Lemma 88.11.1. Let $A$ and $B$ be adic topological rings which have a finitely generated ideal of definition. Let $\varphi : A \to B$ be a continuous ring homomorphism. The following are equivalent:

1. $\varphi$ is adic and $B$ is topologically of finite type over $A$,

2. $\varphi$ is taut and $B$ is topologically of finite type over $A$,

3. there exists an ideal of definition $I \subset A$ such that the topology on $B$ is the $I$-adic topology and there exist an ideal of definition $I' \subset A$ such that $A/I' \to B/I'B$ is of finite type,

4. for all ideals of definition $I \subset A$ the topology on $B$ is the $I$-adic topology and $A/I \to B/IB$ is of finite type,

5. there exists an ideal of definition $I \subset A$ such that the topology on $B$ is the $I$-adic topology and $B$ is in the category (88.2.0.2),

6. for all ideals of definition $I \subset A$ the topology on $B$ is the $I$-adic topology and $B$ is in the category (88.2.0.2),

7. $B$ as a topological $A$-algebra is the quotient of $A\{ x_1, \ldots , x_ r\}$ by a closed ideal,

8. $B$ as a topological $A$-algebra is the quotient of $A[x_1, \ldots , x_ r]^\wedge$ by a closed ideal where $A[x_1, \ldots , x_ r]^\wedge$ is the completion of $A[x_1, \ldots , x_ r]$ with respect to some ideal of definition of $A$, and

Moreover, these equivalent conditions define a local property of morphisms of $\text{WAdm}^{adic*}$ as defined in Formal Spaces, Remark 87.21.4.

Proof. Taut ring homomorphisms are defined in Formal Spaces, Definition 87.5.1. Adic ring homomorphisms are defined in Formal Spaces, Definition 87.6.1. The lemma follows from a combination of Formal Spaces, Lemmas 87.29.6, 87.29.7, and 87.23.1. We omit the details. To be sure, there is no difference between the topological rings $A[x_1, \ldots , x_ n]^\wedge$ and $A\{ x_1, \ldots , x_ r\}$, see Formal Spaces, Remark 87.28.2. $\square$

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).