Generalizations of Burch Ideals and Ideal-Periodicity

TL;DR

This paper explores the periodicity of minimal free resolutions over certain rings, extending known results from complete intersection rings to Generalized Positive Burch Index Rings, and develops new techniques to explain this phenomenon.

Contribution

It generalizes the concept of ideal-periodicity in minimal free resolutions to a broader class of rings, specifically Generalized Positive Burch Index Rings.

Findings

Sum of n consecutive ideals of minors stabilizes asymptotically.

Periodic behavior observed in computed examples is proven for certain rings.

Develops techniques to understand periodicity in more general algebraic contexts.

Abstract

Consider an infinite minimal free resolution of a module $M$ over a local Noetherian ring $R$. It was shown by Eisenbud that if $R$ is a complete intersection ring, then a minimal resolution is periodic iff it is bounded. Over more general rings, Peeva and Gasharov showed this periodicity does not always hold. However, in every computed example, the sum of $n$ consecutive ideals of minors of matrices in the resolution is fixed for some $n$, asymptotically. We prove this in general for certain Generalized Positive Burch Index Rings, in the sense of Dao, Kobayashi, and Takahashi. In doing so, we develop techniques that begin to explain this periodicity in more generality.