Lemma 35.33.2. Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{ fpqc, fppf, {\acute{e}tale}, smooth, syntomic\}$. Suppose that

1. $\mathcal{P}$ is stable under any base change (see Schemes, Definition 26.18.3),

2. if $Y_ j \to V_ j$, $j = 1, \ldots , m$ have $\mathcal{P}$, then so does $\coprod Y_ j \to \coprod V_ j$, and

3. for any surjective morphism of affines $X \to S$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau$ is fpqc, fppf, étale, smooth, or syntomic, any descent datum $(V, \varphi )$ relative to $X$ over $S$ such that $\mathcal{P}$ holds for $V \to X$ is effective.

Then morphisms of type $\mathcal{P}$ satisfy descent for $\tau$-coverings.

Proof. Let $S$ be a scheme. Let $\mathcal{U} = \{ \varphi _ i : U_ i \to S\} _{i \in I}$ be a $\tau$-covering of $S$. Let $(X_ i, \varphi _{ii'})$ be a descent datum relative to $\mathcal{U}$ and assume that each morphism $X_ i \to U_ i$ has property $\mathcal{P}$. We have to show there exists a scheme $X \to S$ such that $(X_ i, \varphi _{ii'}) \cong (U_ i \times _ S X, can)$.

Before we start the proof proper we remark that for any family of morphisms $\mathcal{V} : \{ V_ j \to S\}$ and any morphism of families $\mathcal{V} \to \mathcal{U}$, if we pullback the descent datum $(X_ i, \varphi _{ii'})$ to a descent datum $(Y_ j, \varphi _{jj'})$ over $\mathcal{V}$, then each of the morphisms $Y_ j \to V_ j$ has property $\mathcal{P}$ also. This is true because of assumption (1) that $\mathcal{P}$ is stable under any base change and the definition of pullback (see Definition 35.31.9). We will use this without further mention.

First, let us prove the lemma when $S$ is affine. By Topologies, Lemma 34.9.8, 34.7.4, 34.4.4, 34.5.4, or 34.6.4 there exists a standard $\tau$-covering $\mathcal{V} : \{ V_ j \to S\} _{j = 1, \ldots , m}$ which refines $\mathcal{U}$. The pullback functor $DD(\mathcal{U}) \to DD(\mathcal{V})$ between categories of descent data is fully faithful by Lemma 35.32.11. Hence it suffices to prove that the descent datum over the standard $\tau$-covering $\mathcal{V}$ is effective. By assumption (2) we see that $\coprod Y_ j \to \coprod V_ j$ has property $\mathcal{P}$. By Lemma 35.31.5 this reduces us to the covering $\{ \coprod _{j = 1, \ldots , m} V_ j \to S\}$ for which we have assumed the result in assumption (3) of the lemma. Hence the lemma holds when $S$ is affine.

Assume $S$ is general. Let $V \subset S$ be an affine open. By the properties of site the family $\mathcal{U}_ V = \{ V \times _ S U_ i \to V\} _{i \in I}$ is a $\tau$-covering of $V$. Denote $(X_ i, \varphi _{ii'})_ V$ the restriction (or pullback) of the given descent datum to $\mathcal{U}_ V$. Hence by what we just saw we obtain a scheme $X_ V$ over $V$ whose canonical descent datum with respect to $\mathcal{U}_ V$ is isomorphic to $(X_ i, \varphi _{ii'})_ V$. Suppose that $V' \subset V$ is an affine open of $V$. Then both $X_{V'}$ and $V' \times _ V X_ V$ have canonical descent data isomorphic to $(X_ i, \varphi _{ii'})_{V'}$. Hence, by Lemma 35.32.11 again we obtain a canonical morphism $\rho ^ V_{V'} : X_{V'} \to X_ V$ over $S$ which identifies $X_{V'}$ with the inverse image of $V'$ in $X_ V$. We omit the verification that given affine opens $V'' \subset V' \subset V$ of $S$ we have $\rho ^ V_{V''} = \rho ^ V_{V'} \circ \rho ^{V'}_{V''}$.

By Constructions, Lemma 27.2.1 the data $(X_ V, \rho ^ V_{V'})$ glue to a scheme $X \to S$. Moreover, we are given isomorphisms $V \times _ S X \to X_ V$ which recover the maps $\rho ^ V_{V'}$. Unwinding the construction of the schemes $X_ V$ we obtain isomorphisms

$V \times _ S U_ i \times _ S X \longrightarrow V \times _ S X_ i$

compatible with the maps $\varphi _{ii'}$ and compatible with restricting to smaller affine opens in $X$. This implies that the canonical descent datum on $U_ i \times _ S X$ is isomorphic to the given descent datum and we win. $\square$

Comment #4899 by Robot0079 on

In 3rd paragraph, we can't reduced to the single morphism-case unless adding some local property on P, since in general coproduct of morphisms satisfying P doesn't satisfy P anymore.

Comment #5176 by on

Thanks very much. The text following Definition 35.33.1 exactly warns the reader for the pitfall you point out in your comment! Fixed here.

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