Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$)
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914603831656448 |
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| author | Itaba, Ayako Minaki, Shu |
| author_facet | Itaba, Ayako Minaki, Shu |
| contents | Let $A=A(α, β)$ be a graded down-up algebra with weights $(\mathrm{deg}\, x, \mathrm{deg}\, y)=(n,m)$ and $β\neq 0$, and $\nabla A$ its Beilinson algebra. Such an algebra $A$ is a 3-dimensional cubic AS-regular algebra by Kirkman--Musson--Passman. Assuming $\mathrm{gcd}\,(n, m)=1$ and $m \geq n$, we extend the previous results on the Hochschild cohomology of $\nabla A$. Known cases include $(n,m) = (1,1)$ (Belmans) and $(n = 1,\,m \geq 2)$ (Itaba--Ueyama). In this paper, we determine the dimensions of the Hochschild cohomology groups of $\nabla A$ in the remaining case $n\geq 2$ and $m\geq 2$ by explicitly constructing the projective resolution and computing the ranks of the arising representation matrices. As a byproduct, for $m>n>1$, we show that the derived category of the noncommutative projective scheme associated to $A$ is not equivalent to the derived category of any smooth projective surface. Moreover, for all $m \geq n \geq 1$, we describe the ring structure of the Hochschild cohomology group $\nabla A$ with respect to the Yoneda product. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_20195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$) Itaba, Ayako Minaki, Shu Rings and Algebras Representation Theory 16E40, 16S38, 16E05, 18G80 Let $A=A(α, β)$ be a graded down-up algebra with weights $(\mathrm{deg}\, x, \mathrm{deg}\, y)=(n,m)$ and $β\neq 0$, and $\nabla A$ its Beilinson algebra. Such an algebra $A$ is a 3-dimensional cubic AS-regular algebra by Kirkman--Musson--Passman. Assuming $\mathrm{gcd}\,(n, m)=1$ and $m \geq n$, we extend the previous results on the Hochschild cohomology of $\nabla A$. Known cases include $(n,m) = (1,1)$ (Belmans) and $(n = 1,\,m \geq 2)$ (Itaba--Ueyama). In this paper, we determine the dimensions of the Hochschild cohomology groups of $\nabla A$ in the remaining case $n\geq 2$ and $m\geq 2$ by explicitly constructing the projective resolution and computing the ranks of the arising representation matrices. As a byproduct, for $m>n>1$, we show that the derived category of the noncommutative projective scheme associated to $A$ is not equivalent to the derived category of any smooth projective surface. Moreover, for all $m \geq n \geq 1$, we describe the ring structure of the Hochschild cohomology group $\nabla A$ with respect to the Yoneda product. |
| title | Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$) |
| topic | Rings and Algebras Representation Theory 16E40, 16S38, 16E05, 18G80 |
| url | https://arxiv.org/abs/2511.20195 |