Productive elements in group cohomology

Let $G$ be a finite group and $k$ be a field of characteristic $p \gt 0$. A cohomology class $\zeta\in H^n (G,k)$ is called productive if it annihilates $\operatorname{Ext}^*_{kG}(L_\zeta,L_\zeta)$. We consider the chain complex $\mathbf{P}(\zeta)$ of projective $kG$-modules which has the homology of an $(n - 1)$-sphere and whose $k$-invariant is $\zeta$ under a certain polarization. We show that $\zeta$ is productive if and only if there is a chain map $\Delta : \mathbf{P}(\zeta)\to \mathbf{P}(\zeta)\otimes \mathbf{P}(\zeta)$ such that $(\operatorname{id} \otimes \epsilon) \Delta \simeq \operatorname{id}$ and $(\epsilon \otimes \operatorname{id}) \Delta \simeq \operatorname{id}$. Using the Postnikov decomposition of $\mathbf{P}(\zeta) \otimes \mathbf{P}(\zeta)$, we prove that there is a unique obstruction for constructing a chain map $\Delta$ satisfying these properties. Studying this obstruction more closely, we obtain theorems of Carlson and Langer on productive elements.

Keywords

group cohomology, chain complex, diagonal approximation

2010 Mathematics Subject Classification

20C20, 20J06, 57S17

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Published 12 July 2011