Cohomology Rings of Finite Groups

Carlson, Jon; Townsley, Lisa; Valeri-Elizondo, Luis; Zhang, Mucheng

doi:10.1007/978-94-017-0215-7

Cited by 68 publications

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“…The theory of support varieties played a key role in the classification of endotrivial modules over p-groups and it is a necessary ingredient in two of the constructions that appear in Sections 5 and 6. The books [5] and [14] serve as references for this material.…”

Section: Preliminariesmentioning

confidence: 99%

The classification of endo-trivial modules

Carlson¹,

Thévenaz

2004

Invent. math.

Self Cite

103

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Abstract. Let G be a finite group and let T (G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We investigate the torsion-free part T F (G) of the group T (G) and look for generators of T F (G). We describe three methods for obtaining generators. Each of them only gives partial answers to the question but we obtain more precise results in some specific cases. We also conjecture that T F (G) can be generated by modules belonging to the principal block and we prove the conjecture in some cases.

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Section: Preliminariesmentioning

confidence: 99%

The classification of endo-trivial modules

Carlson¹,

Thévenaz

2004

Invent. math.

Self Cite

103

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“…Whenever there is more than one place to find a theorem and its proof, we refer to the original paper although it may not be the easiest source to find. Many of the old results that we quote in the paper can also be found in the books by Benson [1], [2] and Carlson [7], [8].…”

Section: Theorem 12 Let ζ ∈ Hmentioning

confidence: 75%

“…Observe that if (H 1 , H 2 ) and (H 1 , H 2 ) are two different choices of homotopies satisfying the equations in (8), then the differences H 1 − H 1 and H 2 − H 2 are chain maps. We will see below that, up to chain maps, the homotopies H 1 and H 2 can be chosen to satisfy certain identities.…”

Section: Lemma 52 There Are Homotopy Equivalencesmentioning

confidence: 99%

Productive elements in group cohomology

Yalcin

2011

Homology, Homotopy and Applications

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Let G be a finite group and k be a field of characteristic p > 0. A cohomology class ζ ∈ H n (G, k) is called productive if it annihilates Ext * kG (L ζ , L ζ ). We consider the chain complex P(ζ) of projective kG-modules which has the homology of an (n − 1)-sphere and whose k-invariant is ζ under a certain polarization. We show that ζ is productive if and only if there is a chain map ∆ : P(ζ) → P(ζ) ⊗ P(ζ) such that (id ⊗ )∆ id and ( ⊗ id)∆ id. Using the Postnikov decomposition of P(ζ) ⊗ P(ζ), we prove that there is a unique obstruction for constructing a chain map ∆ satisfying these properties. Studying this obstruction more closely, we obtain theorems of Carlson and Langer on productive elements. IntroductionLet G be a finite group and k be a field of characteristic p > 0. Let ζ ∈ H n (G, k) denote a nonzero cohomology class of degree n, where n 1. Associated to ζ, there is a unique kG-module homomorphismζ : Ω n k → k and the kG-module L ζ is defined as the kernel of this homomorphism. A cohomology class ζ is called productive if it annihilates the cohomology ring Ext * kG (L ζ , L ζ ). In this paper, we study the conditions for a cohomology class to be productive. Under the usual identification of H n (G, k) with the group U n (k, k) of n-fold kGmodule extensions of k by k, the cohomology class ζ is the extension class of an extension of the form

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“…The element ζ annihilates the cohomology of M if and only if L ζ ⊗ M ∼ = Ω n M ⊕ ΩM ⊕ P where P is some projective module. See Section 9.7 of [8]. From the same source we have that if p > 2 and if ζ ∈ H * (G, k) with n even, then ζ annihilates the cohomology of L ζ .…”

Section: Lemma 34 Suppose That We Have An Exact Sequencementioning

confidence: 92%

“…We use standard facts and notation of the stable module category of kG [7], support varieties [2,8], and of almost split sequences [2].…”

Section: Introductionmentioning

confidence: 99%

Finite generation of Tate cohomology

2011

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Abstract. Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable nonprojective kG-module M , we conjecture that if the Tate cohomologyĤ * (G, M ) of G with coefficients in M is finitely generated over the Tate cohomology ringĤ * (G, k), then the support variety V G (M ) of M is equal to the entire maximal ideal spectrum V G (k). We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of k in the stable Auslander-Reiten quiver for kG, but it is shown to be false in general. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology.

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Cohomology Rings of Finite Groups

Cited by 68 publications

References 0 publications

The classification of endo-trivial modules

The classification of endo-trivial modules

Productive elements in group cohomology

Finite generation of Tate cohomology

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