Homological finiteness conditions for modules over strongly group-graded rings

Cornick, Jonathan; Kropholler, Peter H.

doi:10.1017/s030500410007465x

Cited by 14 publications

(14 citation statements)

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“…Kropholler [30] shows that in the context of modules of type F P ∞ over group rings of groups in a very large class denoted LHF , complete cohomology can be calculated using complete resolutions. See also Cornick and Kropholler [16], [17], Benson [7], [8].…”

Section: ) Applying the Exact Functor F Yields A D(a)-presentationmentioning

confidence: 99%

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Realizability of modules over Tate cohomology

Benson

Krause

Schwede³

2003

Trans. Amer. Math. Soc.

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Abstract. Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology γ G ∈ HH 3,−1Ĥ * (G, k) with the following property. Given a gradedĤ(X, X) vanishes if and only if X is isomorphic to a direct summand ofĤ * (G, M ) for some kG-module M .The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3,−1 H * (A) which is a predecessor for these obstructions.

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Section: ) Applying the Exact Functor F Yields A D(a)-presentationmentioning

confidence: 99%

“…16. Consider an N -exact k-Postnikov system as in (A.7) and let X denote the cokernel of the homomorphism…”

Section: Diagram Of Graded H-modulesmentioning

confidence: 99%

Realizability of modules over Tate cohomology

Benson

Krause

Schwede³

2003

Trans. Amer. Math. Soc.

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“…If H is an H F-group then this can be proved by induction on the ordinal height of H in the H F-hierarchy. The proof proceeds in exactly the same way as the proof of the Vanishing Theorem ( [9], §8).…”

Section: Proof the Connecting Mapsmentioning

confidence: 96%

“…For H F-groups of type FP ∞ this follows from ( [9], Proposition 9.2) by taking the ring R to be ZG and taking the module M to be the trivial ZG-module Z. However we need to strengthen this result in two ways.…”

Section: Proof the Connecting Mapsmentioning

confidence: 99%

Groups with Many Finitary Cohomology Functors

Kropholler

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. For a group G, we study the question of which cohomology functors commute with all small filtered colimit systems of coefficient modules. We say that the functor H n (G, ) is finitary when this is so and we consider the finitary set for G, that is the set of natural numbers for which this holds. It is shown that for the class of groups LHF there is a dichotomy: the finitary set of such a group is either finite or cofinite. We investigate which sets of natural numbers n can arise as finitary sets for suitably chosen G and what restrictions are imposed by the presence of certain kinds of normal or near-normal subgroups. Although the class LHF is large, containing soluble and linear groups, being closed under extensions, subgroups, amalgamated free products, HNN-extensions, there are known to be many not in LHF such as Richard Thompson's group F . Our theory does not extend beyond the class LHF at present and so it is an open problem whether the main conclusions of this paper hold for arbitrary groups. There is a survey of recent developments and open questions. Organizational StatementThis paper lays the foundation stones for a series of papers by the author's former student Martin Hamilton: [13,11,12]. As sometimes happens, this literature has not been published in the order in which it was intended to be read and for this reason I am taking the opportunity of this conference proceedings to include a survey of Hamilton's papers and a discussion of possible future directions. This survey follows and expands upon the spirit of the talk I gave at the meeting.

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“…The first and third authors of the present work introduced a theory in a series of papers [6,8,7,16] the last one of which builds on the ideas of Benson. In view of the original Bieri-Eckmann theory, it seems very natural to consider what happens if one requires some or many cohomology functors to commute with filtered colimits. The behaviour of groups for which this kind of continuity of functors begins after some steps was investigated by the third author and then further in a series of papers by Hamilton [17,11,13,12].…”

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confidence: 99%

Finiteness conditions in the stable module category

Cornick

Emmanouil

Kropholler

et al. 2014

Advances in Mathematics

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We study groups whose cohomology functors commute with filtered colimits in high dimensions. We relate this condition to projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg-Mac Lane spaces with finitely many n-cells for all sufficiently large n. To that end we determine the structure of finitary Gorenstein projective modules. The methods are inspired by representation theory and make use of the stable module category in which morphisms are defined through complete cohomology. In order to carry out these methods we need to restrict ourselves to certain classes of hierarchically decomposable groups.

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Homological finiteness conditions for modules over strongly group-graded rings

Cited by 14 publications

References 10 publications

Realizability of modules over Tate cohomology

Realizability of modules over Tate cohomology

Groups with Many Finitary Cohomology Functors

Finiteness conditions in the stable module category

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