A Polynomial Bound on the Regularity of an Ideal in Terms of Half of the Syzygies

2011/12/01 by McCullough, Jason · 1 citation
#13D02 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1112.0058

Abstract

Let K be a field and let S = K[x₁, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in terms of the previous t_i. As a result, we give bounds on the regularity of S/I in terms of as few as half of the numbers t_i. We also prove related bounds for arbitrary modules. These bounds are often much smaller than the known doubly exponential bound on regularity purely in terms of t₁.

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