The Stacks project

Lemma 34.13.2. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $T$ be an affine scheme which is written as a limit $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ of a directed inverse system of affine schemes.

  1. Let $\mathcal{V} = \{ V_ j \to T\} _{j = 1, \ldots , m}$ be a standard $\tau $-covering of $T$, see Definitions 34.3.4, 34.4.5, 34.5.5, 34.6.5, and 34.7.5. Then there exists an index $i$ and a standard $\tau $-covering $\mathcal{V}_ i = \{ V_{i, j} \to T_ i\} _{j = 1, \ldots , m}$ whose base change $T \times _{T_ i} \mathcal{V}_ i$ to $T$ is isomorphic to $\mathcal{V}$.

  2. Let $\mathcal{V}_ i$, $\mathcal{V}'_ i$ be a pair of standard $\tau $-coverings of $T_ i$. If $f : T \times _{T_ i} \mathcal{V}_ i \to T \times _{T_ i} \mathcal{V}'_ i$ is a morphism of coverings of $T$, then there exists an index $i' \geq i$ and a morphism $f_{i'} : T_{i'} \times _{T_ i} \mathcal{V} \to T_{i'} \times _{T_ i} \mathcal{V}'_ i$ whose base change to $T$ is $f$.

  3. If $f, g : \mathcal{V} \to \mathcal{V}'_ i$ are morphisms of standard $\tau $-coverings of $T_ i$ whose base changes $f_ T, g_ T$ to $T$ are equal then there exists an index $i' \geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$.

In other words, the category of standard $\tau $-coverings of $T$ is the colimit over $I$ of the categories of standard $\tau $-coverings of $T_ i$.

Proof. Let us prove this for $\tau = fppf$. 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, Lemmas 32.8.2 and 32.8.7 the same is true for category of schemes which are affine, flat 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 flat finitely presented morphisms with $V_{j, i}$ affine, and the base change $\coprod _ j T \times _{T_ i} V_{j, i} \to T$ is surjective, then for some $i' \geq i$ the morphism $\coprod T_{i'} \times _{T_ i} V_{j, i} \to T_{i'}$ is surjective. Denote $W_{i'} \subset T_{i'}$, resp. $W \subset T$ the image. Of course $W = T$ by assumption. Since the morphisms are flat and of finite presentation we see that $W_ i$ is a quasi-compact open of $T_ i$, see Morphisms, Lemma 29.25.10. Moreover, $W = T \times _{T_ i} W_ i$ (formation of image commutes with base change). Hence by Limits, Lemma 32.4.11 we conclude that $W_{i'} = T_{i'}$ for some large enough $i'$ and we win.

For $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic\} $ a standard $\tau $-covering is a standard fppf covering. Hence the fully faithfulness of the functor holds. The only issue is to show that given a standard fppf covering $\mathcal{V}_ i$ for some $i$ such that $\mathcal{V}_ i \times _{T_ i} T$ is a standard $\tau $-covering, then $\mathcal{V}_ i \times _{T_ i} T_{i'}$ is a standard $\tau $-covering for all $i' \gg i$. This follows immediately from Limits, Lemmas 32.8.12, 32.8.10, 32.8.9, and 32.8.15. $\square$

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