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Tag 05K0

Chapter 36: More on Morphisms > Section 36.38: Zariski's Main Theorem

Lemma 36.38.3 (Zariski's Main Theorem). Let $f : X \to S$ be a morphism of schemes. Assume $f$ is quasi-finite and separated and assume that $S$ is quasi-compact and quasi-separated. Then there exists a factorization $$ \xymatrix{ X \ar[rd]_f \ar[rr]_j & & T \ar[ld]^\pi \\ & S & } $$ where $j$ is a quasi-compact open immersion and $\pi$ is finite.

Proof. Let $X \to S' \to S$ be as in the conclusion of Lemma 36.38.2. By Properties, Lemma 27.22.13 we can write $\nu_*\mathcal{O}_{S'} = \mathop{\rm colim}\nolimits_{i \in I} \mathcal{A}_i$ as a directed colimit of finite quasi-coherent $\mathcal{O}_X$-algebras $\mathcal{A}_i \subset \nu_*\mathcal{O}_{S'}$. Then $\pi_i : T_i = \underline{\mathop{\rm Spec}}_S(\mathcal{A}_i) \to S$ is a finite morphism for each $i$. Note that the transition morphisms $T_{i'} \to T_i$ are affine and that $S' = \mathop{\rm lim}\nolimits T_i$.

By Limits, Lemma 31.4.11 there exists an $i$ and a quasi-compact open $U_i \subset T_i$ whose inverse image in $S'$ equals $f'(X)$. For $i' \geq i$ let $U_{i'}$ be the inverse image of $U_i$ in $T_{i'}$. Then $X \cong f'(X) = \mathop{\rm lim}\nolimits_{i' \geq i} U_{i'}$, see Limits, Lemma 31.2.2. By Limits, Lemma 31.4.16 we see that $X \to U_{i'}$ is a closed immersion for some $i' \geq i$. (In fact $X \cong U_{i'}$ for sufficiently large $i'$ but we don't need this.) Hence $X \to T_{i'}$ is an immersion. By Morphisms, Lemma 28.3.2 we can factor this as $X \to T \to T_{i'}$ where the first arrow is an open immersion and the second a closed immersion. Thus we win. $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 10874–10887 (see updates for more information).

    \begin{lemma}[Zariski's Main Theorem]
    \label{lemma-quasi-finite-separated-pass-through-finite}
    Let $f : X \to S$ be a morphism of schemes.
    Assume $f$ is quasi-finite and separated and assume that
    $S$ is quasi-compact and quasi-separated. Then there exists
    a factorization
    $$
    \xymatrix{
    X \ar[rd]_f \ar[rr]_j & & T \ar[ld]^\pi \\
    & S &
    }
    $$
    where $j$ is a quasi-compact open immersion and $\pi$ is finite.
    \end{lemma}
    
    \begin{proof}
    Let $X \to S' \to S$ be as in the conclusion of
    Lemma \ref{lemma-quasi-finite-separated-quasi-affine}.
    By
    Properties, Lemma
    \ref{properties-lemma-integral-algebra-directed-colimit-finite}
    we can write
    $\nu_*\mathcal{O}_{S'} = \colim_{i \in I} \mathcal{A}_i$ as a
    directed colimit of finite quasi-coherent $\mathcal{O}_X$-algebras
    $\mathcal{A}_i \subset \nu_*\mathcal{O}_{S'}$. Then
    $\pi_i : T_i = \underline{\Spec}_S(\mathcal{A}_i) \to S$
    is a finite morphism for each $i$.
    Note that the transition morphisms $T_{i'} \to T_i$ are affine
    and that $S' = \lim T_i$.
    
    \medskip\noindent
    By Limits, Lemma \ref{limits-lemma-descend-opens}
    there exists an $i$ and a quasi-compact open
    $U_i \subset T_i$ whose inverse image in $S'$ equals
    $f'(X)$. For $i' \geq i$ let $U_{i'}$ be the inverse image
    of $U_i$ in $T_{i'}$. Then $X \cong f'(X) = \lim_{i' \geq i} U_{i'}$, see
    Limits, Lemma \ref{limits-lemma-directed-inverse-system-has-limit}.
    By Limits, Lemma \ref{limits-lemma-finite-type-eventually-closed} we see that
    $X \to U_{i'}$ is a closed immersion for some $i' \geq i$.
    (In fact $X \cong U_{i'}$ for sufficiently
    large $i'$ but we don't need this.) Hence $X \to T_{i'}$ is an immersion. By
    Morphisms, Lemma \ref{morphisms-lemma-factor-quasi-compact-immersion}
    we can factor this as $X \to T \to T_{i'}$ where the first arrow
    is an open immersion and the second a closed immersion. Thus we win.
    \end{proof}

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