Quillen stratification in equivariant homotopy theory
Fuente:
arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915031749230592 |
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| author | Barthel, Tobias Castellana, Natalia Heard, Drew Naumann, Niko Pol, Luca |
| author_facet | Barthel, Tobias Castellana, Natalia Heard, Drew Naumann, Niko Pol, Luca |
| contents | We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group $G$, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules. Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points.
We then apply our methods to the case of Borel-equivariant Lubin--Tate $E$-theory $\underline{E_n}$, for any finite height $n$ and any finite group $G$, where we obtain a sharper theorem in the form of cohomological stratification. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing $\otimes$-ideals of the category of equivariant modules over $\underline{E_n}$, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_02212 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quillen stratification in equivariant homotopy theory Barthel, Tobias Castellana, Natalia Heard, Drew Naumann, Niko Pol, Luca Algebraic Topology Representation Theory We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group $G$, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules. Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points. We then apply our methods to the case of Borel-equivariant Lubin--Tate $E$-theory $\underline{E_n}$, for any finite height $n$ and any finite group $G$, where we obtain a sharper theorem in the form of cohomological stratification. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing $\otimes$-ideals of the category of equivariant modules over $\underline{E_n}$, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory. |
| title | Quillen stratification in equivariant homotopy theory |
| topic | Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2301.02212 |