k summands of syzygies over rings of positive Burch index via canonical resolutions

Abstract

In recent work, Dao and Eisenbud define the notion of a Burch index, expanding the notion of Burch rings of Dao, Kobayashi, and Takahashi, and show that for any module over a ring of Burch index at least 2, its nth syzygy contains direct summands of the residue field for $n=4$ or 5 and all $n\ge 7$. We investigate how this behavior is explained by the bar resolution formed from appropriate differential graded (dg) resolutions, yielding a new proof that includes all $n\ge 5$, which is sharp. When the module is Golod, we use instead the bar resolution formed from $A_{\infty }$ resolutions to identify such k summands explicitly for all $n\ge 4$ and show that the number of these grows exponentially as the homological degree increases.