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The completion and local cohomology theorems for complex cobordism for all compact Lie groups

Marco La Vecchia

Abstract

We generalize the completion theorem for equivariant MU-module spectra for finite groups or finite extensions of a torus to compact Lie groups using the splitting of global functors proved by Schwede. This proves a conjecture of Greenlees and May.

The completion and local cohomology theorems for complex cobordism for all compact Lie groups

Abstract

We generalize the completion theorem for equivariant MU-module spectra for finite groups or finite extensions of a torus to compact Lie groups using the splitting of global functors proved by Schwede. This proves a conjecture of Greenlees and May.

Paper Structure

This paper contains 14 sections, 5 theorems, 50 equations.

Table of Contents

  1. Overview
  2. Introduction
  3. Organization
  4. Acknowledgment
  5. Notations, Conventions and Facts
  6. Spaces
  7. Universal spaces
  8. Algebra
  9. Spectra
  10. Global and Equivariant Stable Homotopy Categories
  11. Global Complex Cobordism
  12. Classical Statement
  13. Schwede's Splitting
  14. The main Result

Key Result

Theorem 1.1

For any compact Lie group $G$ with a faithful representation in $U(n)$ and any $G$-space $X$ there are spectral sequences where the local homology and cohomology are calculated using a subideal of $J_G$ with $n$-generators and therefore concentrated in degrees $\leq n$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • Definition 3.7
  • Theorem 3.9
  • proof
  • ...and 9 more