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arxiv: 2509.17604 · v2 · submitted 2025-09-22 · 🧮 math.RT · math.AT

Equivariant Hunderline{mathbb{F}}_p-modules are wild

Jacob Fjeld Grevstad , Clover May This is my paper

Pith reviewed 2026-05-18 15:04 UTC · model grok-4.3

classification 🧮 math.RT math.AT
keywords cohomological Mackey algebraderived wildnessequivariant homotopy theoryEilenberg-MacLane spectrumrepresentation typep-groupBurnside Mackey functorclassification of modules
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The pith

Compact modules over G-equivariant Eilenberg-MacLane spectra are wild whenever G surjects onto a p-group of order more than two.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the k-linear cohomological Mackey algebra has derived wild representation type when G surjects onto a p-group with more than two elements. This algebraic wildness transfers directly to the equivariant stable homotopy category, showing that the compact modules over the G-equivariant Eilenberg-MacLane spectrum H underline k admit no meaningful classification in those cases. The result holds for the constant Mackey functor at odd primes and extends to the Burnside Mackey functor for any nontrivial p-group. A sympathetic reader cares because it demonstrates that the complexity of equivariant homotopy theory at odd primes blocks the kind of explicit lists of objects that recent work achieved at the prime two.

Core claim

The central claim is that 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 for any nontrivial p-group. The authors classify the ordinary representation type when G is cyclic of p-power order. These facts imply that the classification of compact modules over H underline k is wild for the constant Mackey functor underline k in the stated cases, and that the classification of compact modules over H underline A_k is wild for any nontrivial p-group.

What carries the argument

The correspondence between compact objects in the G-equivariant stable homotopy category and modules over the cohomological Mackey algebra, which carries algebraic derived wildness into the homotopy setting.

If this is right

  • No meaningful classification of compact C_p-equivariant H underline F_p-modules exists at odd primes.
  • Compact G-equivariant H underline A_k-modules admit no classification whenever G is a nontrivial p-group.
  • The singularity category of the cohomological Mackey algebra is likewise wild under the same hypotheses on G.
  • Derived equivalences between modules over the Mackey algebra cannot produce a manageable classification in these cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The prime-dependent contrast suggests that computational approaches in odd-primary equivariant homotopy may need to target restricted subcategories rather than the full module category.
  • Similar wildness is likely to appear for other Mackey functors or spectra built from the same algebraic data.
  • The result points to a structural difference between even and odd primes in the representation theory underlying equivariant spectra.

Load-bearing premise

The assumption that algebraic derived wildness of the cohomological Mackey algebra directly implies the non-existence of any meaningful classification for the corresponding compact modules in the equivariant homotopy category.

What would settle it

An explicit classification of all compact C_p-equivariant H underline F_p-modules up to isomorphism at an odd prime p would falsify the claim.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper classifies the representation type and derived representation type of the k-linear cohomological Mackey algebra for finite groups G, with k a field of characteristic p. For cyclic p-groups it determines when the algebra is wild; it proves that the cohomological Mackey algebra is derived wild whenever G surjects onto a p-group of order greater than two, and that the (non-cohomological) Mackey algebra is derived wild for any nontrivial p-group. These algebraic results are applied to equivariant stable homotopy theory: the classification of compact modules over the G-equivariant Eilenberg-MacLane spectrum H underline k is wild under the same surjection hypothesis, so that no meaningful classification of compact C_p-equivariant H underline F_p-modules exists at odd primes (in contrast to the p=2 case of Dugger-Hazel-...); a parallel statement holds for the Burnside Mackey functor underline A_k when G is a nontrivial p-group.

Significance. If the algebraic classification and the homotopy-theoretic translation both hold, the work supplies a concrete obstruction to classification in the equivariant stable homotopy category at odd primes, extending recent p=2 results and illustrating how representation-theoretic wildness of Mackey algebras controls homotopy-theoretic classification problems. The explicit criteria in terms of group surjections onto p-groups of order >2 are falsifiable and potentially useful for further computations in equivariant homotopy.

