The Stacks project

85.19 Adic morphisms

Let us make the following definition.

Definition 85.19.1. Let $A$ and $B$ be pre-adic topological rings. A ring homomorphism $\varphi : A \to B$ is adic1 if there exists an ideal of definition $I \subset A$ such that the topology on $B$ is the $I$-adic topology.

The reader easily shows that if $\varphi : A \to B$ is an adic homomorphism of pre-adic rings, then $\varphi $ is continuous and the topology on $B$ is the $I$-adic topology for every ideal of definition $I$ of $A$.

Lemma 85.19.2. 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) follows immediately from the definitions, please see Definition 85.4.11 for the definition of a taut ring maps. Conversely, assume $B$ is complete, $I \subset A$ is a finitely generated ideal of definition, and $\varphi $ is taut. Then Lemma 85.4.15 tells us the topology on $B$ is the $I$-adic topology as desired. This proves (2). The final statement is a trivial 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 85.18.2 the following are equivalent

  1. $f$ is representable by algebraic spaces (in other words, the equivalent conditions of Lemma 85.15.4 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 adic2 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 85.19.3. 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] This may be nonstandard terminology.
[2] Equivalently taut by Lemma 85.19.2.

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