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Given two short exact sequences of modules $$0\to N\to P\to R\to 0$$ $$0\to R\to Q\to M\to 0$$ Denote their extension classes by $e_1\in \mathrm{Ext}^1(R,N)$, $e_2\in\mathrm{Ext}^1(M,R)$.

Suppose $e_1\neq 0,e_2\neq 0$, is it possible that $e_2\cup e_1=0\in \mathrm{Ext}^2(M,N)$?

(On the one hand, it is plausible to have zero-divisor for cup product, on the other hand, not sure why two complicated extension may cancel to a trivial extension by splicing.)

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    What about $0\rightarrow\mathbb{Z}\overset{\times 4}\rightarrow\mathbb{Z}\rightarrow\mathbb{Z}/4\mathbb{Z}\rightarrow 0$ and $0\rightarrow\mathbb{Z}/4\mathbb{Z}\rightarrow\mathbb{Z}/8\mathbb{Z}\rightarrow\mathbb{Z}/2\mathbb{Z}\rightarrow 0$ ? Those are non trivial extensions of abelian groups, but the product in $Ext^2$ is zero since $Ext^2=0$ in $\mathbf{Ab}$. – Roland Apr 16 '17 at 16:30
  • Yes, thank you! –  Apr 16 '17 at 16:32

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