The Lubin-tate Spectrum and Its Homotopy Fixed Point Spectra

This note is a summary of the results of my Ph.D. thesis (plus slight modifications), completed May 9, 2003, under the supervision of Professor Paul Goerss at Northwestern University. Let En be the Lubin-Tate spectrum with En∗ = W (Fpn)[[u1, ..., un−1]][u], where the degree of u is −2 and the complete power series ring over the Witt vectors is in degree zero. Let Gn = Sn o Gal(Fpn/Fp), where Sn is the nth Morava stabilizer group (the automorphism group of the Honda formal group law Γn of height n over Fpn), and let G be a closed subgroup of Gn. Note that Sn, Gn and G are all profinite groups. Morava’s change of rings theorem yields a spectral sequence (1) H∗ c (Gn; π∗(En ∧X)) =⇒ π∗LK(n)(X), where the E2-term is continuous cohomology andX is a finite spectrum (see [7], [1], [5]). Using the Gn-action on En by maps of commutative S-algebras (work of Goerss and Hopkins ([3], [4]), and Hopkins and Miller [8]), Devinatz and Hopkins [2] constructed spectra E n with strongly convergent spectral sequences (2) H∗ c (G; π∗(En ∧X)) =⇒ π∗(E n ∧X). Also, Devinatz and Hopkins showed that En n ∧X ' LK(n)(X). When K is a discrete group and Y is a K-spectrum, there is a homotopy fixed point spectrum Y hK = MapK(EK+, Y ), where EK+ is a free contractible K-space. Also, there is a conditionally convergent spectral sequence E 2 = H (K; πt(Y )) =⇒ πt−s(Y ), where the E2-term is group cohomology [6, §1.1]. Such a spectral sequence is called a descent spectral sequence. This scenario also occurs in another context. Let K be a profinite group. We say that Y is a discrete K-spectrum, if Y is a K-spectrum of simplicial sets such that each simplicial set Yk is a simplicial discrete K-set (that is, for each l ≥ 0, the action map on the l-simplices of Yk,