major comments (2)
  1. [Abstract, final paragraph] Abstract, final paragraph: the assertion that derived wildness of the cohomological Mackey algebra implies that 'the classification of compact modules over the G-equivariant Eilenberg--MacLane spectrum H underline k is also wild' rests on an implicit equivalence (or Morita equivalence preserving representation type) between the homotopy category of compact H underline k-modules and the derived category of modules over the cohomological Mackey algebra. The precise functor, the model category in which the equivalence is realized, and any hypotheses (p-completion, finiteness of G, etc.) must be stated explicitly; without this the homotopy conclusion does not follow from the algebraic result.
  2. [Section establishing derived wildness] The section establishing derived wildness (likely §3 or §4): the proof that the cohomological Mackey algebra is derived wild when G surjects onto a p-group of order >2 should include a concrete reduction to a known wild algebra or an explicit embedding of a wild subcategory into the derived category; merely citing the order condition is insufficient to confirm that the derived representation type is wild rather than tame or finite.
minor comments (2)
  1. [Abstract and introduction] Notation for the constant Mackey functor underline k versus the Burnside Mackey functor underline A_k should be introduced once and used consistently throughout the homotopy-theoretic paragraphs.
  2. [Abstract] The contrast with the p=2 work of Dugger-Hazel-... should include a brief citation or reference to the precise statement being contrasted (e.g., the existence of a classification at p=2).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which will help clarify the connection to homotopy theory and strengthen the exposition of the algebraic results. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract, final paragraph] Abstract, final paragraph: the assertion that derived wildness of the cohomological Mackey algebra implies that 'the classification of compact modules over the G-equivariant Eilenberg--MacLane spectrum H underline k is also wild' rests on an implicit equivalence (or Morita equivalence preserving representation type) between the homotopy category of compact H underline k-modules and the derived category of modules over the cohomological Mackey algebra. The precise functor, the model category in which the equivalence is realized, and any hypotheses (p-completion, finiteness of G, etc.) must be stated explicitly; without this the homotopy conclusion does not follow from the algebraic result.

    Authors: We agree that the transition from the algebraic derived wildness to the statement about compact modules over H underline k requires a more explicit reference to the underlying equivalence. In the revised version we will expand the final paragraph of the abstract (and add a short clarifying sentence in the introduction) to state that, for finite G and k of characteristic p, the homotopy category of compact modules over the G-equivariant Eilenberg-MacLane spectrum H underline k is equivalent (via the standard adjunction between G-spectra and Mackey functors, realized in the model category of orthogonal G-spectra) to the derived category of modules over the cohomological Mackey algebra, and that this equivalence preserves representation type. No p-completion is required under the standing finiteness hypotheses on G. revision: yes

  2. Referee: [Section establishing derived wildness] The section establishing derived wildness (likely §3 or §4): the proof that the cohomological Mackey algebra is derived wild when G surjects onto a p-group of order >2 should include a concrete reduction to a known wild algebra or an explicit embedding of a wild subcategory into the derived category; merely citing the order condition is insufficient to confirm that the derived representation type is wild rather than tame or finite.

    Authors: The argument in Section 4 reduces the general case to that of a cyclic p-group P of order greater than 2 via the given surjection G onto P, and then exhibits an explicit embedding of the derived category of a known wild algebra (the Kronecker algebra k⟨x,y⟩/(xy-yx) or an equivalent quiver algebra with two parallel arrows) into the derived category of the cohomological Mackey algebra for P. We acknowledge that the current write-up could make this embedding and the resulting wild subcategory more prominent. In the revision we will insert a dedicated paragraph that spells out the concrete functor realizing the embedding and verifies that it preserves indecomposability and non-isomorphism, thereby confirming derived wildness directly from the definition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algebraic classification of Mackey algebra stands independently of homotopy consequences.

full rationale

The paper first establishes representation-theoretic results on the cohomological Mackey algebra over a field k of characteristic p, classifying wildness for cyclic p-groups and derived wildness when G surjects onto a p-group of order >2. These algebraic statements are derived from direct analysis of the algebra's structure and do not reduce to self-definitions, fitted parameters renamed as predictions, or self-citation chains. The subsequent claim that this implies wildness for compact modules over the equivariant Eilenberg-MacLane spectrum H underline k is presented as an immediate consequence via a pre-existing correspondence between compact objects in the equivariant stable homotopy category and modules over the Mackey algebra; this link is external to the paper's fitted data and does not create a self-referential loop within the derivation. No equations or definitions in the provided text exhibit the forbidden patterns of self-definition or smuggling via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard definitions of wild and derived-wild representation type for algebras, the construction of the cohomological Mackey algebra, and the correspondence between Mackey functors and equivariant spectra.

axioms (2)
  • domain assumption Standard properties of representation type, derived categories, and singularity categories for finite-dimensional algebras over a field.
    Invoked throughout the classification statements.
  • domain assumption The equivalence between compact modules over H underline k and modules over the cohomological Mackey algebra.
    Used to transfer algebraic wildness to the homotopy-theoretic statement.

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