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.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AQ2. Beware of the difference between the letter 'O' and the digit '0'.