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Tag 04GT

Chapter 10: Commutative Algebra > Section 10.150: Henselization and strict henselization

Lemma 10.150.11. Let $\varphi : R \to S$ be a local map of local rings. Let $S/\mathfrak m_S \subset \kappa^{sep}$ be a separable algebraic closure. Let $S \to S^{sh}$ be the strict henselization of $S$ with respect to $S/\mathfrak m_S \subset \kappa^{sep}$. Let $R \to A$ be an étale ring map and let $\mathfrak q$ be a prime of $A$ lying over $\mathfrak m_R$. Given any commutative diagram $$ \xymatrix{ \kappa(\mathfrak q) \ar[r]_{\phi} & \kappa^{sep} \\ R/\mathfrak m_R \ar[r]^{\varphi} \ar[u] & S/\mathfrak m_S \ar[u] } $$ there exists a unique morphism of rings $f : A \to S^{sh}$ fitting into the commutative diagram $$ \xymatrix{ A \ar[r]_f & S^{sh} \\ R \ar[u] \ar[r]^{\varphi} & S \ar[u] } $$ such that $f^{-1}(\mathfrak m_{S^h}) = \mathfrak q$ and the induced map $\kappa(\mathfrak q) \to \kappa^{sep}$ is the given one.

Proof. This is a special case of Lemma 10.148.11. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 40971–40996 (see updates for more information).

    \begin{lemma}
    \label{lemma-strictly-henselian-functorial-prepare}
    Let $\varphi : R \to S$ be a local map of local rings.
    Let $S/\mathfrak m_S \subset \kappa^{sep}$ be a separable algebraic closure.
    Let $S \to S^{sh}$ be the strict henselization of $S$
    with respect to $S/\mathfrak m_S \subset \kappa^{sep}$.
    Let $R \to A$ be an \'etale ring map and let $\mathfrak q$
    be a prime of $A$ lying over $\mathfrak m_R$.
    Given any commutative diagram
    $$
    \xymatrix{
    \kappa(\mathfrak q) \ar[r]_{\phi} & \kappa^{sep} \\
    R/\mathfrak m_R \ar[r]^{\varphi} \ar[u] & S/\mathfrak m_S \ar[u]
    }
    $$
    there exists a unique morphism of rings
    $f : A \to S^{sh}$ fitting into the commutative diagram
    $$
    \xymatrix{
    A \ar[r]_f & S^{sh} \\
    R \ar[u] \ar[r]^{\varphi} & S \ar[u]
    }
    $$
    such that $f^{-1}(\mathfrak m_{S^h}) = \mathfrak q$ and the induced
    map $\kappa(\mathfrak q) \to \kappa^{sep}$ is the given one.
    \end{lemma}
    
    \begin{proof}
    This is a special case of Lemma \ref{lemma-map-into-henselian}.
    \end{proof}

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