Ravenel's chromatic filtration conjecture for Burnside rings

Let GG be a finite group and let n0n\geqslant0. The Ravenel ideal Jn(G)\mathrm{J}_n(G) consists of virtual finite GG-sets whose fixed-point sets have virtual cardinality zero for every subgroup generated by at most nn elements; let In(G)\mathrm{I}_n(G) denote the corresponding chromatic ideal of the Burnside ring A(G)\mathrm{A}(G). Ravenel's conjecture. For all finite groups GG and all integers n0n\geqslant0,

Jn(G)In(G).\mathrm{J}_n(G)\subseteq\mathrm{I}_n(G).

This conjecture relates the filtration of the Burnside ring defined by subgroup-generation bounds to the chromatic filtration arising from localization at Johnson–Wilson spectra. The source gives no resolution status.

Sources & referencesView supporting material

Primary source

Markus Szymik, “The chromatic filtration of the Burnside category”, arXiv:2002.04877 (2020).

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