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arXiv · 2609.24573

Nonvanishing of cohomology for groups

Tiago Cruz·Rene Marczinzik
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Abstract

Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. 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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Tiago Cruz, Rene Marczinzik. 2026-09-21. Nonvanishing of cohomology for groups. https://arxiv.org/abs/2609.24573

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