Exercise 17, § 3

(a) Let $K$ be a commutative field, $A = K[[X_1, \ldots, X_n]]$ the ring of formal power series in $n$ indeterminates with coefficients in $K$ and $V$ a vector space over $K$. With the notation of § 2, no. 6, Example 1, let $V[[X_1, \ldots, X_n]]$ denote the vector space $V^{n^n}$ with the $A$-module structure defined by

where $w_\alpha = \sum_{\beta + \gamma = \alpha} c_\beta v_\gamma$. Show that, if $V$ admits a basis with $I$ as indexing set, the $A$-module $V[[X_1, \ldots, X_n]]$ is isomorphic to $A'$ if $I$ is finite and to the $A$-module $A$, defined in Exercise 16 if $I$ is infinite. Deduce that $V[[X_1, \ldots, X_n]]$ is a flat $A$-module and is faithfully flat if $V$ is not reduced to 0.

(b) Let $L$ be an extension of the field $K$. Deduce from (a) that the ring of formal power series $L[[X_1, \ldots, X_n]]$ is a faithfully flat module over the ring $K[[X_1, \ldots, X_n]]$. If $L$ has finite rank over $K$, $L[[X_1, \ldots, X_n]]$ is isomorphic to $L \otimes_K K[[X_1, \ldots, X_n]]$.

(c) If $L$ is an algebraic extension of $K$, show that the ring $L[[X_1, \ldots, X_n]]$ is a faithfully flat module over the ring

(consider $L$ as the direct limit of its sub-extensions of finite rank over $K$ and use Chapter I, § 2, no. 7, Proposition 9). Deduce that the ring

is a Noetherian local ring whose completion is identified with $L[[X_1, \ldots, X_n]]$. In order that $\hat{B} = B$, it is necessary and sufficient that $n = 0$ or $[L : K] < +\infty$.