Complexes of modules over hierarchical groups
Abstract
Let F̅ be the group class obtained from the class F of finite groups by iterated applications of Kropholler's operation scriptstylebf LH and Talelli's operation Φ. We describe the structure of the cokenels of acyclic complexes of projective modules over the group algebra of F̅-groups with coefficients in a commutative ring. If G is an F̅-group, we show that any acyclic complex of projective (respectively, flat) ℚG-modules is contractible (respectively, pure acyclic). If G is torsion-free, the same conclusions hold for acyclic complexes of projective or flat ℤG-modules. Analogous results are valid for acyclic complexes of injective modules. We present some applications regarding modules that admit complete resolutions, groups with periodic cohomology after some steps and the relation between Gorenstein and ordinary homological dimensions for modules over group algebras.
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BibTeX
@misc{emmanouil2026complexes,
title = {{Complexes of modules over hierarchical groups}},
author = {Ioannis Emmanouil and Olympia Talelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36823},
eprint = {2609.36823},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36823},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complexes of modules over hierarchical groups
AU - Ioannis Emmanouil
AU - Olympia Talelli
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36823
UR - https://arxiv.org/abs/2609.36823
AB - Let F̅ be the group class obtained from the class F of finite groups by iterated applications of Kropholler's operation scriptstylebf LH and Talelli's operation Φ. We describe the structure of the cokenels of acyclic complexes of projective modules over the group algebra of F̅-groups with coefficients in a commutative ring. If G is an F̅-group, we show that any acyclic complex of projective (respectively, flat) ℚG-modules is contractible (respectively, pure acyclic). If G is torsion-free, the same conclusions hold for acyclic complexes of projective or flat ℤG-modules. Analogous results are valid for acyclic complexes of injective modules. We present some applications regarding modules that admit complete resolutions, groups with periodic cohomology after some steps and the relation between Gorenstein and ordinary homological dimensions for modules over group algebras.
ER -