Fix error in Lemma 0ANU

Thanks to Brian Conrad for pointing this out
http://stacks.math.columbia.edu/tag/0ANU#comment-1973

The mistake was using Lemma 07RD without checking the finiteness
assumption in that lemma. Fixed by replacing property (b) by a stronger
condition where we ask for a confinal system of weak ideals of
definition with the desired property.

Also: added a new lemma which explains that if we work with admissible
topological rings and ideals of definition (in stead of weakly
admissible topological rings and weak ideals of definition), then the
lemma is true as it was formulated before this change...

restricted.tex

@@ -287,13 +287,14 @@ \section{Algebras topologically of finite type}
 \item[(a)] $B \to A$ 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 $JA$,
-\item[(b)] $B \to A$ is taut and $B/J \to A/I$ is of finite type for
-some weak ideal of definition $J \subset B$ with $I \subset A$ the
-closure of $JA$,
+\item[(b)] $B \to A$ 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 $J_\lambda A$,
 \item[(c)] $B \to A$ is taut and $A$ is topologically of finite
 type over $B$,
-\item[(d)] $A$ is isomorphic to a quotient of $B\{x_1, \ldots, x_n\}$
-by a closed ideal.
+\item[(d)] $A$ is isomorphic as a topological $B$-algebra to a quotient of
+$B\{x_1, \ldots, x_n\}$ by a closed ideal.
 \end{enumerate}
 Moreover, these equivalent conditions define a local property,
 i.e., they satisfy Formal Spaces, Axioms (\ref{formal-spaces-item-axiom-1}),
@@ -314,25 +315,28 @@ \section{Algebras topologically of finite type}
 $$
 such that $A_{n + 1}/J_nA_{n + 1} = A_n$ and such that $A = \lim A_n$ as in
 Formal Spaces, Lemma \ref{formal-spaces-lemma-representable-property-rings}.
-We may assume $J = J_1$ by replacing $J_1$ by $J_1 + J$ if necessary.
+For every $m$ there exists a $\lambda$ such that $J_\lambda \subset J_m$.
+Since $B/J_\lambda \to A/I_\lambda$ is of finite type, this implies
+that $B/J_m \to A/I_m$ is of finite type.
 Let $\alpha_1, \ldots, \alpha_n \in A_1$ be generators of $A_1$ over
-$B/J_1 = B/J$. Since $A$ is a countable limit of a system with surjective
+$B/J_1$. Since $A$ is a countable limit of a system with surjective
 transition maps, we can find $a_1, \ldots, a_n \in A$ mapping to
 $\alpha_1, \ldots, \alpha_n$ in $A_1$. By
 Remark \ref{remark-universal-property} we find a continuous map
 $B\{x_1, \ldots, x_n\} \to A$ mapping $x_i$ to $a_i$. This map
-induces surjections $B/J_m[x_1, \ldots, x_n] \to A_m$ by
+induces surjections $(B/J_m)[x_1, \ldots, x_n] \to A_m$ by
 Algebra, Lemma \ref{algebra-lemma-surjective-mod-locally-nilpotent}.
 For $m \geq 1$ we obtain a short exact sequence
 $$
-0 \to K_m \to B/J_m[x_1, \ldots, x_n] \to A_m \to 0
+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 \ref{algebra-lemma-ML-exact-sequence}.
-Since $B\{x_1, \ldots, x_n\} = \lim B/J_m[x_1, \ldots, x_n]$ and $A = \lim A_m$
-we conclude that $B\{x_1, \ldots, x_n\} \to A$ is surjective.
+Since $B\{x_1, \ldots, x_n\} = \lim (B/J_m)[x_1, \ldots, x_n]$
+and $A = \lim A_m$
+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.
 
@@ -346,7 +350,7 @@ \section{Algebras topologically of finite type}
 Formal Spaces, Lemma \ref{formal-spaces-lemma-representable-property-rings}
 in order to prove Formal Spaces, Axioms (\ref{formal-spaces-item-axiom-1})
 and (\ref{formal-spaces-item-axiom-2})
-we may assume both $A \to B$ and $(B')^\wedge \to (A')^\wedge$
+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 $J(B')^\wedge$, $JA$, $J(A')^\wedge$.
@@ -377,6 +381,46 @@ \section{Algebras topologically of finite type}
 We omit the proof of Formal Spaces, Axiom (\ref{formal-spaces-item-axiom-3}).
 \end{proof}
 
+\begin{lemma}
+\label{lemma-quotient-restricted-power-series-admissible}
+In Lemma \ref{lemma-quotient-restricted-power-series}
+if $B$ is admissible (for example adic), then the equivalent conditions
+(a) -- (d) are also equivalent to
+\begin{enumerate}
+\item[(e)] $B \to A$ is taut and $B/J \to A/I$ is of finite type for
+some ideal of definition $J \subset B$ where $I \subset A$ is
+the closure of $JA$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+It is enough to show that (e) implies (a). Let $J' \subset B$ be a weak ideal
+of definition and let $I' \subset A$ be the closure of $J'A$. We have
+to show that $B/J' \to A/I'$ is of finite type. If the corresponding statement
+holds for the smaller weak ideal of definition $J'' = J' \cap J$, then it
+holds for $J'$. Thus we may assume $J' \subset J$. As $J$ is an ideal
+of definition (and not just a weak ideal of definition), we get
+$J^n \subset J'$ for some $n \geq 1$. Thus we can consider the
+diagram
+$$
+\xymatrix{
+0 \ar[r] & I/I' \ar[r] & A/I' \ar[r] & A/I \ar[r] & 0 \\
+0 \ar[r] & J/J' \ar[r] \ar[u] & B/J' \ar[r] \ar[u] & B/J \ar[r] \ar[u] & 0
+}
+$$
+with exact rows. Since $I' \subset A$ is open and since
+$I$ is the closure of $J A$ we see that $I/I' = (J/J') \cdot A/I'$.
+By assumption we can find a surjection $(B/J)[x_1, \ldots, x_n] \to A/I$.
+We can lift this to $(B/J')[x_1, \ldots, x_n] \to A'/I'$.
+Because $J/J'$ is a nilpotent ideal, we may apply part (11) of
+Algebra, Lemma \ref{algebra-lemma-NAK} to the map of $B/J'$-modules
+$$
+(B/J')[x_1, \ldots, x_n] \to A'/I'
+$$
+to see that it is surjective. Thus $A/I'$ is of finite type over $B/J'$
+as desired.
+\end{proof}
+
 \begin{lemma}
 \label{lemma-representable-affine-finite-type}
 Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of

tags/tags

@@ -15454,3 +15454,4 @@
 0CB3,more-morphisms-lemma-nr-minimal-primes
 0CB4,more-morphisms-lemma-nr-branches
 0CB5,more-morphisms-lemma-nr-branches-fibre
+0CB6,restricted-lemma-quotient-restricted-power-series-admissible