The Stacks project

85.17 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 85.4.15.

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

\[ P(\varphi )=\text{``}\varphi \text{ is adic''}=\text{``}\varphi \text{ is taut''} \]

(see above for the equivalence). Since $\textit{WAdm}^{adic*}$ is a full subcategory of $\textit{WAdm}^{count}$ it follows trivially from Lemma 85.16.6 that $P$ is a local property on morphisms of $\textit{WAdm}^{adic*}$, see Remark 85.16.4. Combining Lemmas 85.16.3 and 85.16.7 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

  1. $f$ is representable by algebraic spaces (in other words, the equivalent conditions of Lemma 85.14.4 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 ├ętale, the morphism $U \to V$ corresponds to an adic map in $\textit{WAdm}^{adic*}$ (in other words, the equivalent conditions of Lemma 85.16.3 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*.

Definition 85.17.1. 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.

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Your email address will not be published.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'.