The Stacks project

Remark 87.28.2. Let $A$ be a ring and let $I \subset A$ be an ideal. If $A$ is $I$-adically complete, then the $I$-adic completion $A[x_1, \ldots , x_ r]^\wedge $ of $A[x_1, \ldots , x_ r]$ is the restricted power series ring over $A$ as a ring. However, it is not clear that $A[x_1, \ldots , x_ r]^\wedge $ is $I$-adically complete. We think of the topology on $A\{ x_1, \ldots , x_ r\} $ as the limit topology (which is always complete) whereas we often think of the topology on $A[x_1, \ldots , x_ r]^\wedge $ as the $I$-adic topology (not always complete). If $I$ is finitely generated, then $A\{ x_1, \ldots , x_ r\} = A[x_1, \ldots , x_ r]^\wedge $ as topological rings, see Algebra, Lemma 10.96.3.