Lemma 87.21.1. Let A \in \mathop{\mathrm{Ob}}\nolimits (\textit{WAdm}). Let A \to A' be a ring map (no topology). Let (A')^\wedge = \mathop{\mathrm{lim}}\nolimits _{I \subset A\text{ w.i.d}} A'/IA' be the object of \textit{WAdm} constructed in Example 87.19.11.
If A is in \textit{WAdm}^{count}, so is (A')^\wedge .
If A is in \textit{WAdm}^{cic}, so is (A')^\wedge .
If A is in \textit{WAdm}^{weakly\ adic}, so is (A')^\wedge .
If A is in \textit{WAdm}^{adic*}, so is (A')^\wedge .
If A is in \textit{WAdm}^{Noeth} and A' is Noetherian, then (A')^\wedge is in \textit{WAdm}^{Noeth}.
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