Exercise 16, § 4

(a) Let $A$ be an absolutely flat (commutative) ring (Chapter I, § 2, Exercise 17). Show that $X = \operatorname{Spec}(A)$ is a totally disconnected compact space, that every prime ideal $q$ of $A$ is maximal and that $A_q$ is canonically isomorphic to the field $A/p$ (show first that, for all $f \in A$, $V(f)$ is both open and closed in $X$; on the other hand use § 3, Exercise 9).

(b) Show that the mapping $a \mapsto V(a)$ is a bijection of the set of ideals of $A$ onto the set of closed subsets of $X$. Moreover, the following conditions are equivalent: ($a$) $V(a)$ is open; ($\beta$) $a$ is finitely generated; (y) $a$ is generated by an idempotent; ($\delta$) $A/a$ is a projective $A$-module.

(c) Let $\tilde{A}$ be the structure sheaf of rings over $X$. Show that every $\tilde{A}$-Module $\mathscr{F}$ is of the form $\tilde{E}$, where $E$ is an $A$-module defined up to a unique isomorphism (take $E = \Gamma(X, \mathscr{F})$) and use the fact that every point of $X$ admits a fundamental system of neighbourhoods which are both open and closed). For every ideal $a$ of $A$, show that $\operatorname{Hom}_A(a, E)$ is identified with $\Gamma(X - V(a), \tilde{E})$ (note that $X - V(a)$ is the union of the $X_e$, where $e$ runs through the set of idempotents $e \in a$). Deduce that, for $E$ to be an injective $A$-module, it is necessary and sufficient that $\tilde{E}$ be flabby.

(d) Let $A$ be a ring and $\mathfrak{N}$ its nilradical. The following properties are equivalent:

(a) $A/\mathfrak{N}$ is absolutely flat. ($\beta$) $X = \operatorname{Spec}(A)$ is Hausdorff. (y) Every point of $X$ is closed (in other words, every prime ideal of $A$ is maximal). (Use Exercises 9 of §3.)