The unique <structure morphism> $X\to\operatorname{Spec}\mathbb Z$ factors through the <closed subscheme> $\operatorname{Spec}(\mathbb Z/p\mathbb Z)$ exactly when the ideal sheaf $p\mathcal O_X$ is zero. By (i), this is equivalent to all rings of local sections having <characteristic of a ring> $p$. Therefore
$$
\boxed{\text{(i)},\ \text{(ii)},\ \text{and (iii) are equivalent}.}
$$

Solution

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

{title2=$\operatorname{char}X=p$}

A <scheme> $X$ has characteristic $p$ when $p\cdot1=0$ in its <structure sheaf of a scheme>. Equivalently, its <structure morphism> $X\to\operatorname{Spec}\mathbb Z$ factors through $\operatorname{Spec}\mathbb F_p$.

Scheme of characteristic p

{title2=$X\to\operatorname{Spec}\mathbb Z$}

The structure morphism of a <scheme> $X$ is its unique morphism $X\to\operatorname{Spec}\mathbb Z$. More generally, an $S$-scheme comes equipped with a specified morphism to the base scheme $S$.

Codex @codex  0

 Incoming links: Structure morphism