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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2403.20074 |
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| _version_ | 1866916184122720256 |
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| author | Itagaki, Tomohiro Nakamoto, Kazunori Torii, Takeshi |
| author_facet | Itagaki, Tomohiro Nakamoto, Kazunori Torii, Takeshi |
| contents | Let ${\rm N}_m(R) = \{ (a_{ij}) \in {\rm M}_m(R) \mid a_{11} = a_{22} = \cdots = a_{mm} \mbox{ and } a_{ij} = 0 \mbox{ for any } i > j \}$ for a commutative ring $R$. Then ${\rm N}_m(R)$ is a quadratic monomial algebra over $R$. We calculate ${\rm HH}^{\ast}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$ as $R$-modules. We also determine the $R$-algebra structure of the Hochschild cohomology ring ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$. For $m \ge 3$, ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$ is an infinitely generated algebra over $R$ and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hochschild cohomology of the quadratic monomial algebra ${\rm N}_m$ Itagaki, Tomohiro Nakamoto, Kazunori Torii, Takeshi Rings and Algebras Primary 16E40, Secondary 16S37, 18G40 Let ${\rm N}_m(R) = \{ (a_{ij}) \in {\rm M}_m(R) \mid a_{11} = a_{22} = \cdots = a_{mm} \mbox{ and } a_{ij} = 0 \mbox{ for any } i > j \}$ for a commutative ring $R$. Then ${\rm N}_m(R)$ is a quadratic monomial algebra over $R$. We calculate ${\rm HH}^{\ast}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$ as $R$-modules. We also determine the $R$-algebra structure of the Hochschild cohomology ring ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$. For $m \ge 3$, ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$ is an infinitely generated algebra over $R$ and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket. |
| title | Hochschild cohomology of the quadratic monomial algebra ${\rm N}_m$ |
| topic | Rings and Algebras Primary 16E40, Secondary 16S37, 18G40 |
| url | https://arxiv.org/abs/2403.20074 |