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≥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≥5, which is sharp. When the module is Golod, we use instead the bar resolution formed from A∞ resolutions to identify such k summands explicitly for all n≥4 and show that the number of these grows exponentially as the homological degree increases.
Original language | English (US) |
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Pages (from-to) | 657-672 |
Number of pages | 16 |
Journal | Journal of Algebra |
Volume | 666 |
DOIs | |
State | Published - Mar 15 2025 |
Keywords
- Bar resolution
- Burch index
- Dg and A-infinity structures
- Syzygy
ASJC Scopus subject areas
- Algebra and Number Theory