Hochschild cohomology and extensions of triangulated categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913742345732096 |
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| author | Lehmann, Alessandro Lowen, Wendy |
| author_facet | Lehmann, Alessandro Lowen, Wendy |
| contents | We define a notion of categorical first order deformations for (enhanced) triangulated categories. For a category $\mathcal{T}$, we show that there is a bijection between $\operatorname{HH}^2(\mathcal{T})$ and the set of categorical deformations of $\mathcal{T}$. We show that in the case of curved deformations of dg algebras considered in arXiv:2406.04945, the $1$-derived category of the deformation (introduced in arXiv:24020.8660) is a categorical deformation of the derived category of the base; the Hochschild class identified by this deformation is shown to restrict to the class defining the deformation of the algebra. As an application, we give a conceptual proof of the fact that (for a smooth base) the filtered derived category of a dg deformation yields a categorical resolution of the classical derived category. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13700 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hochschild cohomology and extensions of triangulated categories Lehmann, Alessandro Lowen, Wendy Algebraic Geometry Algebraic Topology Category Theory K-Theory and Homology 18G80, 16E40 (Primary) 18G70 18N25 (Secondary) We define a notion of categorical first order deformations for (enhanced) triangulated categories. For a category $\mathcal{T}$, we show that there is a bijection between $\operatorname{HH}^2(\mathcal{T})$ and the set of categorical deformations of $\mathcal{T}$. We show that in the case of curved deformations of dg algebras considered in arXiv:2406.04945, the $1$-derived category of the deformation (introduced in arXiv:24020.8660) is a categorical deformation of the derived category of the base; the Hochschild class identified by this deformation is shown to restrict to the class defining the deformation of the algebra. As an application, we give a conceptual proof of the fact that (for a smooth base) the filtered derived category of a dg deformation yields a categorical resolution of the classical derived category. |
| title | Hochschild cohomology and extensions of triangulated categories |
| topic | Algebraic Geometry Algebraic Topology Category Theory K-Theory and Homology 18G80, 16E40 (Primary) 18G70 18N25 (Secondary) |
| url | https://arxiv.org/abs/2503.13700 |