The Stacks project

Lemma 37.43.4. With notation and hypotheses as in Lemma 37.43.3. Assume moreover that $f$ is locally of finite presentation. Then we can choose the factorization such that $T$ is finite and of finite presentation over $S$.

Proof. By Limits, Lemma 32.9.8 we can write $T = \mathop{\mathrm{lim}}\nolimits T_ i$ where all $T_ i$ are finite and of finite presentation over $Y$ and the transition morphisms $T_{i'} \to T_ i$ are closed immersions. By Limits, Lemma 32.4.11 there exists an $i$ and an open subscheme $U_ i \subset T_ i$ whose inverse image in $T$ is $X$. By Limits, Lemma 32.4.16 we see that $X \cong U_ i$ for large enough $i$. Replacing $T$ by $T_ i$ finishes the proof. $\square$

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