The Stacks project

Situation 87.21.2. Let $P$ be a property of morphisms of $\textit{WAdm}^{count}$. Consider commutative diagrams
\begin{equation} \label{formal-spaces-equation-localize} \vcenter { \xymatrix{ A \ar[r] & (A')^\wedge \\ B \ar[r] \ar[u]^\varphi & (B')^\wedge \ar[u]_{\varphi '} } } \end{equation}

satisfying the following conditions

  1. $A$ and $B$ are objects of $\textit{WAdm}^{count}$,

  2. $A \to A'$ and $B \to B'$ are ├ętale ring maps,

  3. $(A')^\wedge = \mathop{\mathrm{lim}}\nolimits A'/IA'$, resp. $(B')^\wedge = \mathop{\mathrm{lim}}\nolimits B'/JB'$ where $I \subset A$, resp. $J \subset B$ runs through the weakly admissible ideals of definition of $A$, resp. $B$,

  4. $\varphi : B \to A$ and $\varphi ' : (B')^\wedge \to (A')^\wedge $ are continuous.

By Lemma 87.21.1 the topological rings $(A')^\wedge $ and $(B')^\wedge $ are objects of $\textit{WAdm}^{count}$. We say $P$ is a local property if the following axioms hold:

  1. for any diagram ( we have $P(\varphi ) \Rightarrow P(\varphi ')$,

  2. for any diagram ( with $A \to A'$ faithfully flat we have $P(\varphi ') \Rightarrow P(\varphi )$,

  3. if $P(B \to A_ i)$ for $i = 1, \ldots , n$, then $P(B \to \prod _{i = 1, \ldots , n} A_ i)$.

Axiom (3) makes sense as $\textit{WAdm}^{count}$ has finite products.

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