Another characterization of rings of weak dimension <= 1

Namely, that the local rings are valuation rings

more-algebra.tex

@@ -3263,7 +3263,7 @@ \section{Content ideals}
 $u(x) \not \in I'N$. Since $I/I'$ is a nonzero finite $A$-module
 (Lemma \ref{lemma-content-finitely-generated}) there is a nonzero map
 $\chi : I/I' \to A/\mathfrak m$ of $A$-modules
-by Nakayama's lemma (Algebra, Lemma \ref{lemma-NAK}).
+by Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}).
 Since $I$ is the content ideal of $x$ we see that
 $x \not \in I''M$ where $I'' = \Ker(\chi)$.
 Hence $x$ is not in the kernel of the map
@@ -17605,7 +17605,8 @@ \section{Weakly \'etale ring maps}
 \item $A$ has weak dimension $\leq 1$,
 \item every ideal of $A$ is flat,
 \item every finitely generated ideal of $A$ is flat,
-\item every submodule of a flat $A$-module is flat.
+\item every submodule of a flat $A$-module is flat, and
+\item every local ring of $A$ is a valuation ring.
 \end{enumerate}
 \end{lemma}
 
@@ -17626,7 +17627,7 @@ \section{Weakly \'etale ring maps}
 Every ideal is the union of the finitely generated ideals
 contained in it. Hence (3) implies (2) by
 Algebra, Lemma \ref{algebra-lemma-colimit-flat}.
-Thus (3) and (2) are equivalent.
+Thus (3) $\Leftrightarrow$ (2).
 
 \medskip\noindent
 Assume (2). Suppose that $N \subset M$ with $M$ a flat $A$-module.
@@ -17641,7 +17642,33 @@ \section{Weakly \'etale ring maps}
 the module $N$ has a finite filtration by the submodules
 $R^{\oplus j} \cap N$ whose subquotients are ideals.
 By (2) these ideals are flat and hence $N$ is flat by
-Algebra, Lemma \ref{algebra-lemma-flat-ses}.
+Algebra, Lemma \ref{algebra-lemma-flat-ses}. Thus (2) $\Rightarrow$ (4).
+
+\medskip\noindent
+Assume $A$ satisfies (1) and let $\mathfrak p \subset A$ be a
+prime ideal. By
+Lemmas \ref{lemma-when-weakly-etale} and \ref{lemma-weak-dimension-goes-up}
+we see that $A_\mathfrak p$ satisfies (1). We will show $A$ is a valuation ring
+if $A$ is a local ring satisfying (3). Let $f \in \mathfrak m$
+be a nonzero element. Then $(f)$ is a flat nonzero module generated by
+one element. Hence it is a free $A$-module by
+Algebra, Lemma \ref{algebra-lemma-finite-flat-local}.
+It follows that $f$ is a nonzerodivisor and $A$ is a domain.
+If $I \subset A$ is a finitely generated ideal, then we similarly
+see that $I$ is a finite free $A$-module, hence (by considering the
+rank) free of rank $1$ and $I$ is a principal ideal. Thus $A$ is a
+valuation ring by
+Algebra, Lemma \ref{algebra-lemma-characterize-valuation-ring}.
+Thus (1) $\Rightarrow$ (5).
+
+\medskip\noindent
+Assume (5). Let $I \subset A$ be a finitely generated ideal.
+Then $I_\mathfrak p \subset A_\mathfrak p$ is a finitely generated ideal
+in a valuation ring, hence principal
+(Algebra, Lemma \ref{algebra-lemma-characterize-valuation-ring}), hence flat.
+Thus $I$ is flat by
+Algebra, Lemma \ref{algebra-lemma-flat-localization}.
+Thus (5) $\Rightarrow$ (3). This finishes the proof of the lemma.
 \end{proof}
 
 \begin{lemma}