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

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Abstract

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 \(1\le i \le \frac{c+1}{2}\), \(\beta _i(M) \ge 2\beta _0(M){c \atopwithdelims ()i}\).

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Acknowledgements

We thank Daniel Erman for inspiring this project and for the many conversations about Boij–Söderberg theory over the years. We thank Craig Huneke for the suggestion to consider how the ideas in Erman’s paper might be adapted to the study of the sum of the betti numbers rather than individual ones. A portion of this research was conducted at the Fields Institute and the second author warmly thanks them for their hospitality during that period. Finally, we are grateful for helpful conversations with David Eisenbud, Srikanth Iyengar, Anurag Singh, Mark Walker, and to the anonymous referees whose comments helped improve an earlier draft of this paper.

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  1. University of San Diego, San Diego, CA, USA

    Adam Boocher

  2. University of Arkansas, Fayetteville, AR, USA

    Derrick Wigglesworth

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  1. Adam Boocher
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  2. Derrick Wigglesworth
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Correspondence to Adam Boocher.

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Boocher, A., Wigglesworth, D. Large lower bounds for the betti numbers of graded modules with low regularity. Collect. Math. 72, 393–410 (2021). https://doi.org/10.1007/s13348-020-00292-4

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  • DOI: https://doi.org/10.1007/s13348-020-00292-4

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