Hochschild cohomology and extensions of triangulated categories

Alessandro Lehmann; Wendy Lowen

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arXiv · 2503.13700

Hochschild cohomology and extensions of triangulated categories

Alessandro Lehmann·Wendy Lowen

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Abstract

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.

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BibTeX RIS

Alessandro Lehmann, Wendy Lowen. 2025-03-17. Hochschild cohomology and extensions of triangulated categories. https://arxiv.org/abs/2503.13700

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