Abstract
We formulate a version of Hopkins’ chromatic splitting conjecture for an arbitrary structured ring spectrum R, and prove it whenever \(\pi _*R\) is Noetherian. As an application, these results provide a new local-to-global principle in the modular representation theory of finite groups.
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Notes
We warn the reader that \({R}//{\mathfrak {p}^{(s)}}\) is not the derived quotient with respect to the ideal \(\mathfrak {p}^s\).
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Acknowledgements
We would like to thank John Greenlees, Henning Krause, and Hal Sadofsky for helpful discussions, as well as the referee for many useful suggestions and corrections. Moreover, we are grateful to the Max Planck Institute for Mathematics for its hospitality, funding a week-long visit of the third-named author in June 2016. The first-named author was partially supported by the DNRF92.
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Barthel, T., Heard, D. & Valenzuela, G. The algebraic chromatic splitting conjecture for Noetherian ring spectra. Math. Z. 290, 1359–1375 (2018). https://doi.org/10.1007/s00209-018-2066-5
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DOI: https://doi.org/10.1007/s00209-018-2066-5