| 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 Mapc(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 Gn, this profinite group is the semi-direct product of the Morava stabilizer group Sn with the Galois group of the field extension Fpn/ Fp.;The objects K(n)--the n-th Morava K-theory spectrum, En--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 En-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 pi*( LK(n)( En ∧ X)). For any discrete Gn-spectrum Y , (Y)fGn is used to denote a fibrant replacement of Y in the category of discrete Gn-spectra. Given any tower of generalized Moore spectra {Mi}i≥0 such that LK(n)(E n ∧ X) = holimi≥0 En ∧ X ∧ Mi, each Xi denotes a certain fibrant discrete Gn-spectrum that is weakly equivalent to E n ∧ X ∧ Mi. We produce a long exact sequence in which for any s ≥ 0 the s-th row has Es,t 2, the E2-term of the K( n)-local En-Adams spectral sequence Es,t2(X) ⇒ pi t(LK(n)( X)), as the middle term, Hscts( Gn; limi≥0pit( Xi)), the cohomology of continuous cochains with coefficients in the Gn-module limi≥0pi t(Xi), as the term on the right, and Hs(lim1i≥0Mapc(G *n, pit+1( Xi))) 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 En-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 Hs(lim1 i≥0Mapc(G*n, pi t+1(Xi))) 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, --). |
