The Stacks project

Lemma 31.6.3. Let $f : X \to S$ be a morphism of schemes. If for every directed limit $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ of affine schemes over $S$ the map

\[ \mathop{\mathrm{colim}}\nolimits \mathop{Mor}\nolimits _ S(T_ i, X) \longrightarrow \mathop{Mor}\nolimits _ S(T, X) \]

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 $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$. According to Algebra, Lemma 10.126.3 it suffices to show that

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ S(V)}(\mathcal{O}_ X(U), A_ i) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ S(V)}(\mathcal{O}_ X(U), A) \]

is surjective. Consider the schemes $T_ i = \mathop{\mathrm{Spec}}(A_ i)$. 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 $T := \mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{lim}}\nolimits _ i T_ i$. The formula above becomes in terms of morphism sets of schemes

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{Mor}\nolimits _ V(T_ i, U) \to \mathop{Mor}\nolimits _ V(\mathop{\mathrm{lim}}\nolimits _ i T_ i, U) \]

We first observe that $\mathop{Mor}\nolimits _ V(T_ i, U) = \mathop{Mor}\nolimits _ S(T_ i, U)$ and $\mathop{Mor}\nolimits _ V(T, U) = \mathop{Mor}\nolimits _ S(T, U)$. Hence we have to show that

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{Mor}\nolimits _ S(T_ i, U) \to \mathop{Mor}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits _ i T_ i, U) \]

is surjective and we are given that

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{Mor}\nolimits _ S(T_ i, X) \to \mathop{Mor}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits _ i T_ i, X) \]

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. $\square$


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