= Vanishing cup product from an open cover
{title2=$\alpha_i|_{U_i}=0\ \Longrightarrow\ \prod_i\alpha_i=0$}
Let $U_1,\ldots,U_k$ be an open cover of $Y$, and suppose $\alpha_i\in H^{d_i}(Y;R)$ restricts to zero on $U_i$. The pair <long exact sequence> lifts each $\alpha_i$ to $H^{d_i}(Y,U_i;R)$. Their <relative cup product> lies in $H^{\sum d_i}(Y,\bigcup_iU_i;R)=H^{\sum d_i}(Y,Y;R)=0$, so $\alpha_1\smile\cdots\smile\alpha_k=0$. This proves characteristic-class products vanish from local trivializations without cancelling possible zero divisors in the coefficient ring.
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