Linear resolutions and quasi-linearity of monomial ideals
DOI:
https://doi.org/10.7146/math.scand.a-136634Abstract
We introduce the notion of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and clarify all the quasi-linear monomial ideals generated in degree $2$. We also introduce the notion of a strongly linear monomial over a monomial ideal and prove that if $\mathbf {u}$ is a monomial strongly linear over $I$ then $I$ has a linear resolution (respectively is quasi-linear) if and only if $I+\mathbf {u}\mathfrak {p}$ has a linear resolution (respectively is quasi-linear). Here $\mathfrak {p}$ is any monomial prime ideal.
References
Bigdeli, M., Herzog, J., Yazdan Pour, A. A., and Zaare-Nahandi, R., Simplicial orders and chordality, J. Algebraic Comb. 45 (2017), no. 4, 1021–1039. https://doi.org/10.1007/s10801-016-0733-3
Bigdeli, M., and Yazdan Pour, A. A. Multigraded minimal free resolutions of simplicial subclutters, J. Combin. Theory Ser. A 178 (2021), Paper No. 105339, 28 pp. https://doi.org/10.1016/j.jcta.2020.105339
bibitem BYZ Bigdeli, M., Yazdan Pour, A. A., and Zaare-Nahandi, R., Stability of Betti numbers under reduction processes: towards chordality of clutters, J. Combin. Theory Ser. A 145 (2017), 129–149. https://doi.org/10.1016/j.jcta.2016.06.001
Björner, A., and Wachs, M. L., Shellable nonpure complexes and posets II, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3945–3975. https://doi.org/10.1090/S0002-9947-97-01838-2
Dao, H., Doolittle, J., and Lyle, J., Minimal Cohen-Macaulay simplicial complexes, SIAM J. Discrete Math. 34 (2020), no. 3, 1602–1608. https://doi.org/10.1137/19M1275164
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
Fröberg, R., On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988), 57–70, Banach Center Publ., 26, Part 2, PWN, Warsaw, 1990.
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 Zheng, X., Monomial ideals whose powers have a linear resolution, Math. Scand. 95 (2004), no. 1, 23–32. https://doi.org/10.7146/math.scand.a-14446
Herzog, J., Reiner, V., and Welker, V., Componentwise linear ideals and Golod rings, Michigan Math. J. 46 (1999), no. 2, 211–223. https://doi.org/10.1307/mmj/1030132406
Herzog, J., and Takayama, Y., Resolutions by mapping cones, Homology Homotopy Appl. 4 (2002), no. 2, part 2, 277–294. https://doi.org/10.4310/hha.2002.v4.n2.a13
Peeva, I., Graded syzygies, Algebra and Applications, 14. Springer-Verlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-177-6
