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

TL;DR

The paper addresses why the residue field $k$ appears as direct summands in syzygies of finitely generated modules over rings with positive Burch index. It develops a unified framework using bar resolutions built from dg and $A_\infty$-resolutions to produce explicit cycles (Burch cycles) that yield $k$-summands in high syzygies, extending previous Dao--Eisenbud results. In the general case with $\mathrm{Burch}(\varphi)\ge 2$, it proves a uniform existence of at least one $k$-summand in $\mathrm{syz}_{q+1}^R M$ for all $q\ge 4$, while in the Golod setting the bar resolution is minimal and yields an explicit exponential lower bound on the number of such summands, growing with homological degree. These results provide sharp, constructive insights into the structure and growth of syzygies over rings of positive Burch index, revealing both universal and growth-rate aspects of residue-field summands.

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.

Theorems & Definitions (26)

• Theorem A
• Theorem B
• Definition 1.3
• Definition 1.4
• Proposition 1.5
• proof
• Definition 1.8
• Definition 1.10
• Proposition 1.11: Bur-15, Proposition 3.6
• Lemma 2.2