2025/09/22 by Jacob Fjeld Grevstad, Grevstad, Jacob Fjeld, Clover May +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2509.17604
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 \underlinek, the classification of compact modules over the G-equivariant Eilenberg--MacLane spectrum H\underlinek 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 H\underline\mathbbFp-modules exists at odd primes. For the Burnside Mackey functor \underlineAk, there is no classification of compact G-equivariant H\underlineAk-modules whenever G is a nontrivial p-group.