Examples at low primes and large heights

To apply asymptotic constancy, we require fp-type n ring spectra that both satisfy Lichtenbaum–Quillen and admit locally unipotent Z-actions. For our study of the telescope conjecture, these Z-actions should be related to cyclotomic extensions. At primes p ≥ 5 and height n + 1 = 2, we may use the Adams summand BP⟨1⟩, the computations of Ausoni–Rognes, and geometrically defined Adams operations. At primes p < 5 and height n+1 = 2, we may still use geometrically defined Adams operations on the Adams summand, but need a replacement for the Ausoni–Rognes proof of Lichtenbaum–Quillen.
At a general prime and height, this talk will summarize how BP⟨n⟩ can be constructed as an E3-ring with (E1 ⊗ A2)-Adams operations, and how one may prove the Lichtenbaum–Quillen property for it.

Suggested references: [BHLS23, §5,§7]