On homological properties of the category of $\mathbb{F}_1$-representations over a linear quiver of type $\mathbb{A}_n$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913234161762304 |
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| author | Fu, Changjian Ran, Longjun Yang, Liang |
| author_facet | Fu, Changjian Ran, Longjun Yang, Liang |
| contents | Let $Q$ be a quiver of type $\mathbb{A}_n$ with linear orientation and $\operatorname{rep}(Q,\mathbb{F}_1)$ the category of representations of $Q$ over the virtual field $\mathbb{F}_1$.It is proved that $\operatorname{rep}(Q,\mathbb{F}_1)$ has global dimension $2$ whenever $n\geq 3$ and it is hereditary if $n\leq 2$. As a consequence, the Euler form $\langle L, M\rangle=\sum_{i=0}^\infty (-1)^i\operatorname{dim} \operatorname{Ext}^i(L,M)$ is well-defined. However, it does not descend to the Grothendieck group of $\operatorname{rep}(Q,\mathbb{F}_1)$. This yields negative answers to questions raised by Szczesny in [IMRN, Vol. 2012, No. 10, pp. 237-2404]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06136 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On homological properties of the category of $\mathbb{F}_1$-representations over a linear quiver of type $\mathbb{A}_n$ Fu, Changjian Ran, Longjun Yang, Liang Representation Theory Category Theory Rings and Algebras Let $Q$ be a quiver of type $\mathbb{A}_n$ with linear orientation and $\operatorname{rep}(Q,\mathbb{F}_1)$ the category of representations of $Q$ over the virtual field $\mathbb{F}_1$.It is proved that $\operatorname{rep}(Q,\mathbb{F}_1)$ has global dimension $2$ whenever $n\geq 3$ and it is hereditary if $n\leq 2$. As a consequence, the Euler form $\langle L, M\rangle=\sum_{i=0}^\infty (-1)^i\operatorname{dim} \operatorname{Ext}^i(L,M)$ is well-defined. However, it does not descend to the Grothendieck group of $\operatorname{rep}(Q,\mathbb{F}_1)$. This yields negative answers to questions raised by Szczesny in [IMRN, Vol. 2012, No. 10, pp. 237-2404]. |
| title | On homological properties of the category of $\mathbb{F}_1$-representations over a linear quiver of type $\mathbb{A}_n$ |
| topic | Representation Theory Category Theory Rings and Algebras |
| url | https://arxiv.org/abs/2309.06136 |