Localization and completion theorems for \(MU\)-module spectra (Q1268031)

For \(G\) a finite or a finite extension of a torus and \(M\) any module over \(MU\) the authors prove localization and completion theorems for the computation of \(M_*(BG)\) and \(M^*(BG)\). The computation is expressed in terms of spectral sequences whose respective \(E_2\) terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring \(MU^G_*\) and its module \(M^G_*\). The proof is based on a new norm map in equivariant stable homotopy theory and the construction involves a new concept of a global \({\mathfrak T}_*\)-functor with smash product. The paper has eleven paragraphs. First, in the introduction, the authors give the statements of results. They give their completion theorem for module over \(MU_G\). Then, they emphasize that Thom isomorphisms and Euler classes are essential to the strategy of the proof.