Exercise 1, § 3

(a) Let $K$ be a commutative field of characteristic $p > 0$ and $B$ the ring of formal power series with coefficients in $K$ in two infinite systems of indeterminates $(X_n), (Y_n)$ with only a finite number of terms of given degree ($§ 2$, Exercise 12); $B$ is a Hausdorff local ring with the $n$-adic topology, $m$ being its maximal ideal. Let $b$ be the closed ideal of $B$ generated by the monomials $Y_i X_j$ for $i \geq 0, 0 \leq j \leq i$; let $A$ be the local ring $B/b$, which is Hausdorff with the $n$-adic topology, where $n = m/b$ is its maximal ideal. Let $c$ be the element of $A$ equal to the class mod. $b$ of $\sum_{n=1}^\infty X_n^{p^n}$; show that, for $k \geq 3$, $n^k n (Ac) \not\subset n^2 c$ (and a fortiori the relation

cannot hold).

(b) Let $C'$ be the commutative ring with underlying set $A \times A$, with multiplication $(a, x)(a', x') = (aa', ax' + a'x)$; let $C$ be the subring $A \times (Ac)$ of $C'$; $C$ and $C'$ are local rings, whose maximal ideals we denote by $r$ and $r'$, and $C'$ is a finitely generated $C$-module. Show that $r = r' n C$ but that the topology induced on $C$ by the $r'$-adic topology is not the $r$-adic topology.