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Nonvanishing of cohomology for groups
Abstract
Let G be a finite group. We prove that the order of the abelianization G/G' divides the order of H2m(G,ℤ) for every m≥1. As a consequence, for a field K, nonvanishing of Ext1KG(K,K) implies nonvanishing of ExtnKG(K,K) for every n≥1. This answers a conjecture by Erdmann, Klász and Marczinzik. We use this to show that the Hochschild cohomology of KG is nonzero in every nonnegative degree whenever the characteristic of K divides |G|.
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BibTeX
@misc{cruz2026nonvanishing,
title = {{Nonvanishing of cohomology for groups}},
author = {Tiago Cruz and Rene Marczinzik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24573},
eprint = {2609.24573},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24573},
}
RIS
TY - UNPB
TI - Nonvanishing of cohomology for groups
AU - Tiago Cruz
AU - Rene Marczinzik
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24573
UR - https://arxiv.org/abs/2609.24573
AB - Let G be a finite group. We prove that the order of the abelianization G/G' divides the order of H^2m(G,ℤ) for every m≥1. As a consequence, for a field K, nonvanishing of Ext^1_KG(K,K) implies nonvanishing of Ext^n_KG(K,K) for every n≥1. This answers a conjecture by Erdmann, Klász and Marczinzik. We use this to show that the Hochschild cohomology of KG is nonzero in every nonnegative degree whenever the characteristic of K divides |G|.
ER -