The Stacks project

Proof. By Limits, Lemma 32.10.1 the category of schemes of finite presentation over $T$ is the colimit over $I$ of the categories of finite presentation over $T_ i$. By Limits, Lemma 32.8.2 the same is true for category of schemes which are affine and of finite presentation over $T$. To finish the proof of the lemma it suffices to show that if $\{ V_{j, i} \to T_ i\} _{j = 1, \ldots , m}$ is a finite family of finitely presented morphisms with $V_{j, i}$ affine, and the base change family $\{ T \times _{T_ i} V_{j, i} \to T\}$ is an h covering, then for some $i' \geq i$ the family $\{ T_{i'} \times _{T_ i} V_{j, i} \to T_{i'}\}$ is an h covering. To see this we use Lemma 38.34.5 to choose a finitely presented, proper, surjective morphism $Y \to T$ and a finite affine open covering $Y = \bigcup _{k = 1, \ldots , n} Y_ k$ such that $\{ Y_ k \to T\} _{k = 1, \ldots , n}$ refines $\{ T \times _{T_ i} V_{j, i} \to T\}$. Using the arguments above and Limits, Lemmas 32.13.1, 32.8.15, and 32.4.11 we can find an $i' \geq i$ and a finitely presented, surjective, proper morphism $Y_{i'} \to T_{i'}$ and an affine open covering $Y_{i'} = \bigcup _{k = 1, \ldots , n} Y_{i', k}$ such that moreover $\{ Y_{i', k} \to Y_{i'}\}$ refines $\{ T_{i'} \times _{T_ i} V_{j, i} \to T_{i'}\}$. It follows that this last mentioned family is a h covering and the proof is complete. $\square$