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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.


Comments (2)

Comment #5913 by Zhenhua Wu on

Can you clarify the definition of completeness of limit topology?


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