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

• Boocher, Adam ^[1] ; Wigglesworth, Derrick ^[2]
  1. [1] University of San Diego

     University of San Diego

     Estados Unidos

     
  2. [2] University of Arkansas, Fayetteville, AR, USA
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 2, 2021, págs. 393-410
• Idioma: inglés
• DOI: 10.1007/s13348-020-00292-4
• Texto completo no disponible (Saber más ...)
• Resumen

  • Suppose that M is a finitely-generated graded module (generated in degree 0) of codimension c≥3 over a polynomial ring and that the regularity of M is at most 2a−2 where a≥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 β0(M)(2c+2c−1). Additionally, under the same hypothesis on the regularity, we establish the surprising fact that if c≥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≤i≤c+12, βi(M)≥2β0(M)(ci).