A group action on the input ring or category induces an action on thealgebraic $K$-theory spectrum. However, a shortcoming of this naive approach toequivariant algebraic $K$-theory is, for example, that the map of spectra with$G$-action induced by a $G$-map of $G$-rings is not equivariant. We define aversion of equivariant algebraic $K$-theory which encodes a group action on theinput in a functorial way to produce a $genuine$ algebraic $K$-theory$G$-spectrum for a finite group $G$. The main technical work lies in studyingcoherent actions on the input category. A payoff of our approach is that itbuilds a unifying framework for equivariant topological $K$-theory, Atiyah'sReal $K$-theory, and existing statements about algebraic $K$-theory spectrawith $G$-action. We recover the map from the Quillen-Lichtenbaum conjecture andthe representational assembly map studied by Carlsson and interpret them fromthe perspective of equivariant stable homotopy theory.
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机译：输入环或类别上的组动作引起代数$ K $-理论谱上的动作。但是，这种对等变代数$ K $理论的幼稚方法的缺点是，例如，由$ G $环的$ G $映射诱导的具有$ G $作用的频谱图不是等变的。我们定义了等变代数$ K $-理论的反函数，该函数以函数的方式对输入进行群操作，从而为有限组$ G $产生$$$$代数$ K $-理论$ G $-谱。主要技术工作在于研究输入类别上的连贯动作。我们的方法的好处是，它为等变拓扑$ K $理论，Atiyah的Real $ K $理论以及关于带有$ G $作用的代数$ K $理论谱的现有陈述建立了一个统一的框架。我们从Quillen-Lichtenbaum猜想和Carlsson研究的代表性装配图恢复地图，并从等变稳定同伦理论的角度对其进行解释。
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