Wilson spaces, snaith constructions, and elliptic orientations

Abstract

We construct a canonical family of even periodic E ∞ -ring spectra, with exactly one member of the family for every prime p and chromatic height n . At height 1 our construction is due to Snaith, who built complex K -theory from CP ∞ . At height 2 we replace CP ∞ with a p -local retract of BU ⟨ 6 ⟩ , producing a new theory that orients elliptic, but not generic, height 2 Morava E -theories. In general our construction exhibits a kind of redshift, whereby BP ⟨ n − 1 ⟩ is used to produce a height n theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the K ( n ) -localization of our height n ring to work of Peterson and Westerland building E n h S G ± from K ( Z , n + 1 ) .