Resolution and Betti numbers of vertex cover ideals
DOI:
https://doi.org/10.7146/math.scand.a-158301Abstract
The vertex cover ideal $J(G)$ of a finite graph $G$ is studied. We characterize when a Cohen–Macaulay vertex cover ideal $J(G)$ has a Scarf minimal free resolution. Furthermore, by using both combinatorial and topological techniques, the graded Betti number $\beta _{i,i+j}(J(G))$, where $i$ and $j$ are the projective dimension and the regularity of $J(G)$, is computed, when $G$ is either a path or a cycle.
References
Alilooee, A., and Faridi, S., On the resolution of path ideals of cycles, Comm. Algebra 43 (2015), no. 12, 5413–5433. https://doi.org/10.1080/00927872.2013.837170
Banerjee, A., Beyarslan, S. K., and Hà, H. T., Regularity of edge ideals and their powers, Advances in algebra, 17–52, Springer Proc. Math. Stat., 277, Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-11521-0_2
Bayer, D., Peeva, I., and Sturmfels, B., Monomial resolutions, Math. Res. Lett. 5 (1998), no. 1–2, 31–46. https://doi.org/10.4310/MRL.1998.v5.n1.a3
Bouchat, R., Hà, H. T., and O'Keefe, A., Path ideals of rooted trees and their graded Betti numbers, J. Combin. Theory Ser. A 118 (2011), no. 8, 2411–2425. https://doi.org/10.1016/j.jcta.2011.06.007
Constantinescu, A., Pournaki, M. R., Seyed Fakhari, S. A., Terai, N., and Yassemi, S., Cohen-Macaulayness and limit behavior of depth for powers of cover ideals, Comm. Algebra 43 (2015), no. 1, 143–157. https://doi.org/10.1080/00927872.2014.897550
Eagon, J. A., and Reiner, V., Resolutions of Stanley–Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), no. 3, 265–275. https://doi.org/10.1016/S0022-4049(97)00097-2
Faridi, S., Hà, H. T., Hibi, T., and Morey S., Scarf complexes of graphs and their powers, arXiv:2403.05439. https://doi.org/10.48550/arXiv.2403.05439
Francisco, C., and Van Tuyl, A., Sequentially Cohen–Macaulay edge ideals, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2327–2337. https://doi.org/10.1090/S0002-9939-07-08841-7
Fröberg, R., On Stanley–Reisner rings with linear resolution, Topics in algebra, Part 2 (Warsaw, 1988), 57–70, Banach Center Publ., 26, Part 2, PWN, Warsaw, 1990.
Hà, H. T., and Van Tuyl, A., Powers of componentwise linear ideals: the Herzog-Hibi-Ohsugi conjecture and related problems, Res. Math. Sci. 9 (2022), no. 2, Paper No. 22, 26 pp. https://doi.org/10.1007/s40687-022-00316-4
He, J., and Van Tuyl, A., Algebraic properties of the path ideal of a tree, Comm. Algebra 38 (2010), no. 5, 1725–1742. https://doi.org/10.1080/00927870902998166
Herzog, J., and Hibi, T., Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141–153. https://doi.org/10.1017/S0027763000006930
Herzog, J., and Hibi, T., Monomial ideals, Graduate Texts in Mathematics, 260. Springer-Verlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-106-6
Herzog, J., Hibi, T., and Ohsugi, H., Powers of componentwise linear ideals, Combinatorial aspects of commutative algebra and algebraic geometry, 49–60, Abel Symp., 6, Springer, Berlin, 2011. https://doi.org/10.1007/978-3-642-19492-4_4
Nevo, E., and Peeva, I., $C_4$-free edge ideals, J. Algebraic Combin. 37 (2013), no. 2, 243–248. https://doi.org/10.1007/s10801-012-0364-2
