The Stacks project

87.23 Adic morphisms

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 87.23.1. 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 87.6.4. Part (2) is Lemma 87.6.5. 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 87.22.2 the following are equivalent

  1. $f$ is representable by algebraic spaces (in other words, the equivalent conditions of Lemma 87.19.4 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 87.23.2. 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 87.23.1.

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