A lower bound on levels with applications to Koszul Complexes

TL;DR

This work proves a sharp lower bound on the $R$-level of a finite free complex with $I$-power torsion homology: $\mathrm{level}^R F \geq \dim R-\dim R/I+1$, under the stated torsion hypotheses, and shows this bound is optimal via concrete examples. The authors connect the result to the level of Koszul complexes, deriving new, tight lower bounds for $\mathrm{level}^R K(\underline{x};R)$ and showing equality cases for Lech-independent sequences. The proof leverages balanced big Cohen–Macaulay algebras, derived completion techniques, and base-change arguments rooted in Evans–Griffith’s version of the New Intersection Theorem, together with standard level- and depth-tools. They also develop a free-rank–based lower bound for levels of dg-algebras and demonstrate the limits of replacing the dimension-difference bound with superheight or bigheight in this setting, including detailed tensor-nilpotent and fiberwise-zero map analysis.

Abstract

In this paper, we establish a lower bound on the level of a perfect complex with power torsion homology on positive degrees and a power torsion minimal generator for zero homology. Examples are provided to demonstrate that the bound is optimal. This result is applied to improve existing lower bounds on the level of a Koszul complex on various classes of sequences.

Theorems & Definitions (18)

• Theorem \ref{theorem_level_ineq}
• Proposition 2.5
• Proposition 2.6
• Definition 2.8
• Proposition 2.9
• Proposition 2.10
• Theorem 3.1
• proof
• Proposition 3.2
• proof