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Tag 04AK

Chapter 58: Limits of Algebraic Spaces > Section 58.3: Morphisms of finite presentation

Proposition 58.3.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. The morphism $f$ is a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition 55.28.1.
  2. The morphism $f : X \to Y$ is limit preserving as a transformation of functors, see Definition 58.3.1.

Proof. Assume (1). Let $T$ be a scheme and let $y \in Y(T)$. We have to show that $T \times_Y X$ is limit preserving over $T$ in the sense of Definition 58.3.1. Hence we are reduced to proving that if $X$ is an algebraic space which is locally of finite presentation over $S$ as an algebraic space, then it is limit preserving as a functor $X : (\textit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. To see this choose a presentation $X = U/R$, see Spaces, Definition 53.9.3. It follows from Morphisms of Spaces, Definition 55.28.1 that both $U$ and $R$ are schemes which are locally of finite presentation over $S$. Hence by Limits, Proposition 31.6.1 we have $$ U(T) = \mathop{\rm colim}\nolimits U(T_i), \quad R(T) = \mathop{\rm colim}\nolimits R(T_i) $$ whenever $T = \mathop{\rm lim}\nolimits_i T_i$ in $(\textit{Sch}/S)_{fppf}$. It follows that the presheaf $$ (\textit{Sch}/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad W \longmapsto U(W)/R(W) $$ is limit preserving. Hence by Lemma 58.3.6 its sheafification $X = U/R$ is limit preserving too.

Assume (2). Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Next, choose a scheme $U$ and a surjective étale morphism $U \to V \times_Y X$. By Lemma 58.3.4 the transformation of functors $V \times_Y X \to V$ is limit preserving. By Morphisms of Spaces, Lemma 55.38.8 the morphism of algebraic spaces $U \to V \times_Y X$ is locally of finite presentation, hence limit preserving as a transformation of functors by the first part of the proof. By Lemma 58.3.3 the composition $U \to V \times_Y X \to V$ is limit preserving as a transformation of functors. Hence the morphism of schemes $U \to V$ is locally of finite presentation by Limits, Proposition 31.6.1 (modulo a set theoretic remark, see last paragraph of the proof). This means, by definition, that (1) holds.

Set theoretic remark. Let $U \to V$ be a morphism of $(\textit{Sch}/S)_{fppf}$. In the statement of Limits, Proposition 31.6.1 we characterize $U \to V$ as being locally of finite presentation if for all directed inverse systems $(T_i, f_{ii'})$ of affine schemes over $V$ we have $U(T) = \mathop{\rm colim}\nolimits V(T_i)$, but in the current setting we may only consider affine schemes $T_i$ over $V$ which are (isomorphic to) an object of $(\textit{Sch}/S)_{fppf}$. So we have to make sure that there are enough affines in $(\textit{Sch}/S)_{fppf}$ to make the proof work. Inspecting the proof of (2) $\Rightarrow$ (1) of Limits, Proposition 31.6.1 we see that the question reduces to the case that $U$ and $V$ are affine. Say $U = \mathop{\rm Spec}(A)$ and $V = \mathop{\rm Spec}(B)$. By construction of $(\textit{Sch}/S)_{fppf}$ the spectrum of any ring of cardinality $\leq |B|$ is isomorphic to an object of $(\textit{Sch}/S)_{fppf}$. Hence it suffices to observe that in the "only if" part of the proof of Algebra, Lemma 10.126.3 only $A$-algebras of cardinality $\leq |B|$ are used. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-limits.tex and is located in lines 446–459 (see updates for more information).

    \begin{proposition}
    \label{proposition-characterize-locally-finite-presentation}
    Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic
    spaces over $S$. The following are equivalent:
    \begin{enumerate}
    \item The morphism $f$ is a morphism of algebraic spaces which is
    locally of finite presentation, see
    Morphisms of Spaces,
    Definition \ref{spaces-morphisms-definition-locally-finite-presentation}.
    \item The morphism $f : X \to Y$ is limit preserving as
    a transformation of functors, see
    Definition \ref{definition-locally-finite-presentation}.
    \end{enumerate}
    \end{proposition}
    
