Localization and completion theorems for MU-module spectra

• 成果类型: Article
• 署名作者: Greenlees, JPC; May, JP
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2952455
• 发表日期: 1997
• 页码: 509-544
• 关键词: brown-peterson cohomology; equivariant stable-homotopy; elementary p-groups; morava k-theories; finite-groups; classifying-spaces; HOMOLOGY; bordism; ring

摘要

Let G be a finite extension of a torus. WVorking 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*(BG) and M*(BG). The G-spectrum MUG that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum S-G, and there is an MUG-module M-G whose underlying MU-module is nl. This allows the use of topological analogues of constructions in commutative algebra. The computation of M*(BG) and M*(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*(G) and its module M*(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 L*-functor with smash product.