Contents

# Contents

## Idea

Rector 74 proved the analog of the Atiyah-Segal completion theorem with topological K-theory replaced by the algebraic K-theory $K \mathbb{F}_p$ of a prime field $\mathbb{F}_p$.

This theorem further amplifies the analogy to the Segal-Carlsson completion theorem for stable cohomotopy and the interpretation of the latter as the algebraic K-theory over the “field with one element” (Guillot 06):

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation ring
$KU_G(\ast) \simeq R_{\mathbb{C}}(G)$
Atiyah-Segal completion theorem
$R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant)
complex cobordism cohomology
MU$MU_G(\ast)$completion theorem for complex cobordism cohomology
$MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant)
algebraic K-theory
$K \mathbb{F}_p$representation ring
$(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$
Rector completion theorem
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

## References

The result is due to

• D. L. Rector, Modular characters and K-theory with coefficients in a finite field, Journal of Pure and Applied Algebra Volume 4, Issue 2, April 1974, Pages 137-158 (doi:10.1016/0022-4049(74)90019-X)

Its interpretation as amplifying the identification of stable cohomotopy with the algebraic K-theory of the “field with one element” is highlighted in

Created on September 11, 2018 at 06:16:52. See the history of this page for a list of all contributions to it.