Lemma 85.4.16. Let $B \to A$ and $B \to C$ be continuous homomorphisms of linearly topologized rings.

1. If $A$ and $C$ are weakly pre-admissible, then $A \widehat{\otimes }_ B C$ is weakly admissible.

2. If $A$ and $C$ are pre-admissible, then $A \widehat{\otimes }_ B C$ is admissible.

3. If $A$ and $C$ have a countable fundamental system of open ideals, then $A \widehat{\otimes }_ B C$ has a countable fundamental system of open ideals.

4. If $A$ and $C$ are pre-adic and have finitely generated ideals of definition, then $A \widehat{\otimes }_ B C$ is adic and has a finitely generated ideal of definition.

5. If $A$ and $C$ are pre-adic Noetherian rings and $B/\mathfrak b \to A/\mathfrak a$ is of finite type where $\mathfrak a \subset A$ and $\mathfrak b \subset B$ are the ideals of topologically nilpotent elements, then $A \widehat{\otimes }_ B C$ is adic Noetherian.

Proof. Let $I_\lambda \subset A$, $\lambda \in \Lambda$ and $J_\mu \subset C$, $\mu \in M$ be fundamental systems of open ideals, then by definition

$A \widehat{\otimes }_ B C = \mathop{\mathrm{lim}}\nolimits _{\lambda , \mu } A/I_\lambda \otimes _ B C/J_\mu$

with the limit topology. Thus a fundamental system of open ideals is given by the kernels $K_{\lambda , \mu }$ of the maps $A \widehat{\otimes }_ B C \to A/I_\lambda \otimes _ B C/J_\mu$. Note that $K_{\lambda , \mu }$ is the closure of the ideal $I_\lambda (A \widehat{\otimes }_ B C) + J_\mu (A \widehat{\otimes }_ B C)$. Finally, we have a ring homomorphism $\tau : A \otimes _ B C \to A \widehat{\otimes }_ B C$ with dense image.

Proof of (1). If $I_\lambda$ and $J_\mu$ consist of topologically nilpotent elements, then so does $K_{\lambda , \mu }$ by Lemma 85.4.10. Hence $A \widehat{\otimes }_ B C$ is weakly admissible by definition.

Proof of (2). Assume for some $\lambda _0$ and $\mu _0$ the ideals $I = I_{\lambda _0} \subset A$ and $J_{\mu _0} \subset C$ are ideals of definition. Thus for every $\lambda$ there exists an $n$ such that $I^ n \subset I_\lambda$. For every $\mu$ there exists an $m$ such that $J^ m \subset J_\mu$. Then

$\left(I(A \widehat{\otimes }_ B C) + J(A \widehat{\otimes }_ B C)\right)^{n + m} \subset I_\lambda (A \widehat{\otimes }_ B C) + J_\mu (A \widehat{\otimes }_ B C)$

It follows that the open ideal $K = K_{\lambda _0, \mu _0}$ satisfies $K^{n + m} \subset K_{\lambda , \mu }$. Hence $K$ is an ideal of definition of $A \widehat{\otimes }_ B C$ and $A \widehat{\otimes }_ B C$ is admissible by definition.

Proof of (3). If $\Lambda$ and $M$ are countable, so is $\Lambda \times M$.

Proof of (4). Assume $\Lambda = \mathbf{N}$ and $M = \mathbf{N}$ and we have finitely generated ideals $I \subset A$ and $J \subset C$ such that $I_ n = I^ n$ and $J_ n = J^ n$. Then

$I(A \widehat{\otimes }_ B C) + J(A \widehat{\otimes }_ B C)$

is a finitely generated ideal and it is easily seen that $A \widehat{\otimes }_ B C$ is the completion of $A \otimes _ B C$ with respect to this ideal. Hence (4) follows from Algebra, Lemma 10.96.3.

Proof of (5). Let $\mathfrak c \subset C$ be the ideal of topologically nilpotent elements. Since $A$ and $C$ are adic Noetherian, we see that $\mathfrak a$ and $\mathfrak c$ are ideals of definition (details omitted). From part (4) we already know that $A \widehat{\otimes }_ B C$ is adic and that $\mathfrak a(A \widehat{\otimes }_ B C) + \mathfrak c(A \widehat{\otimes }_ B C)$ is a finitely generated ideal of definition. Since

$A \widehat{\otimes }_ B C / \left(\mathfrak a(A \widehat{\otimes }_ B C) + \mathfrak c(A \widehat{\otimes }_ B C)\right) = A/\mathfrak a \otimes _{B/\mathfrak b} C/\mathfrak c$

is Noetherian as a finite type algebra over the Noetherian ring $C/\mathfrak c$ we conclude by Algebra, Lemma 10.97.5. $\square$

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