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 $\textit{WAdm}^{Noeth}$.

## Comments (0)