    \begin{proof}
    Assume (1). Let $T$ be a scheme and let $y \in Y(T)$. We have to show that
    $T \times_Y X$ is limit preserving over $T$ in the sense of
    Definition \ref{definition-locally-finite-presentation}.
    Hence we are reduced to proving that if $X$ is an algebraic space which
    is locally of finite presentation over $S$ as an algebraic space, then it
    is limit preserving as a functor
    $X : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$.
    To see this choose a presentation $X = U/R$, see
    Spaces, Definition \ref{spaces-definition-presentation}.
    It follows from
    Morphisms of Spaces,
    Definition \ref{spaces-morphisms-definition-locally-finite-presentation}
    that both $U$ and $R$ are schemes which are locally of finite presentation
    over $S$. Hence by
    Limits, Proposition
    \ref{limits-proposition-characterize-locally-finite-presentation}
    we have
    $$
    U(T) = \colim U(T_i), \quad
    R(T) = \colim R(T_i)
    $$
    whenever $T = \lim_i T_i$ in $(\Sch/S)_{fppf}$. It follows
    that the presheaf
    $$
    (\Sch/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad
    W \longmapsto U(W)/R(W)
    $$
    is limit preserving. Hence by
    Lemma \ref{lemma-sheafify-finite-presentation}
    its sheafification $X = U/R$ is limit preserving too.
    
    \medskip\noindent
    Assume (2). Choose a scheme $V$ and a surjective \'etale morphism
    $V \to Y$. Next, choose a scheme $U$ and a surjective \'etale morphism
    $U \to V \times_Y X$. By
    Lemma \ref{lemma-base-change-locally-finite-presentation}
    the transformation of functors $V \times_Y X \to V$ is limit preserving. By
    Morphisms of Spaces,
    Lemma \ref{spaces-morphisms-lemma-etale-locally-finite-presentation}
    the morphism of algebraic spaces $U \to V \times_Y X$ is locally
    of finite presentation, hence limit preserving as
    a transformation of functors by the first part of the proof. By
    Lemma \ref{lemma-composition-locally-finite-presentation}
    the composition $U \to V \times_Y X \to V$ is limit preserving
    as a transformation of functors. Hence
    the morphism of schemes $U \to V$ is locally of finite presentation by
    Limits, Proposition
    \ref{limits-proposition-characterize-locally-finite-presentation}
    (modulo a set theoretic remark, see last paragraph of the proof).
    This means, by definition, that (1) holds.
    
    \medskip\noindent
    Set theoretic remark. Let $U \to V$ be a morphism of
    $(\Sch/S)_{fppf}$. In the statement of
    Limits, Proposition
    \ref{limits-proposition-characterize-locally-finite-presentation}
    we characterize $U \to V$ as being locally of finite presentation
    if for {\it all} directed inverse systems $(T_i, f_{ii'})$ of affine schemes
    over $V$ we have $U(T) = \colim V(T_i)$, but in the current setting
    we may only consider affine schemes $T_i$ over $V$ which are (isomorphic to)
    an object of $(\Sch/S)_{fppf}$. So we have to make sure that there
    are enough affines in $(\Sch/S)_{fppf}$ to make the proof work.
    Inspecting the proof of (2) $\Rightarrow$ (1) of
    Limits, Proposition
    \ref{limits-proposition-characterize-locally-finite-presentation}
    we see that the question reduces to the case that $U$ and $V$ are affine.
    Say $U = \Spec(A)$ and $V = \Spec(B)$. By construction
    of $(\Sch/S)_{fppf}$ the spectrum of any ring of cardinality
    $\leq |B|$ is isomorphic to an object of $(\Sch/S)_{fppf}$.
    Hence it suffices to observe that in the "only if" part of the proof of
    Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation}
    only $A$-algebras of cardinality $\leq |B|$ are used.
    \end{proof}

    Comments (2)

    Comment #2298 by Eric Ahlqvist on November 7, 2016 a 1:26 pm UTC

    Hi! I think you should switch $X$ and $Y$ in the fiber product on the first line of the proof: $T\times_XY$ should be $T\times_YX$?

    Comment #2324 by Johan (site) on December 12, 2016 a 12:43 am UTC

    Thanks, fixed here.

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