A short computation of the Rouquier dimension for a cycle of projective lines
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913576906653696 |
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| author | Hanlon, Andrew Hicks, Jeff |
| author_facet | Hanlon, Andrew Hicks, Jeff |
| contents | Given a dg category $\mathcal C$, we introduce a new class of objects (weakly product bimodules) in $\mathcal C^{op}\otimes \mathcal C$ generalizing product bimodules. We show that the minimal generation time of the diagonal by weakly product bimodules provides an upper bound for the Rouquier dimension of $\mathcal C$. As an application, we give a purely algebro-geometric proof of a result of Burban and Drozd that the Rouquier dimension of the derived category of coherent sheaves on an $n$-cycle of projective lines is one. Our approach explicitly gives the generator realizing the minimal generation time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_05753 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A short computation of the Rouquier dimension for a cycle of projective lines Hanlon, Andrew Hicks, Jeff Algebraic Geometry 14F08, 18G80 Given a dg category $\mathcal C$, we introduce a new class of objects (weakly product bimodules) in $\mathcal C^{op}\otimes \mathcal C$ generalizing product bimodules. We show that the minimal generation time of the diagonal by weakly product bimodules provides an upper bound for the Rouquier dimension of $\mathcal C$. As an application, we give a purely algebro-geometric proof of a result of Burban and Drozd that the Rouquier dimension of the derived category of coherent sheaves on an $n$-cycle of projective lines is one. Our approach explicitly gives the generator realizing the minimal generation time. |
| title | A short computation of the Rouquier dimension for a cycle of projective lines |
| topic | Algebraic Geometry 14F08, 18G80 |
| url | https://arxiv.org/abs/2311.05753 |