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The stable module category and model structures for hierarchically defined groups
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abstract
In this work we construct a compactly generated tensor-triangulated stable category for a large class of infinite groups, including those in Kropholler's hierarchy $\mathrm{LH}\mathfrak{F}$. This can be constructed as the homotopy category of a certain model category structure, which we show is Quillen equivalent to several other model categories, including those constructed by Bravo, Gillespie, and Hovey in their work on stable module categories for general rings. We also investigate the compact objects in this category. In particular, we give a characterisation of those groups of finite global Gorenstein AC-projective dimension such that the trivial representation $\mathbb{Z}$ is compact.
Forward citations
Cited by 3 Pith papers
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The Balmer spectrum and telescope conjecture for infinite groups
For H1F groups of type FP∞, the Balmer spectrum of dualisable objects is Proj(H*(G,k)) and the telescope conjecture holds, while infinite free products give Stone-Cech spectra and failures of the telescope conjecture ...
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Total acyclicity of complexes over group algebras
The authors define a class of groups for which Gorenstein projective, flat, and injective modules over the group algebra behave as their classical counterparts, and prove this class is closed under Kropholler's LH and...
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Group class operations and homological conditions
Groups obtained by closing finite or weakly Gorenstein regular groups under the LH and Φ operations have virtually Gorenstein group algebras and satisfy Moore's conjecture.
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