Separable commutative algebras and Galois theory in stable homotopy theories
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909625582878720 |
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| author | Naumann, Niko Pol, Luca |
| author_facet | Naumann, Niko Pol, Luca |
| contents | We relate two different proposals to extend the étale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra introduced by Balmer. We show that finite covers are precisely those separable commutative algebras with underlying dualizable module, which have a locally constant and finite degree function. We then use Galois theory to classify separable commutative algebras in numerous categories of interest. Examples include the category of modules over a connective $\mathbb{E}_\infty$-ring $R$ which is either connective or even periodic with $π_0(R)$ regular Noetherian in which $2$ acts invertibly, the stable module category of a finite group of $p$-rank one and the derived category of a qcqs scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01259 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Separable commutative algebras and Galois theory in stable homotopy theories Naumann, Niko Pol, Luca Algebraic Topology Representation Theory We relate two different proposals to extend the étale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra introduced by Balmer. We show that finite covers are precisely those separable commutative algebras with underlying dualizable module, which have a locally constant and finite degree function. We then use Galois theory to classify separable commutative algebras in numerous categories of interest. Examples include the category of modules over a connective $\mathbb{E}_\infty$-ring $R$ which is either connective or even periodic with $π_0(R)$ regular Noetherian in which $2$ acts invertibly, the stable module category of a finite group of $p$-rank one and the derived category of a qcqs scheme. |
| title | Separable commutative algebras and Galois theory in stable homotopy theories |
| topic | Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2305.01259 |