# nLab equivariant algebraic K-theory

Contents

cohomology

### Theorems

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

The equivariant cohomology/equivariant spectrum-version of algebraic K-theory.

## Properties

### Rector completion theorem

(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
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$
• Z. Fiedorowicz, H. Hauschild, Peter May, theorem 0.4 of Equivariant algebraic K-theory, Equivariant algebraic K-theory, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 (pdf)

• Henning Hauschild, Stefan Waner, theorem 0.1 of The equivariant Dold theorem mod $k$ and the Adams conjecture, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. (euclid:1256065410)

• Kuzuhisa Shimakawa, Note on the equivariant $K$-theory spectrum, Publ. RIMS, Kyoto Univ. 29 (1993), 449-453 (pdf, doi)

• Christopher French, theorem 2.4 in The equivariant $J$–homomorphism for finite groups at certain primes, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 (euclid:1513797069)