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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2511.13267 |
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| _version_ | 1866917222770802688 |
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| author | Lu, Dancheng Wang, Zexin Zhu, Guangjun |
| author_facet | Lu, Dancheng Wang, Zexin Zhu, Guangjun |
| contents | The homological shift algebra and the projective dimension function of complementary edge ideals are investigated. Let $G$ be a connected graph, and let $I$ be its complementary edge ideal. For bipartite graphs $G$, we show that the projective dimension of $I^s$ increases strictly with $s$ until reaching its maximum value. For trees and cycles, explicit expressions for the projective dimension of $I^s$ are provided, along with detailed descriptions of their homological shift algebras. In particular, it is shown that the $i$-th homological shift algebra of such ideals is generated in degree at most $i$. Additionally, we prove that if $G$ is a tree, then the homological shift ideal $\mathrm{HS}_i(I^i)$, when divided by a suitable monomial, becomes a Veronese-type ideal, and every Veronese-type ideal arises in this manner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13267 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homological shifts of powers a complementary edge ideal Lu, Dancheng Wang, Zexin Zhu, Guangjun Commutative Algebra The homological shift algebra and the projective dimension function of complementary edge ideals are investigated. Let $G$ be a connected graph, and let $I$ be its complementary edge ideal. For bipartite graphs $G$, we show that the projective dimension of $I^s$ increases strictly with $s$ until reaching its maximum value. For trees and cycles, explicit expressions for the projective dimension of $I^s$ are provided, along with detailed descriptions of their homological shift algebras. In particular, it is shown that the $i$-th homological shift algebra of such ideals is generated in degree at most $i$. Additionally, we prove that if $G$ is a tree, then the homological shift ideal $\mathrm{HS}_i(I^i)$, when divided by a suitable monomial, becomes a Veronese-type ideal, and every Veronese-type ideal arises in this manner. |
| title | Homological shifts of powers a complementary edge ideal |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2511.13267 |