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arxiv: 2605.27244 · v1 · pith:F5DQBU4Qnew · submitted 2026-05-26 · 🧮 math.CT · math.AG· math.RT

Residual regularity in tensor triangular geometry

Emmy Van Rooy This is my paper

Pith reviewed 2026-06-29 14:19 UTC · model grok-4.3

classification 🧮 math.CT math.AGmath.RT
keywords residual regularitytensor triangulated categoriesfinite separable extensionspermutation modulesderived categoriesfinite groupsclassification
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The pith

Residual regularity descends and ascends via finite separable extensions, classifying all finite groups whose derived category of permutation modules satisfies the property.

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

The paper defines residual regularity as a new notion of regularity for tensor triangulated categories. It proves that this property descends and ascends along finite separable extensions. It applies the result to classify all finite groups for which the derived category of permutation modules is residually regular. A sympathetic reader would care because the stability gives a practical way to move the property between categories and the classification identifies exactly when it holds in this representation-theoretic setting.

Core claim

We introduce residual regularity as a new notion of regularity for tensor triangulated categories. We show that residual regularity descends and ascends via finite separable extensions and we classify all finite groups whose derived category of permutation modules is residually regular.

What carries the argument

Residual regularity, the newly defined property of tensor triangulated categories that is shown to transfer along finite separable extensions and to classify the relevant groups.

Load-bearing premise

The definition of residual regularity must pick out a meaningful and non-vacuous property on the tensor triangulated categories under study.

What would settle it

A finite separable extension where residual regularity fails to descend or ascend, or a finite group outside the classified list whose derived category of permutation modules is residually regular.

read the original abstract

We investigate a new notion of regularity for tensor triangulated categories, called residual regularity. We show that residual regularity descends and ascends via finite separable extensions and we classify all finite groups whose derived category of permutation modules is residually regular.

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

0 major / 1 minor

Summary. The manuscript introduces a new notion of residual regularity for tensor triangulated categories. It proves that this property descends and ascends along finite separable extensions and classifies all finite groups whose derived category of permutation modules satisfies residual regularity.

Significance. If residual regularity is a meaningful and non-vacuous property, the stability results under finite separable extensions and the classification for permutation-module categories would constitute a useful contribution to tensor triangular geometry, particularly for understanding regularity phenomena in derived categories of group representations. The classification result would serve as evidence that the definition is not vacuous.

minor comments (1)
  1. The abstract does not indicate whether the definition of residual regularity is accompanied by concrete examples or computations that would allow readers to verify non-vacuousness independently of the classification theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. The recommendation is marked uncertain, which appears to hinge on whether residual regularity is a meaningful notion. The classification of all finite groups whose derived permutation module categories are residually regular provides concrete evidence that the property is non-vacuous and distinguishes interesting examples, supporting the utility of the descent/ascent results under finite separable extensions.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a new definition of residual regularity and derives its stability under finite separable extensions together with a classification of finite groups for which the derived category of permutation modules satisfies the property. These steps are internal to the definition and its consequences; no load-bearing claim reduces to a fitted parameter, self-citation chain, or renaming of prior results. The classification demonstrates non-vacuousness rather than presupposing it, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, background axioms, or invented entities; the new notion itself is a definition rather than a postulated object with independent evidence.

pith-pipeline@v0.9.1-grok · 5541 in / 962 out tokens · 34048 ms · 2026-06-29T14:19:34.848298+00:00 · methodology

discussion (0)

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Reference graph

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