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 $ext{Spec }k$, $BG$ is a motivic étale classifying space introduced by Voevodsky and Morel, and $I_G$ is the augmentation ideal of $K_0^G(k)$.
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机译：在拓扑中，有一个著名的定理Atiyah，Hirzebruch和Segal，其中指出，对于一个连通的紧凑Lie群$ G $，存在同构$ widehat {R（G）} cong K ^ 0（BG）$ ，其中$ BG $是$ G $的分类空间。在本文中，我们考虑该定理的代数类似物。对于在字段$ k $上的拆分归约组$ G $，我们证明存在自然同构 [ widehat {K_n ^ G（k）} _ {I_G} cong K_n（BG），]其中$ K_n ^ G（k）$是Thomason的$ G $等价$ K $-$ text {Spec} k $理论，$ BG $是Voevodsky和Morel引入的动机分类空间，$ I_G $是$ K_0 ^ G（k）$的增广理想。
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