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Lower Bounds on Betti Numbers

Abstract

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.

Keywords

  • Free resolutions
  • Betti numbers
  • Buchsbaum–Eisenbud–Horrocks Conjecture
  • Total Rank Conjecture

2020 Mathematics Subject Classification

  • Primary: 13D02

Dedicated to David Eisenbud on the occasion of his 75th birthday.

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Notes

  1. 1.

    Really, we mean the graded Betti numbers of M, to be defined in Sect. 3.

  2. 2.

    Fun fact: in astronomy, a syzygy is an alignment of three or more celestial objects.

  3. 3.

    This means that no more than 3 lie on a plane and no more than 5 on a conic.

  4. 4.

    This is an abuse of notation since the expression “complete intersection” typically refers to a ring, not a module.

  5. 5.

    This is the term used by Herzog and Kühl.

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Acknowledgements

We thank Daniel Erman and Josh Pollitz for their detailed comments on a previous version of this survey. We also thank Mark Walker and Tim Römer for helpful correspondence, and Juan Migliore for alerting us to a typo in a previous version of the paper. The second author was partially supported by NSF grant DMS-2001445, now DMS-2140355.

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Boocher, A., Grifo, E. (2021). Lower Bounds on Betti Numbers. In: Peeva, I. (eds) Commutative Algebra. Springer, Cham. https://doi.org/10.1007/978-3-030-89694-2_2

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