Homology, Homotopy and Applications

Volume 16 (2014)

Number 2

Algebraic analogue of the Atiyah completion theorem

Pages: 289 – 306

DOI: https://dx.doi.org/10.4310/HHA.2014.v16.n2.a16

Authors

Alisa Knizel (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Alexander Neshitov (Department of Mathematics and Statistics, University of Ottawa, Ontario, Canada; and the St. Petersburg Department of Steklov Mathematical Institute, St. Petersburg, Russia)

Abstract

In topology there is a well-known theorem of Atiyah, Hirzebruch, and Segal which states that for a connected compact Lie group $G$ there is an isomorphism $\widehat{R(G)} \cong K^0(BG)$, where $BG$ is the classifying space of $G$. In the present paper we consider an algebraic analogue of this theorem. For a split reductive group $G$ over a field $k$, we prove that there is a natural isomorphism\[\widehat{K_n^G(k)}_{I_G} \cong K_n(BG),\]where $K_n^G(k)$ is Thomason’s $G$-equivariant $K$-theory of $\text{Spec }k$, $BG$ is a motivic étale classifying space introduced by Voevodsky and Morel, and $I_G$ is the augmentation ideal of $K_0^G(k)$.

Keywords

classifying space, representation ring, Atiyah-Segal theorem

2010 Mathematics Subject Classification

19E08, 20G15

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Published 30 November 2014