The Stacks project

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

9. add more here.

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