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The naive-commutative structure on rational equivariant K-theory May, Clover


The uniqueness of complex $K$-theory as an $E_\infty$ ring spectrum was shown by Baker and Richter in 2005 using obstruction theory. Working rationally, we show for any finite abelian group this extends uniquely to a naive-commutative ring structure for equivariant $K$-theory. The proof involves finding the image of $K$-theory in the algebraic model of Barnes, Greenlees, and Kedziorek given by rational CDGAs with an action of the Weyl group. Despite lacking an explicit description of the CDGAs corresponding to $K$-theory, we compute the homology from the homotopy of the geometric fixed-points and prove formality. This is joint work with Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, and Magdalena Kedziorek.

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