Permutation modules and cohomological singularity

Balmer, Paul; Gallauer, Martin

doi:10.4171/cmh/534

Cited by 6 publications

(4 citation statements)

References 15 publications

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“…For example, for a discrete commutative ring A, the tt-category Fun(BG, Perf(A)) is equivalent to the bounded derived category of A[G]-representations whose underlying A-module is perfect. A result of Rouquier, Mathew [Tre15], and Balmer-Gallauer [BG22a] then implies that ψ(A) is an equivalence for any regular Noetherian A.…”

Section: Stratification For Cochainsmentioning

confidence: 92%

Quillen stratification in equivariant homotopy theory

Barthel¹,

Castellana²,

Heard³

et al. 2023

Preprint

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We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group G, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules. Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points.We then apply our methods to the case of Borel-equivariant Lubin-Tate E-theory En, for any finite height n and any finite group G, where we obtain a sharper theorem in the form of cohomological stratification. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing ⊗-ideals of the category of equivariant modules over En, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.

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Section: Stratification For Cochainsmentioning

confidence: 92%

Quillen stratification in equivariant homotopy theory

Barthel¹,

Castellana²,

Heard³

et al. 2023

Preprint

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“…Recollection. In [BG22a], we prove that the homotopy category of injective RGmodules, with coefficients in any regular ring R (e.g. our field k), is a localization of DPerm(G; R).…”

Section: Conservativity Via Modular Fixed-pointsmentioning

confidence: 99%

“…Following Neeman-Thomason, the above localization (2.13) is the compact part of a finite localization of the corresponding 'big' tt-categories T(G) K Inj(kG), the homotopy category of complexes of injectives. See [BG22a,Remark 4.21]. We return to this localization of big categories in Recollection 5.7.…”

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confidence: 99%

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

Balmer¹,

Gallauer²

2022

Preprint

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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.Permutation modules. Among the easiest representations to construct, permutation modules are simply the k-linearizations k(X) of G-sets X. And yet they play an important role in subjects as varied as derived equivalences [Ric96], Mackey functors [Yos83], or equivariant homotopy theory [MNN17], to name a few. The authors' original interest stems from yet another connection, namely the one with Voevodsky's theory of motives [Voe00], specifically Artin motives. For a gentle introduction to these ideas, we refer the reader to [BG21].We consider a 'small' tensor triangulated category, the homotopy category (1.2) K(G) := K b perm(G; k)

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“…For the identification of the two categories see[BG22a].) But Ẑp is a free pro-p-group and therefore has p-cohomological dimension one [RZ10, Corollary 7.5.2].…”

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confidence: 99%