Let n ≥ 1 be any integer and let p be a prime number. For a profinite group G and any discrete abelian group M, we use Map^c(G, M ) to denote the abelian group of continuous functions from G to M. For the most part, our interests lie in a particular profinite group known as the extended Morava stabilizer group. Denoted by G_n, this profinite group is the semi-direct product of the Morava stabilizer group Sn with the Galois group of the field extension F_pⁿ/ F_p.

The objects K(n)–the n-th Morava K-theory spectrum, E_n–the Lubin-Tate spectrum, and E(n)–the Johnson-Wilson spectrum, are essential to this dissertation. By using the setting of symmetric spectra, we provide a cohomological approximation of the E₂-term of the K( n)-local E_n-Adams spectral sequence. Given any spectrum X, LK( n)(X) denotes the Bousfield localization of X with respect to K(n), while E _*^∨(X) denotes π_*( L_K(_n)( E_n ∧ X)). For any discrete Gn-spectrum Y , (Y)_fGn is used to denote a fibrant replacement of Y in the category of discrete G_n-spectra. Given any tower of generalized Moore spectra {M_i}_i≥0 such that L_K(_n)(E _n ∧ X) = holim_i≥0 E_n ∧ X ∧ M_i, each X_i denotes a certain fibrant discrete G_n-spectrum that is weakly equivalent to E _n ∧ X ∧ M_i. We produce a long exact sequence in which for any s ≥ 0 the s-th row has E^s,t ₂, the E₂-term of the K( n)-local E_n-Adams spectral sequence E^s,t₂(X) ⇒ π _t(L_K(_n)( X)), as the middle term, H^s_cts( G_n; lim_i≥0π_t( X_i)), the cohomology of continuous cochains with coefficients in the G_n-module lim_i≥0π _t(X_i), as the term on the right, and H^s(lim1_i≥0Map^c(G ^*_n, π_t+1( X_i))) as the term on the left. This result provides a tool for generalizing most known instances in which the E ₂-term of the K(n)-local E_n-Adams spectral sequence is the continuous cohomology, and we maintain that this theorem has the potential to provide a generalization of all remaining known instances.

The term H^s(lim¹ _i≥0Map^c(G^*_n, π _t+1(X_i))) plays a vital role in this dissertation, and in an attempt to simplify it, we provide an analysis of the relationship between the first derived functor of the inverse limit of non-negatively-graded towers of abelian groups and the functor Map ^c(G, –).