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Tag 05WQ

Henselization is compatible with quotients.

Lemma 10.150.10. Let $R$ be a local ring with henselization $R^h$. Let $I \subset \mathfrak m_R$. Then $R^h/IR^h$ is the henselization of $R/I$.

Proof. This is a special case of Lemma 10.150.9. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 40956–40964 (see updates for more information).

\begin{lemma}
\label{lemma-quotient-henselization}
\begin{slogan}
Henselization is compatible with quotients.
\end{slogan}
Let $R$ be a local ring with henselization $R^h$.
Let $I \subset \mathfrak m_R$.
Then $R^h/IR^h$ is the henselization of $R/I$.
\end{lemma}

\begin{proof}
This is a special case of
Lemma \ref{lemma-quasi-finite-henselization}.
\end{proof}

Comment #2805 by Minseon Shin on September 13, 2017 a 9:04 pm UTC

Suggested slogan: Henselization is compatible with quotients.

There are also 3 comments on Section 10.150: Commutative Algebra.

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