This section matches the occasionally used notion of an “adic morphism” $f : X \to Y$ of locally adic* formal algebraic spaces $X$ and $Y$ on the one hand with representability of $f$ by algebraic spaces and on the other hand with our notion of taut continuous ring homomorphisms. First we recall that tautness is equivalent to adicness for adic rings with finitely generated ideal of definition.

Lemma 86.23.1. Let $A$ and $B$ be pre-adic topological rings. Let $\varphi : A \to B$ be a continuous ring homomorphism.

1. If $\varphi$ is adic, then $\varphi$ is taut.

2. If $B$ is complete, $A$ has a finitely generated ideal of definition, and $\varphi$ is taut, then $\varphi$ is adic.

In particular the conditions “$\varphi$ is adic” and “$\varphi$ is taut” are equivalent on the category $\textit{WAdm}^{adic*}$.

Proof. Part (1) is Lemma 86.6.4. Part (2) is Lemma 86.6.5. The final statement is a consequence of (1) and (2). $\square$

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally adic* formal algebraic spaces over $S$. By Lemma 86.22.2 the following are equivalent

1. $f$ is representable by algebraic spaces (in other words, the equivalent conditions of Lemma 86.19.4 hold),

2. 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 adic1 map in $\textit{WAdm}^{adic*}$.

In this situation we will say that $f$ is an adic morphism (the formal definition is below). This notion/terminology will only be defined/used for morphisms between formal algebraic spaces which are locally adic* since otherwise we don't have the equivalence between (1) and (2) above.

Definition 86.23.2. 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.

[1] Equivalently taut by Lemma 86.23.1.

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