Exercise 17, § 4

\* (a) Let X be a totally disconnected compact space and $\mathscr{O}$ a sheaf of rings over X such that, for all $x \in X, \mathscr{O}$, is a commutative field. Show that, if $A = \Gamma(X, \mathscr{O})$, the ring A is absolutely flat and its spectrum (with the sheaf of rings $\tilde{A}$) is canonically identified with X (with the sheaf $\mathscr{O}$). (Observe that for all $f \in A$ the set of $x \in X$ such that $f_x = 0$ is both open and closed in X and deduce the existence of $g \in A$ such that $f = gf^2$.) \*

(b) Let X be a totally disconnected compact space, K a (commutative) field and A the ring of locally constant mappings from X to K. Show that A is absolutely flat (argue directly or use (a)).

(c) Let A be the ring of real-valued step functions defined on $I = [0, 1[$, whose points of discontinuity are of the form $k/2^n$ ($n \in \mathbf{N},\ 0 \leq k < 2^n$) and which are right continuous. Let $\mathfrak{g}$ be the set of finite unions of half-open intervals $[x, y[ \subset I$ whose extremities are of the form $k/2^n$. For every $f \in A$, the set $Z(f) = \overline{f(0)}$ belongs to $\mathbf{3}$. If $a$ is an ideal of A, the set 23 of $Z(f)$, where $f$ runs through $a$, is such that: (1) the intersection of two sets of 23 belongs to $\mathfrak{B}$; (2) every set of $\mathfrak{g}$ containing a set of 23 belongs to $\mathfrak{B}$. Obtain the converse. For $a$ to be prime, it is necessary and sufficient that 23 be the base of a filter which converges to a point of $[0, 1]$. Show that A is an absolutely flat reduced ring and that, for every maximal ideal m of A, $A_m$ is isomorphic to the field $\mathbf{R}$. Describe the space $\mathrm{Spec}(A)$.