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Tag 094Z

Chapter 36: More on Morphisms > Section 36.53: Weakly étale morphisms

Lemma 36.53.11. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent

  1. $f$ is weakly étale, and
  2. for $x \in X$ the local ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ induces an isomorphism on strict henselizations.

Proof. Let $x \in X$ be a point with image $y = f(x)$ in $Y$. Choose a separable algebraic closure $\kappa^{sep}$ of $\kappa(x)$. Let $\mathcal{O}_{X, x}^{sh}$ be the strict henselization corresponding to $\kappa^{sep}$ and $\mathcal{O}_{Y, y}^{sh}$ the strict henselization relative to the separable algebraic closure of $\kappa(y)$ in $\kappa^{sep}$. Consider the commutative diagram $$ \xymatrix{ \mathcal{O}_{X, x} \ar[r] & \mathcal{O}_{X, x}^{sh} \\ \mathcal{O}_{Y, y} \ar[u] \ar[r] & \mathcal{O}_{Y, y}^{sh} \ar[u] } $$ local homomorphisms of local rings, see Algebra, Lemma 10.150.12. Since the strict henselization is a filtered colimit of étale ring maps, More on Algebra, Lemma 15.85.14 shows the horizontal maps are weakly étale. Moreover, the horizontal maps are faithfully flat by More on Algebra, Lemma 15.41.1.

Assume $f$ weakly étale. By Lemma 36.53.2 the left vertical arrow is weakly étale. By More on Algebra, Lemmas 15.85.9 and 15.85.11 the right vertical arrow is weakly étale. By More on Algebra, Theorem 15.85.25 we conclude the right vertical map is an isomorphism.

Assume $\mathcal{O}_{Y, y}^{sh} \to \mathcal{O}_{X, x}^{sh}$ is an isomorphism. Then $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}^{sh}$ is weakly étale. Since $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^{sh}$ is faithfully flat we conclude that $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is weakly étale by More on Algebra, Lemma 15.85.10. Thus (2) implies (1) by Lemma 36.53.2. $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 15792–15802 (see updates for more information).

    \begin{lemma}
    \label{lemma-weakly-etale-strictly-henselian-local-rings}
    Let $f : X \to Y$ be a morphism of schemes.
    The following are equivalent
    \begin{enumerate}
    \item $f$ is weakly \'etale, and
    \item for $x \in X$ the local ring map
    $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ induces an isomorphism
    on strict henselizations.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Let $x \in X$ be a point with image $y = f(x)$ in $Y$.
    Choose a separable algebraic closure $\kappa^{sep}$ of $\kappa(x)$.
    Let $\mathcal{O}_{X, x}^{sh}$ be the strict henselization
    corresponding to $\kappa^{sep}$ and $\mathcal{O}_{Y, y}^{sh}$
    the strict henselization relative to the separable algebraic
    closure of $\kappa(y)$ in $\kappa^{sep}$.
    Consider the commutative diagram
    $$
    \xymatrix{
    \mathcal{O}_{X, x} \ar[r] & \mathcal{O}_{X, x}^{sh} \\
    \mathcal{O}_{Y, y} \ar[u] \ar[r] & \mathcal{O}_{Y, y}^{sh} \ar[u]
    }
    $$
    local homomorphisms of local rings, see
    Algebra, Lemma \ref{algebra-lemma-strictly-henselian-functorial}.
    Since the strict henselization is a filtered colimit of \'etale
    ring maps, More on Algebra, Lemma \ref{more-algebra-lemma-when-weakly-etale}
    shows the horizontal maps are weakly \'etale.
    Moreover, the horizontal maps are faithfully flat by
    More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization}.
    
    \medskip\noindent
    Assume $f$ weakly \'etale. By Lemma \ref{lemma-check-weakly-etale-stalks}
    the left vertical arrow is weakly \'etale. By
    More on Algebra, Lemmas \ref{more-algebra-lemma-composition-weakly-etale} and
    \ref{more-algebra-lemma-weakly-etale-permanence}
    the right vertical arrow is weakly \'etale. By
    More on Algebra, Theorem \ref{more-algebra-theorem-olivier}
    we conclude the right vertical map is an isomorphism.
    
    \medskip\noindent
    Assume $\mathcal{O}_{Y, y}^{sh} \to \mathcal{O}_{X, x}^{sh}$ is an isomorphism.
    Then $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}^{sh}$ is weakly \'etale.
    Since $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^{sh}$ is faithfully
    flat we conclude that $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$
    is weakly \'etale by
    More on Algebra, Lemma \ref{more-algebra-lemma-go-down}.
    Thus (2) implies (1) by Lemma \ref{lemma-check-weakly-etale-stalks}.
    \end{proof}

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