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arXiv · 2401.00142

$k$ Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions

Michael DeBellevue·Claudia Miller
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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 $n$th syzygy contains direct summands of the residue field for $n=4$ or $5$ and all $n\geq 7$. 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 $n\geq 5$, 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 $n\geq 4$ and show that the number of these grows exponentially as the homological degree increases.

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Michael DeBellevue, Claudia Miller. 2023-12-30. $k$ Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions. https://arxiv.org/abs/2401.00142

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