Productive elements in group cohomology

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

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