## 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:

- The morphism $f$ is a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition 55.28.1.
- 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

alldirected 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}
```

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