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

2019/03/29 by Adam Boocher, Boocher, Adam, Derrick Wigglesworth +1
Mathematics · #13D02 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1903.12503

openalex publication_date 2019/03/29 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

Suppose that M is a finitely-generated graded module of codimension c\≥\n3 over a polynomial ring and that the regularity of M is at most 2a-2\nwhere a\≥ 2 is the minimal degree of a first syzygy of M. Then we show\nthat the sum of the betti numbers of M is at least \β₀(M)(2^c +\n2^c-1). In addition, if c \≥ 9 then for each 1\≤ i\≤ lceil\nc/2 rceil, we show \β_i(M)\≥ 2\β₀(M)c choose i.\n

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