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The Stacks project

Lemma 76.34.4. With notation and hypotheses as in Lemma 76.34.3. 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 Y.

Proof. By Limits of Spaces, Lemma 70.11.3 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 of Spaces, Lemma 70.5.7 there exists an i and an open subscheme U_ i \subset T_ i whose inverse image in T is X. By Limits of Spaces, Lemma 70.5.12 we see that X \cong U_ i for large enough i. Replacing T by T_ i finishes the proof. \square


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