Asymptotic behavior of invariants of syzygies of maximal Cohen-Macaulay modules

2024/12/08 by Tony J. Puthenpurakal, Puthenpurakal, Tony J., Samarendra Sahoo +1
Mathematics · #Commutative Algebra and Its Applications #Algebraic structures and combinatorial models #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2412.05860

Abstract

Let (A,\mathfrakm) be a complete intersection ring of codimension c≥ 2 and dimension d≥ 1. Let M be a finitely generated maximal Cohen-Macaulay A-module. Set M_i=Syz^A_i(M). Let e^\mathfrakm_i(M) be the i-th Hilbert coefficient of M with respect to \mathfrakm. We prove for all i≫0, the function i↦ e^\mathfrakm_j(M_i) is a quasi-polynomial type with period 2 and degree cx(M)-1 for j=0,1, where cx(M) is the complexity of M. For cx(M)=2, we prove lim_{n→ ∞}\dfrace^\mathfrakm₁(M_2n+j)n≥ lim_{n→ ∞}\dfrace^\mathfrakm₀(M_2n+j)n-lim_{n→ ∞}\dfracμ(M_2n+j)n for j=0,1. When equality holds, we prove that the Castelnuovo-Mumford regularity of the associated graded ring of M_i with respect to the maximal ideal \mathfrakm is bounded for all i≥ 0.

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