Lemma 87.29.6. Let $B \to A$ be an arrow of $\textit{WAdm}^{count}$, see Section 87.21. The following are equivalent

$B \to A$ is taut and $B/J \to A/I$ is of finite type for every weak ideal of definition $J \subset B$ where $I \subset A$ is the closure of $JA$,

$B \to A$ is taut and $B/J_\lambda \to A/I_\lambda $ is of finite type for a cofinal system $(J_\lambda )$ of weak ideals of definition of $B$ where $I_\lambda \subset A$ is the closure of $J_\lambda A$,

$B \to A$ is taut and $A$ is topologically of finite type over $B$,

$A$ is isomorphic as a topological $B$-algebra to a quotient of $B\{ x_1, \ldots , x_ n\} $ by a closed ideal.

Moreover, these equivalent conditions define a local property, i.e., they satisfy Axioms (1), (2), (3).

**Proof.**
The implications (a) $\Rightarrow $ (b), (c) $\Rightarrow $ (a), (d) $\Rightarrow $ (c) are straightforward from the definitions. Assume (b) holds and let $J \subset B$ and $I \subset A$ be as in (a). Choose a commutative diagram

\[ \xymatrix{ A \ar[r] & \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 \\ B \ar[r] \ar[u] & \ldots \ar[r] & B/J_3 \ar[r] \ar[u] & B/J_2 \ar[r] \ar[u] & B/J_1 \ar[u] } \]

such that $A_{n + 1}/J_ nA_{n + 1} = A_ n$ and such that $A = \mathop{\mathrm{lim}}\nolimits A_ n$ as in Lemma 87.22.1. For every $m$ there exists a $\lambda $ such that $J_\lambda \subset J_ m$. Since $B/J_\lambda \to A/I_\lambda $ is of finite type, this implies that $B/J_ m \to A/I_ m$ is of finite type. Let $\alpha _1, \ldots , \alpha _ n \in A_1$ be generators of $A_1$ over $B/J_1$. Since $A$ is a countable limit of a system with surjective transition maps, we can find $a_1, \ldots , a_ n \in A$ mapping to $\alpha _1, \ldots , \alpha _ n$ in $A_1$. By Remark 87.28.1 we find a continuous map $B\{ x_1, \ldots , x_ n\} \to A$ mapping $x_ i$ to $a_ i$. This map induces surjections $(B/J_ m)[x_1, \ldots , x_ n] \to A_ m$ by Algebra, Lemma 10.126.9. For $m \geq 1$ we obtain a short exact sequence

\[ 0 \to K_ m \to (B/J_ m)[x_1, \ldots , x_ n] \to A_ m \to 0 \]

The induced transition maps $K_{m + 1} \to K_ m$ are surjective because $A_{m + 1}/J_ mA_{m + 1} = A_ m$. Hence the inverse limit of these short exact sequences is exact, see Algebra, Lemma 10.86.4. Since $B\{ x_1, \ldots , x_ n\} = \mathop{\mathrm{lim}}\nolimits (B/J_ m)[x_1, \ldots , x_ n]$ and $A = \mathop{\mathrm{lim}}\nolimits A_ m$ we conclude that $B\{ x_1, \ldots , x_ n\} \to A$ is surjective and open. As $A$ is complete the kernel is a closed ideal. In this way we see that (a), (b), (c), and (d) are equivalent.

Let a diagram (87.21.2.1) as in Situation 87.21.2 be given. By Example 87.24.7 the maps $A \to (A')^\wedge $ and $B \to (B')^\wedge $ satisfy (a), (b), (c), and (d). Moreover, by Lemma 87.22.1 in order to prove Axioms (1) and (2) we may assume both $B \to A$ and $(B')^\wedge \to (A')^\wedge $ are taut. Now pick a weak ideal of definition $J \subset B$. Let $J' \subset (B')^\wedge $, $I \subset A$, $I' \subset (A')^\wedge $ be the closure of $J(B')^\wedge $, $JA$, $J(A')^\wedge $. By what was said above, it suffices to consider the commutative diagram

\[ \xymatrix{ A/I \ar[r] & (A')^\wedge /I' \\ B/J \ar[r] \ar[u]^{\overline{\varphi }} & (B')^\wedge /J' \ar[u]_{\overline{\varphi }'} } \]

and to show (1) $\overline{\varphi }$ finite type $\Rightarrow \overline{\varphi }'$ finite type, and (2) if $A \to A'$ is faithfully flat, then $\overline{\varphi }'$ finite type $\Rightarrow \overline{\varphi }$ finite type. Note that $(B')^\wedge /J' = B'/JB'$ and $(A')^\wedge /I' = A'/IA'$ by the construction of the topologies on $(B')^\wedge $ and $(A')^\wedge $. In particular the horizontal maps in the diagram are étale. Part (1) now follows from Algebra, Lemma 10.6.2 and part (2) from Descent, Lemma 35.14.2 as the ring map $A/I \to (A')^\wedge /I' = A'/IA'$ is faithfully flat and étale.

We omit the proof of Axiom (3).
$\square$

## Comments (4)

Comment #1973 by Brian Conrad on

Comment #1977 by Johan on

Comment #2012 by Johan on

Comment #2024 by Johan on