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Large lower bounds for the betti numbers of graded modules with low regularity
Collectanea Mathematica ( IF 1.1 ) Pub Date : 2020-06-18 , DOI: 10.1007/s13348-020-00292-4
Adam Boocher , Derrick Wigglesworth

Suppose that M is a finitely-generated graded module (generated in degree 0) of codimension \(c\ge 3\) over a polynomial ring and that the regularity of M is at most \(2a-2\) where \(a\ge 2\) is the minimal degree of a first syzygy of M. Then we show that the sum of the betti numbers of M is at least \(\beta _0(M)(2^c + 2^{c-1})\). Additionally, under the same hypothesis on the regularity, we establish the surprising fact that if \(c \ge 9\) then the first half of the betti numbers are each at least twice the bound predicted by the Buchsbaum-Eisenbud-Horrocks rank conjecture: for \(1\le i \le \frac{c+1}{2}\), \(\beta _i(M) \ge 2\beta _0(M){c \atopwithdelims ()i}\).



中文翻译:

具有低规则性的渐变模块的贝蒂数的较大下界

假设M是在多项式环上的余维\(c \ ge 3 \)的有限生成的渐变模块(以0度生成),并且M的正则性最多为\(2a-2 \),其中\(a \ ge 2 \)M的第一次合音的最小程度。然后我们证明M的贝蒂数之和至少为\(\ beta _0(M)(2 ^ c + 2 ^ {c-1})\)。此外,在关于正则性的相同假设下,我们建立了一个令人惊讶的事实,即如果\(c \ ge 9 \),则贝蒂数的前半部至少每个都是布赫斯鲍姆-艾森德-霍罗克斯等级猜想预测的界的两倍:为\(1 \ le i \ le \ frac {c + 1} {2} \)\(\ beta _i(M)\ ge 2 \ beta _0(M){c \ atopwithdelims()i} \)

更新日期:2020-06-18
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