Extremal Behavior of ideals of minors

TL;DR

This paper investigates the periodicity of ideals of minors in minimal free resolutions over specific local rings, establishing conditions under which these ideals become eventually periodic or powers of the maximal ideal, extending known results.

Contribution

It proves that ideals of minors are eventually 2-periodic or become powers of the maximal ideal in certain fiber product or Gorenstein rings, and studies the transfer of this periodicity between rings.

Findings

Ideals of minors are eventually 2-periodic in specified rings.

Ideals of minors become powers of the maximal ideal when embedding dimension is at least 3.

Periodic behavior can transfer from quotient rings to original rings via super-regular elements.

Abstract

Let $(R,\mathfrak m,\mathsf k)$ be either a fiber product or an artinian stretched Gorenstein ring, with $\operatorname{ch}(\mathsf k)\neq 2$ in the latter case. We prove that the ideals of minors of the minimal free resolution of any finitely generated $R$-module are eventually 2-periodic. Moreover, if the embedding dimension of $R$ is at least 3, eventually the ideals of minors become the powers of the maximal ideal, yielding the 1-periodicity. These are analogs of results obtained over complete intersections and Golod rings by Brown, Dao, and Sridhar. We also study the transfer of periodicity between rings. Specifically, we prove that for any local ring $(R,\mathfrak m)$, if $x\in \mathfrak m$ is a super-regular element and $M$ is an $R/(x)$ module whose ideals of minors are asymptotically the powers of the maximal ideal over $R/(x)$, then the same holds for the ideals of minors of $M$ over $R$.