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Tag 0CM0

Chapter 31: Limits of Schemes > Section 31.6: Limits and morphisms of finite presentation

Lemma 31.6.3. Let $f : X \to S$ be a morphism of schemes. If for every directed limit $T = \mathop{\rm lim}\nolimits_{i \in I} T_i$ of affine schemes over $S$ the map $$ \mathop{\rm colim}\nolimits \mathop{\rm Mor}\nolimits_S(T_i, X) \longrightarrow \mathop{\rm Mor}\nolimits_S(T, X) $$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition 31.6.1 parts (2) and (3) it suffices to check surjectivity of the map.

Proof. The proof is exactly the same as the proof of the implication ''(2) implies (1)'' in Proposition 31.6.1. Choose any affine opens $U \subset X$ and $V \subset S$ such that $f(U) \subset V$. We have to show that $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation. Let $(A_i, \varphi_{ii'})$ be a directed system of $\mathcal{O}_S(V)$-algebras. Set $A = \mathop{\rm colim}\nolimits_i A_i$. According to Algebra, Lemma 10.126.3 it suffices to show that $$ \mathop{\rm colim}\nolimits_i \mathop{\rm Hom}\nolimits_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A_i) \to \mathop{\rm Hom}\nolimits_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A) $$ is surjective. Consider the schemes $T_i = \mathop{\rm Spec}(A_i)$. They form an inverse system of $V$-schemes over $I$ with transition morphisms $f_{ii'} : T_i \to T_{i'}$ induced by the $\mathcal{O}_S(V)$-algebra maps $\varphi_{i'i}$. Set $T := \mathop{\rm Spec}(A) = \mathop{\rm lim}\nolimits_i T_i$. The formula above becomes in terms of morphism sets of schemes $$ \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_V(T_i, U) \to \mathop{\rm Mor}\nolimits_V(\mathop{\rm lim}\nolimits_i T_i, U) $$ We first observe that $\mathop{\rm Mor}\nolimits_V(T_i, U) = \mathop{\rm Mor}\nolimits_S(T_i, U)$ and $\mathop{\rm Mor}\nolimits_V(T, U) = \mathop{\rm Mor}\nolimits_S(T, U)$. Hence we have to show that $$ \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_S(T_i, U) \to \mathop{\rm Mor}\nolimits_S(\mathop{\rm lim}\nolimits_i T_i, U) $$ is surjective and we are given that $$ \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_S(T_i, X) \to \mathop{\rm Mor}\nolimits_S(\mathop{\rm lim}\nolimits_i T_i, X) $$ is surjective. Hence it suffices to prove that given a morphism $g_i : T_i \to X$ over $S$ such that the composition $T \to T_i \to X$ ends up in $U$ there exists some $i' \geq i$ such that the composition $g_{i'} : T_{i'} \to T_i \to X$ ends up in $U$. Denote $Z_{i'} = g_{i'}^{-1}(X \setminus U)$. Assume each $Z_{i'}$ is nonempty to get a contradiction. By Lemma 31.4.8 there exists a point $t$ of $T$ which is mapped into $Z_{i'}$ for all $i' \geq i$. Such a point is not mapped into $U$. A contradiction. $\square$

    The code snippet corresponding to this tag is a part of the file limits.tex and is located in lines 1401–1412 (see updates for more information).

    \begin{lemma}
    \label{lemma-surjection-is-enough}
    Let $f : X \to S$ be a morphism of schemes. If for every directed limit
    $T = \lim_{i \in I} T_i$ of affine schemes over $S$ the map
    $$
    \colim \Mor_S(T_i, X) \longrightarrow \Mor_S(T, X)
    $$
    is surjective, then $f$ is locally of finite presentation.
    In other words, in
    Proposition \ref{proposition-characterize-locally-finite-presentation}
    parts (2) and (3) it suffices to check surjectivity of the map.
    \end{lemma}
    
    \begin{proof}
    The proof is exactly the same as the proof of the implication
    ``(2) implies (1)'' in
    Proposition \ref{proposition-characterize-locally-finite-presentation}.
    Choose any affine opens $U \subset X$ and $V \subset S$ such that
    $f(U) \subset V$. We have to show that
    $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation.
    Let $(A_i, \varphi_{ii'})$ be a directed system of
    $\mathcal{O}_S(V)$-algebras. Set $A = \colim_i A_i$.
    According to
    Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation}
    it suffices to show that
    $$
    \colim_i \Hom_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A_i) \to
    \Hom_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A)
    $$
    is surjective. Consider the schemes $T_i = \Spec(A_i)$. They
    form an inverse system of $V$-schemes over $I$
    with transition morphisms $f_{ii'} : T_i \to T_{i'}$
    induced by the $\mathcal{O}_S(V)$-algebra maps $\varphi_{i'i}$.
    Set $T := \Spec(A) = \lim_i T_i$.
    The formula above becomes in terms of morphism sets of schemes
    $$
    \colim_i \Mor_V(T_i, U) \to \Mor_V(\lim_i T_i, U)
    $$
    We first observe that
    $\Mor_V(T_i, U) = \Mor_S(T_i, U)$
    and
    $\Mor_V(T, U) = \Mor_S(T, U)$.
    Hence we have to show that
    $$
    \colim_i \Mor_S(T_i, U) \to
    \Mor_S(\lim_i T_i, U)
    $$
    is surjective and we are given that
    $$
    \colim_i \Mor_S(T_i, X) \to
    \Mor_S(\lim_i T_i, X)
    $$
    is surjective.
    Hence it suffices to prove that given a morphism $g_i : T_i \to X$ over $S$
    such that the composition $T \to T_i \to X$ ends up in $U$ there exists some
    $i' \geq i$ such that the composition $g_{i'} : T_{i'} \to T_i \to X$ ends up
    in $U$. Denote $Z_{i'} = g_{i'}^{-1}(X \setminus U)$.
    Assume each $Z_{i'}$ is nonempty
    to get a contradiction. By Lemma \ref{lemma-limit-closed-nonempty}
    there exists a point $t$ of $T$ which is mapped into $Z_{i'}$ for all
    $i' \geq i$. Such a point is not mapped into $U$. A contradiction.
    \end{proof}

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