Abstract
Let $G$ be a finite extension of a torus. Working with highly structured ring and module spectra, let $M$ be any module over $MU$; examples include all of the standard homotopical $MU$-modules, such as the Brown-Peterson and Morava $K$-theory spectra. We shall prove localization and completion theorems for the computation of $M_\ast(BG)$ and $M^\ast(BG)$. The $G$-spectrum $MU_G$ that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum $S_G$, and there is an $MU_G$-module $M_G$ whose underlying $MU$-module is $M$. This allows the use of topological analogues of constructions in commutative algebra. The computation of $M_\ast(BG)$ and $M^\ast(BG)$ 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 $MU_\ast^G$ and its module $M_\ast^G$. The central feature of the proof is a new norm map in equivariant stable homotopy theory, the construction of which involves the new concept of a global $\mathcal{I}_\ast$-functor with smash product.
