The Stacks project

Lemma 86.11.1. 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 (86.2.0.2),

  6. for all ideals of definition $I \subset A$ the topology on $B$ is the $I$-adic topology and $B$ is in the category (86.2.0.2),

  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 85.17.4.

Proof. Taut ring homomorphisms are defined in Formal Spaces, Definition 85.4.11. Adic ring homomorphisms are defined in Formal Spaces, Definition 85.19.1. The lemma follows from a combination of Formal Spaces, Lemmas 85.25.6, 85.25.7, and 85.19.2. 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 85.24.2. $\square$


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