Algebraic analogue of the Atiyah completion theorem

Alisa Knizel ; Alexander Neshitov

Homology, homotopy, and applications, Tome 16 (2014) no. 2, pp. 289-306.

Résumé

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 $K_0^G(k)$.

Détail

DOI : 10.4310/HHA.2014.v16.n2.a16

Classification : 19E08, 20G15
Keywords: classifying space, representation ring, Atiyah-Segal theorem