Definition 84.17.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. Assume $X$ and $Y$ are locally adic*. We say $f$ is an *adic morphism* if $f$ is representable by algebraic spaces. See discussion above.

## 84.17 Adic morphisms

Suppose that $\varphi : A \to B$ is a continuous map between adic topological rings. One says $\varphi $ is *adic* if there exists an ideal of definition $I \subset A$ such that the topology on $B$ is $I$-adic. However, this is not a good notion unless we assume $A$ has a finitely generated ideal of definition. In this case, the condition is equivalent to $\varphi $ being taut, see Lemma 84.4.15.

Let $P$ be the property of morphisms $\varphi : A \to B$ of $\textit{WAdm}^{adic*}$ defined by

(see above for the equivalence). Since $\textit{WAdm}^{adic*}$ is a full subcategory of $\textit{WAdm}^{count}$ it follows trivially from Lemma 84.16.6 that $P$ is a local property on morphisms of $\textit{WAdm}^{adic*}$, see Remark 84.16.4. Combining Lemmas 84.16.3 and 84.16.7 we obtain the result stated in the next paragraph.

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally adic* formal algebraic spaces over $S$. Then the following are equivalent

$f$ is representable by algebraic spaces (in other words, the equivalent conditions of Lemma 84.14.4 hold),

for every commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to an adic map in $\textit{WAdm}^{adic*}$ (in other words, the equivalent conditions of Lemma 84.16.3 hold with $P$ as above).

In this situation we will sometimes say that *$f$ is an adic morphism*. Here it is understood that this notion is only defined for morphisms between formal algebraic spaces which are locally adic*.

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