Equivariant HFp-modules are wild

Let k be an arbitrary field of characteristic p and let G be a finite group. We investigate the representation type, derived representation type, and singularity category of the k-linear (cohomological) Mackey algebra. We classify when the cohomological Mackey algebra is wild for G a cyclic p-group. Furthermore, we show the cohomological Mackey algebra is derived wild whenever G surjects onto a p-group of order more than two, and the Mackey algebra is derived wild whenever G is a nontrivial p-group. Derived wildness has some immediate consequences in equivariant homotopy theory. In particular, for the constant Mackey functor k, the classification of compact modules over the G-equivariant Eilenberg–MacLane spectrum Hk is also wild whenever G surjects onto a p-group of order more than two. Thus, in contrast to recent work at the prime 2 by Dugger, Hazel, and the second author, no meaningful classification of compact Cp-equivariant HFp-modules exists at odd primes. For the Burnside Mackey functor Ak, there is no classification of compact G-equivariant HAk-modules whenever G is a nontrivial p-group.

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