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← On the Balmer spectrum of derived Mackey functors

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A result of Devinatz-Hopkins-Smith describes the spectrum of prime ideals of finite spectra. Under this identification the information encoded by the zeroth and infinite chromatic levels can be identified with the spectrum of the derived category of integers, which by a result of Hopkins-Neeman is just equivalent to Spec(Z). It turns out that in the equivariant context neither the spectrum of the usual derived category of Mackey functors nor the spectrum of the Burnside ring play the role of Spec(Z). Given a finite group G, we show that Kaledin's category of derived G-Mackey functors describes the zeroth and infinite chromatic levels of the Balmer spectrum of finite G-spectra. We compute the Balmer spectrum of derived G-Mackey functors. Along the way we will identify Kaledin's category with the homotopy category of the stable infinity category of HZ-linear spectral Mackey functors in the sense of Barwick. This is all joint with B. Sanders and C. Wimmer.

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