Rings of formal and restricted power series. Preparation theorems. Hensel lemma

Abstract

Let A[[Xl,..., X n ]] be the ring of formal power series in n indeterminates over the ring A; the elements of A[[X1,..., X n ]] are the series f = f 0 + f1+ ··· + f n + ··· where each f i is a homogeneous polynomial of degree i in X1,..., X n . The ring A[[X1,..., X n ]] is complete for the (X1,..., X n )-topology. In fact, if (g n )(g n ∈ A [X1,..., X n ]) is a Cauchy sequence, the series g0 + (g1 - g0) + ···+ (g n +1 - g n ) + ··· ∈ A[[X1,..., X n ]] is the limit of (g n ); furthermore A[[X1,..., X n ]] is Hausdorff, as \( \cap {({X_1},...,{X_n})^m} = 0 \). The (Xl,..., X n )-adic topology of \( \mathop A\limits^m \left[ {\left[ {{X_{\text{l}}},...,{X_n}} \right]} \right] \) induces the (Xl,..., X n )-topology on the polynomial ring A[X1,..., X n ] which is a subring of A[[X1,..., X n ]]; every series may be approximated by polynomials, so the ring A[Xl,..., X n ] is dense in A[[Xl,..., X n ]]. Thus we have the following proposition.