Exercise 14, § 3

Let $K$ be a commutative field, $C$ the ring $K[X, Y]$, $p$ the maximal ideal $CX + CY$ of $C$, $A$ the local ring $C_p$, $m$ its maximal ideal and $\hat{A}$ the completion of $A$, which is identified with the ring of formal power series $K[[X, Y]]$ (no. 4, Proposition 8). Let $B$ be the subring of $\hat{A}$ consisting of the formal power series of the type $Xf(X, Y) + (P(Y)/Q(Y))$, where $f \in \hat{A}$ and $P$ and $Q$ are two polynomials of $K[Y]$ such that $Q(0) \neq 0$.

(a) Show that $A \subset B$, that $B$ is a local ring whose maximal ideal $n$ is equal to $Bm$ and that $n^k = B \cap m^k$ for every integer $k \geq 1$; $B$ is therefore everywhere dense in $\hat{A}$ and the topology induced on $B$ by the $m$-adic topology on $\hat{A}$ is the $n$-adic topology.

(b) Show that in $B$ the ideal $b$ generated by all the elements of the form $Xf(Y)$, where $f(Y) \in K[[Y]]$, is not finitely generated (cf. Algebra, Chapter V, § 5, Exercise 13) and therefore $B$ is not Noetherian, although $B/n$ and $\mathrm{gr}(B) = \mathrm{gr}(\hat{A})$ are Noetherian and the ideal $n$ is finitely generated; $b = B \cap \hat{A}X$ and $b$ is the closure in $B$ of the principal ideal (not closed) $BX$; finally $B$ is not a flat $A$-module and $\hat{B} = \hat{A}$ is not a flat $B$-module (use Chapter I, § 3, no. 5, Proposition 9).

(c) Let $f(Y)$ be an invertible formal power series in $K[[Y]]$, which is not an element of $K(Y)$. If $c$ is the ideal of $B$ generated by $X$ and $Xf(Y)$, show that on $c$ the $n$-adic topology is strictly finer than the topology induced on $c$ by the $n$-adic topology on $B$. Show that the canonical mapping $\hat{B} \otimes_B c \to \hat{c}$ (where $\hat{c}$ is the completion of $c$ with respect to the $n$-adic topology) is not injective (consider the images of $f(Y) \otimes X$ and $1 \otimes Xf(Y)$; to show that these two elements are distinct in $\hat{B} \otimes_B c$, consider this tensor product as a quotient of the tensor product $\hat{B} \otimes_{K(Y)} \hat{c}$).

(d) Let $f_1(Y), f_2(Y)$ be two invertible formal power series in $K[[Y]]$ such that $1, f_1$ and $f_2$ are linearly independent over $K(Y)$. Let $c_1, c_2$ be the principal ideals of $B$ generated respectively by $Xf_1(Y)$ and $Xf_2(Y)$; the $n$-adic topologies on $c_1$ and $c_2$ are respectively identical with the topologies induced by that on $B$ and the closures of these ideals in $\hat{A} = \hat{B}$ are both identical with the principal ideal $\hat{A}X$ of $\hat{A}$. Deduce that the closure of $c_1 \cap c_2$ in $\hat{A}$ is not equal to $\hat{c}_1 \cap \hat{c}_2$ and that the closure of $c_1 : c_2$ is not equal to $\hat{c}_1 : \hat{c}_2$.