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Tag 09E2

Chapter 10: Commutative Algebra > Section 10.156: Nagata rings

Lemma 10.156.18. Let $(A, \mathfrak m)$ be a Noetherian local domain which is Nagata and has fraction field of characteristic $p$. If $a \in A$ has a $p$th root in $A^\wedge$, then $a$ has a $p$th root in $A$.

Proof. Consider the ring extension $A \subset B = A[x]/(x^p - a)$. If $a$ does not have a $p$th root in $A$, then $B$ is a domain whose completion isn't reduced. This contradicts our earlier results, as $B$ is a Nagata ring (Proposition 10.156.15) and hence analytically unramified by Lemma 10.156.13. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 43653–43658 (see updates for more information).

    \begin{lemma}
    \label{lemma-nagata-pth-roots}
    Let $(A, \mathfrak m)$ be a Noetherian local domain which is Nagata
    and has fraction field of characteristic $p$. If $a \in A$ has a
    $p$th root in $A^\wedge$, then $a$ has a $p$th root in $A$.
    \end{lemma}
    
    \begin{proof}
    Consider the ring extension $A \subset B = A[x]/(x^p - a)$.
    If $a$ does not have a $p$th root in $A$, then $B$ is a domain
    whose completion isn't reduced. This contradicts our earlier
    results, as $B$ is a Nagata ring
    (Proposition \ref{proposition-nagata-universally-japanese})
    and hence analytically unramified by
    Lemma \ref{lemma-local-nagata-domain-analytically-unramified}.
    \end{proof}

    Comments (2)

    Comment #2461 by Takumi Murayama (site) on March 23, 2017 a 3:12 pm UTC

    The second sentence of the statement has an extra "is": it should probably read "…then $a$ has a $p$th root in $A$." In the proof, the phrase "as $B$ is a Nagata" should probably say "…is Nagata" or "…is a Nagata ring".

    Comment #2498 by Johan (site) on April 13, 2017 a 11:23 pm UTC

    Yes indeed, thanks. Fixed here.

    There are also 2 comments on Section 10.156: Commutative Algebra.

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