Exercise 21, § 4

Let $(A_\alpha, \phi_{\beta\alpha})$ be a direct system of rings whose indexing set is directed, $A = \varprojlim A_\alpha$, and $\phi_\alpha : A_\alpha \to A$ the canonical homomorphism. If we write $X_\alpha = \operatorname{Spec}(A_\alpha)$, $(X_\alpha, {}^a\phi_{\beta\alpha})$ is an inverse system of topological spaces and, if $X = \operatorname{Spec}(A)$, the ${}^a\phi_\alpha : X \to X_\alpha$, form an inverse system of continuous mappings. Show that $u = \varprojlim {}^a\phi_\alpha$ is a homeomorphism of $X$ onto $\varprojlim X_\alpha$. (Prove first that, if, for all $a$, $p_\alpha$ is a prime ideal of $A_\alpha$, such that $p_\alpha = \phi_{\beta\alpha}^{-1}(p_\beta)$ for all $a \leq \beta$, the union $p$ of the $\phi_\alpha(p_\alpha)$ is a prime ideal of $A$; conversely, every prime ideal $p$ of $A$ is the union of the $\phi_\alpha(p_\alpha)$, where $p_\alpha = \phi_\alpha(p)$. Finally, if $f_\alpha \in A_\alpha$ and $f = \phi_\alpha(f_\alpha)$, note that the relation $p \in X$, is equivalent to