Vector space summands of lower syzygies
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arXiv
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| Format: | Preprint |
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2024
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| author | Asgharzadeh, Mohsen DeBellevue, Michael Dey, Souvik Nasseh, Saeed Takahashi, Ryo |
| author_facet | Asgharzadeh, Mohsen DeBellevue, Michael Dey, Souvik Nasseh, Saeed Takahashi, Ryo |
| contents | In this paper, we investigate problems concerning when the residue field $k$ of a local ring $(R,\frak m$, $k)$ appears as a direct summand of syzygy modules, from two perspectives. First, we prove that the following conditions are equivalent: (i) $k$ is a direct summand of second syzygies of all non-free finitely generated $R$-modules; (ii) $k$ is a direct summand of third syzygies of all non-free finitely generated $R$-modules; (iii) $k$ is a direct summand of $\frak m$. We also prove various consequences of these conditions.
The second point of this article is to investigate for what artinian local rings $R$ the dual $E^*=Hom_R(E_R(k),R)$ of the injective envelope of the residue field, which is also a second syzygy, is a $k$-vector space. Using the notion of Eliahou-Kervaire resolution, we introduce a large class of artinian local rings that satisfy this condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vector space summands of lower syzygies Asgharzadeh, Mohsen DeBellevue, Michael Dey, Souvik Nasseh, Saeed Takahashi, Ryo Commutative Algebra 13D02, 13E10, 13H10 In this paper, we investigate problems concerning when the residue field $k$ of a local ring $(R,\frak m$, $k)$ appears as a direct summand of syzygy modules, from two perspectives. First, we prove that the following conditions are equivalent: (i) $k$ is a direct summand of second syzygies of all non-free finitely generated $R$-modules; (ii) $k$ is a direct summand of third syzygies of all non-free finitely generated $R$-modules; (iii) $k$ is a direct summand of $\frak m$. We also prove various consequences of these conditions. The second point of this article is to investigate for what artinian local rings $R$ the dual $E^*=Hom_R(E_R(k),R)$ of the injective envelope of the residue field, which is also a second syzygy, is a $k$-vector space. Using the notion of Eliahou-Kervaire resolution, we introduce a large class of artinian local rings that satisfy this condition. |
| title | Vector space summands of lower syzygies |
| topic | Commutative Algebra 13D02, 13E10, 13H10 |
| url | https://arxiv.org/abs/2410.12227 |