Every restriction map of the <structure sheaf of a scheme> is a unital <ring homomorphism>. Thus $p\cdot1=0$ in $\Gamma(X,\mathcal O_X)$ implies the same identity in $\Gamma(U,\mathcal O_X)$ for every open $U$. This proves (ii)$\Rightarrow$(i), so (i) and (ii) are equivalent.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 2 / a / ii / Solution

 Solution (source code)