Positselski duality in $\infty$-categories
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912694483812352 |
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| author | Aambø, Torgeir |
| author_facet | Aambø, Torgeir |
| contents | We introduce the notion of a contramodule over a cocommutative coalgebra in a presentably symmetric monoidal $\infty$-category $\mathcal{C}$, and prove a symmetric monoidal $\infty$-categorical version of Positselski's comodule-contramodule correspondence when the coalgebra is coidempotent. This gives a new perspective on, and a new proof of local duality -- in the sense of Hovey--Palmieri--Strickland and Dwyer--Greenlees -- whenever $\mathcal{C}$ is stable and compactly generated. We further consider an analog of Positselski's definition of contramodules over topological rings in the $\infty$-categorical setting, and show that the two perspectives on contramodules are equivalent. As examples we describe the categories of $K(n)$-local spectra, $T(n)$-local spectra and the derived complete category of a ring $R$, as categories of contramodules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04060 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positselski duality in $\infty$-categories Aambø, Torgeir Algebraic Topology Category Theory We introduce the notion of a contramodule over a cocommutative coalgebra in a presentably symmetric monoidal $\infty$-category $\mathcal{C}$, and prove a symmetric monoidal $\infty$-categorical version of Positselski's comodule-contramodule correspondence when the coalgebra is coidempotent. This gives a new perspective on, and a new proof of local duality -- in the sense of Hovey--Palmieri--Strickland and Dwyer--Greenlees -- whenever $\mathcal{C}$ is stable and compactly generated. We further consider an analog of Positselski's definition of contramodules over topological rings in the $\infty$-categorical setting, and show that the two perspectives on contramodules are equivalent. As examples we describe the categories of $K(n)$-local spectra, $T(n)$-local spectra and the derived complete category of a ring $R$, as categories of contramodules. |
| title | Positselski duality in $\infty$-categories |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2411.04060 |