stacks/stacks-project

Change order of two lemmas in algebra

 @@ -8885,27 +8885,8 @@ \section{Flat modules and flat ring maps} Immediate from Lemma \ref{lemma-ff-rings}. \end{proof} \begin{lemma} \label{lemma-flat-going-down} Let $R \to S$ be flat. Let $\mathfrak p \subset \mathfrak p'$ be primes of $R$. Let $\mathfrak q' \subset S$ be a prime of $S$ mapping to $\mathfrak p'$. Then there exists a prime $\mathfrak q \subset \mathfrak q'$ mapping to $\mathfrak p$. \end{lemma} \begin{proof} Namely, consider the flat local ring map $R_{\mathfrak p'} \to S_{\mathfrak q'}$. By Lemma \ref{lemma-local-flat-ff} this is faithfully flat. By Lemma \ref{lemma-ff-rings} there is a prime mapping to $\mathfrak p R_{\mathfrak p'}$. The inverse image of this prime in $S$ does the job. \end{proof} \noindent The property of $R \to S$ described in the lemma is called the going down property''. See Definition \ref{definition-going-up-down}. We finish with some remarks on flatness and localization. Flatness meshes well with localization. \begin{lemma} \label{lemma-flat-localization} @@ -8976,6 +8957,27 @@ \section{Flat modules and flat ring maps} way from the last statement (proofs omitted). \end{proof} \begin{lemma} \label{lemma-flat-going-down} Let $R \to S$ be flat. Let $\mathfrak p \subset \mathfrak p'$ be primes of $R$. Let $\mathfrak q' \subset S$ be a prime of $S$ mapping to $\mathfrak p'$. Then there exists a prime $\mathfrak q \subset \mathfrak q'$ mapping to $\mathfrak p$. \end{lemma} \begin{proof} By Lemma \ref{lemma-flat-localization} the local ring map $R_{\mathfrak p'} \to S_{\mathfrak q'}$ is flat. By Lemma \ref{lemma-local-flat-ff} this local ring map is faithfully flat. By Lemma \ref{lemma-ff-rings} there is a prime mapping to $\mathfrak p R_{\mathfrak p'}$. The inverse image of this prime in $S$ does the job. \end{proof} \noindent The property of $R \to S$ described in the lemma is called the going down property''. See Definition \ref{definition-going-up-down}. \begin{lemma} \label{lemma-colimit-faithfully-flat} Let $R$ be a ring. Let $\{S_i, \varphi_{ii'}\}$ be a directed system of