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Paul Balmer

Publications and source records attributed to Paul Balmer.

At least 19 recordsLinked to original sources
arXiv2026

Mackeyfication of equivariant categories I

Colloquially speaking, `equivariant categories' refer to families of additive categories $\mathcal{A}(G)$ depending 2-functorially on a finite group $G$. We construct approximations of equivariant categories by Mackey 2-functors, both on the left and on the right. The idea is to enlarge $\mathcal{A}$ in a minimal way to make induction appear. These `mackeyfications' are inspired by Boltje's work with ordinary Mackey 1-functors. We also relate our left and right mackeyfications via a mark transformation. Finally we discuss examples.

math.RT↗
arXiv2026

Permutation, stabilization and decomposition

Informed by our understanding of the tt-geometry of permutation modules, we investigate the proper definition of the `stable permutation category' of a finite group. Then we prove that this category decomposes over cyclic and generalized quaternion groups and only in those cases.

math.RT↗
arXiv2025

Patch-density in tensor-triangular geometry

The spectrum of a tensor-triangulated category carries a compact Hausdorff topology, called the constructible topology, also known as the patch topology. We prove that patch-dense subsets detect tt-ideals and we prove that any infinite family of tt-functors that detects nilpotence provides such a patch-dense subset. We review several applications and examples in algebra, in topology and in the representation theory of profinite groups.

math.CT↗
arXiv2025

The Tate Intermediate Value Theorem

We explain how the gluing of a closed piece of the tensor-triangular spectrum with its open complement hinges on the support of the Tate ring.

math.AC↗
arXiv2024

Perfect complexes and completion

Let $\hat{R}$ be the $I$-adic completion of a commutative ring $R$ with respect to a finitely generated ideal $I$. We give a necessary and sufficient criterion for the category of perfect complexes over $\hat{R}$ to be equivalent to the subcategory of dualizable objects in the derived category of $I$-complete complexes of $R$-modules. Our criterion is always satisfied when $R$ is noetherian. When specialized to $R$ local and noetherian and to $I$ the maximal ideal, our theorem recovers a recent result of Benson, Iyengar, Krause and Pevtsova.

math.AC↗
arXiv2024

The spectrum of Artin motives

We analyze the tt-geometry of derived Artin motives, via modular representation theory of profinite groups. To illustrate our methods, we discuss Artin motives over a finite field, in which case we also prove stratification.

math.AG↗
arXiv2023

The geometry of permutation modules

We consider the derived category of permutation modules for a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the spectrum of its compact objects, by reducing the problem to elementary abelian groups and then by using a twisted form of cohomology to express the spectrum locally in terms of the graded endomorphism ring of the unit. Together, these results yield a classification of thick and of localizing ideals.

math.RT↗
arXiv2022

The tt-geometry of permutation modules. Part I: Stratification

We consider the derived category of permutation modules over a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the set underlying the tt-spectrum of compact objects, and discuss several examples.

math.RT↗
arXiv2021

Relative stable categories and birationality

We propose a general method to construct new triangulated categories, relative stable categories, as additive quotients of a given one. This construction enhances results of Beligiannis, particularly in the tensor-triangular setting. We prove a birationality result showing that the original category and its relative stable quotient are equivalent on some open piece of their spectrum.

math.CT↗
arXiv2021

Cohomological Mackey 2-functors

We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in arXiv:1808.04902, obtained by modding out the so-called cohomological relations. This categorifies Yoshida's Theorem for ordinary cohomological Mackey functors, and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

math.CT↗
arXiv2021

Green equivalences in equivariant mathematics

We establish Green equivalences for all Mackey 2-functors, without assuming Krull-Schmidt. By running through the examples of Mackey 2-functors, we recover all variants of the Green equivalence and Green correspondence known in representation theory and obtain new ones in several other contexts. Such applications include equivariant stable homotopy theory in topology and equivariant sheaves in geometry.

math.CT↗
arXiv2020

Finite permutation resolutions

We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.

math.RT↗
arXiv2020

Permutation modules and cohomological singularity

We define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient ring. It detects which representations are controlled by permutation modules.

math.RT↗
arXiv2020

Mackey 2-functors and Mackey 2-motives

We study collections of additive categories $\mathcal{M}(G)$, indexed by finite groups $G$ and related by induction and restriction in a way that categorifies usual Mackey functors. We call them `Mackey 2-functors'. We provide a large collection of examples in particular thanks to additive derivators. We prove the first properties of Mackey 2-functors, including separable monadicity of restriction to subgroups. We then isolate the initial such structure, leading to what we call `Mackey 2-motives'. We also exhibit a convenient calculus of morphisms in Mackey 2-motives, by means of string diagrams. Finally, we show that the 2-endomorphism ring of the identity of $G$ in this 2-category of Mackey 2-motives is isomorphic to the so-called crossed Burnside ring of $G$.

math.RT↗
arXiv2020

Resolutions by permutation modules

We prove that, up to adding a complement, every modular representation of a finite group admits a finite resolution by permutation modules.

math.RT↗