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Tag 053E

Chapter 15: More on Algebra > Section 15.23: Flatness and finiteness conditions

Lemma 15.23.6. Let $A$ be a valuation ring. Let $A \to B$ be a ring map of finite type. Let $M$ be a finite $B$-module.

  1. If $B$ is flat over $A$, then $B$ is a finitely presented $A$-algebra.
  2. If $M$ is flat as an $A$-module, then $M$ is finitely presented as a $B$-module.

Proof. We are going to use that an $A$-module is flat if and only if it is torsion free, see Lemma 15.20.10. By Algebra, Lemma 10.56.10 we can find a graded $A$-algebra $S$ with $S_0 = A$ and generated by finitely many elements in degree $1$, an element $f \in S_1$ and a finite graded $S$-module $N$ such that $B \cong S_{(f)}$ and $M \cong N_{(f)}$. If $M$ is torsion free, then we can take $N$ torsion free by replacing it by $N/N_{tors}$, see Lemma 15.20.2. Similarly, if $B$ is torsion free, then we can take $S$ torsion free by replacing it by $S/S_{tors}$. Hence in case (1), we may apply Lemma 15.23.4 to see that $S$ is a finitely presented $A$-algebra, which implies that $B = S_{(f)}$ is a finitely presented $A$-algebra. To see (2) we may first replace $S$ by a graded polynomial ring, and then we may apply Lemma 15.23.3 to conclude. $\square$

    The code snippet corresponding to this tag is a part of the file more-algebra.tex and is located in lines 4953–4962 (see updates for more information).

    \begin{lemma}
    \label{lemma-flat-finite-type-valuation-ring-finite-presentation}
    Let $A$ be a valuation ring. Let $A \to B$ be a ring map of finite type.
    Let $M$ be a finite $B$-module.
    \begin{enumerate}
    \item If $B$ is flat over $A$, then $B$ is a finitely presented $A$-algebra.
    \item If $M$ is flat as an $A$-module, then $M$ is finitely presented
    as a $B$-module.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    We are going to use that an $A$-module is flat if and only if it is
    torsion free, see
    Lemma \ref{lemma-valuation-ring-torsion-free-flat}.
    By
    Algebra, Lemma \ref{algebra-lemma-homogenize}
    we can find a graded $A$-algebra $S$ with $S_0 = A$ and generated
    by finitely many elements in degree $1$, an element $f \in S_1$ and a
    finite graded $S$-module $N$ such that $B \cong S_{(f)}$ and
    $M \cong N_{(f)}$. If $M$ is torsion free, then we can take $N$ torsion
    free by replacing it by $N/N_{tors}$, see
    Lemma \ref{lemma-torsion}.
    Similarly, if $B$ is torsion free, then we can take
    $S$ torsion free by replacing it by $S/S_{tors}$.
    Hence in case (1), we may apply
    Lemma \ref{lemma-flat-graded-finite-type-finite-presentation}
    to see that $S$ is a finitely presented
    $A$-algebra, which implies that $B = S_{(f)}$ is a finitely
    presented $A$-algebra. To see (2) we may first replace $S$ by
    a graded polynomial ring, and then we may apply
    Lemma \ref{lemma-flat-graded-finite-type-finite-presentation-module}
    to conclude.
    \end{proof}

    Comments (1)

    Comment #2756 by Kestutis Cesnavicius on August 3, 2017 a 9:36 am UTC

    For the sake of keeping track of references, it may be worthwhile to mention that part (1) of this lemma is Thoerem 3 in Nagata "Finitely generated rings over a valuation ring."

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