Extremal Behavior of ideals of minors

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 $R$.