecture Series , Vol. 8 (American Mathematical Society, 1996). Google Scholar - 24. , Rees algebras of edge ideals, Comm. Algebra 23 (1995) 3513–3524. Web of Science, Google Scholar
- 25. , Monomial Algebras,
Monographs and Textbooks in Pure and Applied Mathematics , Vol. 238 (Marcel Dekker Inc., New York, 2001). Google Scholar