Large lower bounds for the betti numbers of graded modules with low regularity

Lower bounds on Betti numbers

“…For example, Erman showed [40] that if M is a graded module of small regularity (in terms of the degrees of the first syzygies), then not only is the BEH Conjecture 4.1 true, but in fact β i (M) β 0 (M) c i . The first author and Wigglesworth [12] then extended Erman's work to say that under the same low regularity hypothesis, β(M) β 0 (M)(2 c + 2 c−1 ). This stronger bound asserts that on average, each Betti number β i (M) is at least 1.5 times β 0 (M) c i .…”

“…Charalambous, Evans, and Miller [31] asked if in fact we must have β(M) 2 c + 2 c−1 , and proved that this holds when M is either a graded module small codimension (c 4), or a multigraded module of finite length (meaning c = n) for arbitrary c [30,29]. More evidence towards this larger bound for Betti numbers has recently been found, including [11,12].…”

“…Erman's proof uses general Boij-Söderberg techniques to reduce studying the Betti tables of arbitrary modules to the study of pure modules and then use a degeneration argument to supply the required numerical bound. These techniques were pushed even further in [12], where it is shown that in fact the sum of the Betti numbers is 50% larger: Theorem 5.19 (Boocher-Wigglesworth, 2020 [12]). Let M be a graded R-module of codimension c 3 generated in degree 0 and let a 2 be the minimal degree of a first syzygy of…”