Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology
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| Format: | Preprint |
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2025
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| _version_ | 1866918005704753152 |
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| author | Scherotzke, Sarah Sibilla, Nicolò Tomasini, Paolo |
| author_facet | Scherotzke, Sarah Sibilla, Nicolò Tomasini, Paolo |
| contents | This paper establishes a unifying framework for various forms of twisted Hochschild homology by comparing two definitions of elliptic Hochschild homology: one introduced by Moulinos--Robalo--Toën and the other by Sibilla--Tomasini. Central to our approach is a new Fourier--Mukai duality for formal groups. We prove that when $\widehat{E}$ is the formal group associated to an elliptic curve $E$, the resulting $\widehat{E}$-Hochschild homology coincides with the mapping stack construction of Sibilla--Tomasini. This identification also recovers ordinary and Hodge Hochschild homology as degenerate limits corresponding to nodal and cuspidal cubics, respectively. Building on this, we introduce global versions of elliptic Hochschild homology over the moduli stacks of elliptic and cubic curves, which interpolate between these theories and suggest a universal form of TMF-Hochschild homology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00172 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology Scherotzke, Sarah Sibilla, Nicolò Tomasini, Paolo Algebraic Geometry Algebraic Topology K-Theory and Homology 14A30 (primary), 19D55 (secondary) This paper establishes a unifying framework for various forms of twisted Hochschild homology by comparing two definitions of elliptic Hochschild homology: one introduced by Moulinos--Robalo--Toën and the other by Sibilla--Tomasini. Central to our approach is a new Fourier--Mukai duality for formal groups. We prove that when $\widehat{E}$ is the formal group associated to an elliptic curve $E$, the resulting $\widehat{E}$-Hochschild homology coincides with the mapping stack construction of Sibilla--Tomasini. This identification also recovers ordinary and Hodge Hochschild homology as degenerate limits corresponding to nodal and cuspidal cubics, respectively. Building on this, we introduce global versions of elliptic Hochschild homology over the moduli stacks of elliptic and cubic curves, which interpolate between these theories and suggest a universal form of TMF-Hochschild homology. |
| title | Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology 14A30 (primary), 19D55 (secondary) |
| url | https://arxiv.org/abs/2505.00172 |