Generalizations of Burch Ideals and Ideal-Periodicity

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.