Complexes of injective<i>kG</i>-modules

Krause, Henning

doi:10.2140/ant.2008.2.1

Cited by 32 publications

(31 citation statements)

References 23 publications

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“…The stable category StMod(kG) of a finite group G is tensor triangulated, where the tensor product is M ⊗ k N with diagonal G-action and the unit is k. The homotopy category K(Inj kG) is also tensor triangulated with the same tensor product, but here the unit is ik, the injective resolution of k. In either case, the unit generates the category when G is a p-group; see [8] for details.…”

Section: Stratificationsmentioning

confidence: 99%

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Stratifying modular representations of finite groups

2011

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We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

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Section: Stratificationsmentioning

confidence: 99%

“…In [8], it was proved that if G is a finite p-group, then there is an equivalence of triangulated categories…”

Section: The Main Theoremsmentioning

confidence: 99%

Stratifying modular representations of finite groups

2011

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“…In order to capture the part of C that admits a good notion of homotopy groups, we will now introduce and study cellular objects in stable categories. Special cases of these ideas have previously appeared in various contexts, for example in motivic homotopy theory [DI05,Tot15] or work in progress by Casacuberta and White, in modular representation theory [BK08], and in equivariant KK-theory [Del10].…”

Section: 2mentioning

confidence: 99%

Local duality for structured ring spectra

Barthel

Heard²,

Valenzuela

2018

Journal of Pure and Applied Algebra

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Abstract. We use the abstract framework constructed in our earlier paper [BHV15] to study local duality for Noetherian E∞-ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.

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“…Using the techniques of [BHV18] and [BK08], the chromatic splitting can be transported to the stable module category StMod kG for any finite p-group G. Therefore, we obtain a qualitative description of how the stable module module category is locally built up from its indecomposable layers, at least for compact objects.…”

Section: 3mentioning

confidence: 99%

“…The E ∞ -ring spectrum C * (BG, k) of cochains on G with coefficients in k has homotopy groups π − * C * (BG, k) ∼ = H * (G, k) ∼ = k[ζ 1 , ζ 2 ] with ζ 1 and ζ 2 generators in degree 1. Therefore, C * (BG, k) is Noetherian and there is an equivalence Mod C * (BG,k) ≃ Stable kG as G is a 2-group, where Stable kG is the slight enlargement of the stable module category StMod kG constructed by Benson and Krause [BK08]. We may therefore work within the stable module category StMod kG ; in particular, all our constructions can be restricted to this subcategory of Stable kG .…”

Section: 3mentioning

confidence: 99%