Equivariant $H\underline{\mathbb{F}}_p$-modules are wild
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| Format: | Preprint |
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2025
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| _version_ | 1866910201061310464 |
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| author | Grevstad, Jacob Fjeld May, Clover |
| author_facet | Grevstad, Jacob Fjeld May, Clover |
| contents | 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 $\underline{k}$, the classification of compact modules over the $G$-equivariant Eilenberg--MacLane spectrum $H\underline{k}$ 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 $C_p$-equivariant $H\underline{\mathbb{F}}_p$-modules exists at odd primes. For the Burnside Mackey functor $\underline{A}_k$, there is no classification of compact $G$-equivariant $H\underline{A}_k$-modules whenever $G$ is a nontrivial $p$-group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_17604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant $H\underline{\mathbb{F}}_p$-modules are wild Grevstad, Jacob Fjeld May, Clover Representation Theory Algebraic Topology 16G60 (Primary), 55P91 (Secondary) 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 $\underline{k}$, the classification of compact modules over the $G$-equivariant Eilenberg--MacLane spectrum $H\underline{k}$ 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 $C_p$-equivariant $H\underline{\mathbb{F}}_p$-modules exists at odd primes. For the Burnside Mackey functor $\underline{A}_k$, there is no classification of compact $G$-equivariant $H\underline{A}_k$-modules whenever $G$ is a nontrivial $p$-group. |
| title | Equivariant $H\underline{\mathbb{F}}_p$-modules are wild |
| topic | Representation Theory Algebraic Topology 16G60 (Primary), 55P91 (Secondary) |
| url | https://arxiv.org/abs/2509.17604 |