The Morava Cohomology of Finite General Linear Groups

In this thesis, for all finite heights n and odd primes p, we compute the Morava E-theory and Morava K-theory of general linear groups over finite fields F of order q ≡ 1 mod p. We rephrase the problem in terms of V_*, the graded groupoid of vector spaces over F, and focus on the graded algebra and coalgebra structures induced from the direct sum functor. We use character theory to determine the ranks of E^0BV_* and K^0BV_*, and use this information to reverse engineer the Atiyah-Hirzebruch spectral sequence for K^*BV_* and K_*BV_*. We then use this result in two ways: we deduce that the algebra and coalgebra structures are free commutative and cofree cocommutative respectively, and we identify a lower bound for the nilpotence of the canonical top normalised Chern class in K^0BV_{p^k}. Following this we make use of algebro-geometric and Galois theoretic techniques to determine the indecomposables in Morava E-theory and K-theory, before using this calculation in conjunction with K-local duality and the nilpotence lower bound to determine the primitives of the coalgebra structure in Morava K-theory. Along the way, we show that E^0BV_* and K^0BV_* have structures similar to that of a graded Hopf ring, but with a modified version of the compatibility relation. We call such structures "graded faux Hopf rings".