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Change order of two lemmas in algebra

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aisejohan committed Oct 3, 2019
1 parent c191740 commit 3520110c256438254c4b869fb714567dc3cab699
Showing with 22 additions and 20 deletions.
  1. +22 −20 algebra.tex
@@ -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

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