$k$ Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions
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| Format: | Preprint |
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2023
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| _version_ | 1866917557312684032 |
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| author | DeBellevue, Michael Miller, Claudia |
| author_facet | DeBellevue, Michael Miller, Claudia |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00142 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $k$ Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions DeBellevue, Michael Miller, Claudia Commutative Algebra 13D02 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. |
| title | $k$ Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions |
| topic | Commutative Algebra 13D02 |
| url | https://arxiv.org/abs/2401.00142 |