The Stacks project

Lemma 97.25.3. Let $X$ be an object of $(\textit{Noetherian}/S)_\tau $. If the functor of points $h_ X : (\textit{Noetherian}/S)_\tau ^{opp} \to \textit{Sets}$ is limit preserving, then $X$ is locally of finite presentation over $S$.

Proof. Let $V \subset X$ be an affine open subscheme which maps into an affine open $U \subset S$. We may write $V = \mathop{\mathrm{lim}}\nolimits V_ i$ as a directed limit of affine schemes $V_ i$ of finite presentation over $U$, see Algebra, Lemma 10.127.2. By assumption, the arrow $V \to X$ factors as $V \to V_ i \to X$ for some $i$. After increasing $i$ we may assume $V_ i \to X$ factors through $V$ as the inverse image of $V \subset X$ in $V_ i$ eventually becomes equal to $V_ i$ by Limits, Lemma 32.4.11. Then the identity morphism $V \to V$ factors through $V_ i$ for some $i$ in the category of schemes over $U$. Thus $V \to U$ is of finite presentation; the corresponding algebra fact is that if $B$ is an $A$-algebra such that $\text{id} : B \to B$ factors through a finitely presented $A$-algebra, then $B$ is of finite presentation over $A$ (nice exercise). Hence $X$ is locally of finite presentation over $S$. $\square$


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