Fixed-point homotopy-orbit Landweber exactness conjecture

Let AA be a compact abelian Lie group, let BB be a subgroup of AA, let EE be a Landweber exact spectrum, and let ΦB\Phi^B denote geometric fixed points. For an AA-space or AA-spectrum XX, write ΦB(X)h(A/B)\Phi^B(X)_{h(A/B)} for the homotopy orbits under A/BA/B. Fixed-point homotopy-orbit Landweber exactness conjecture. If EE is a Landweber exact spectrum, then for any subgroup BB of a compact abelian Lie group AA, the cohomology theory

XE(ΦB(X)h(A/B))X\mapsto E^*(\Phi^B(X)_{h(A/B)})

is AA-Landweber exact. Cohomology theories of this form include Strickland's equivariant Lubin–Tate theories. The source does not state a resolution of this conjecture.

Sources & referencesView supporting material

Primary source

Noah Wisdom, “Properties and Examples of A-Landweber Exact Spectra”, arXiv:2401.12227 (2024).

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