Remark 86.28.2. 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.

Comment #5913 by Zhenhua Wu on

Can you clarify the definition of completeness of limit topology?

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