Lemma 37.55.1. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i\in I}$ be an fppf covering, see Topologies, Definition 34.7.1. Let $(V_ i/X_ i, \varphi _{ij})$ be a descent datum relative to $\{ X_ i \to S\}$. If each morphism $V_ i \to X_ i$ is separated and locally quasi-finite, then the descent datum is effective.

Proof. Being separated and being locally quasi-finite are properties of morphisms of schemes which are preserved under any base change, see Schemes, Lemma 26.21.12 and Morphisms, Lemma 29.20.13. Hence Descent, Lemma 35.36.2 applies and it suffices to prove the statement of the lemma in case the fppf-covering is given by a single $\{ X \to S\}$ flat surjective morphism of finite presentation of affines. Say $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi )$ be a descent datum relative to $X$ over $S$ and assume that $\pi : V \to X$ is separated and locally quasi-finite.

Let $W^1 \subset V$ be any affine open. Consider $W = \text{pr}_1(\varphi (W^1 \times _ S X)) \subset V$. Here is a picture

$\xymatrix{ W^1 \times _ S X \ar[rrrrr] \ar[ddd] \ar[rd] & & & & & \varphi (W^1 \times _ S X) \ar[ddd] \ar[ld] \\ & V \times _ S X \ar[rrr]^\varphi \ar[rd] \ar[dd] & & & X \times _ S V \ar[ld] \ar[dd] & \\ & & X \times _ S X \ar[r]^1 \ar[d]_{\text{pr}_0} & X \times _ S X \ar[d]^{\text{pr}_1} & & \\ W^1 \ar[r] & V \ar[r] & X & X & V \ar[l] & W \ar[l] }$

Ok, and now since $X \to S$ is flat and of finite presentation it is universally open (Morphisms, Lemma 29.25.10). Hence we conclude that $W$ is open. Moreover, it is also clearly the case that $W$ is quasi-compact, and $W^1 \subset W$. Moreover, we note that $\varphi (W \times _ S X) = X \times _ S W$ by the cocycle condition for $\varphi$. Hence we obtain a new descent datum $(W, \varphi ')$ by restricting $\varphi$ to $W \times _ S X$. Note that the morphism $W \to X$ is quasi-compact, separated and locally quasi-finite. This implies that it is separated and quasi-finite by definition. Hence it is quasi-affine by Lemma 37.43.2. Thus by Descent, Lemma 35.38.1 we see that the descent datum $(W, \varphi ')$ is effective.

In other words, we find that there exists an open covering $V = \bigcup W_ i$ by quasi-compact opens $W_ i$ which are stable for the descent morphism $\varphi$. Moreover, for each such quasi-compact open $W \subset V$ the corresponding descent data $(W, \varphi ')$ is effective. This means the original descent datum is effective by glueing the schemes obtained from descending the opens $W_ i$, see Descent, Lemma 35.35.13. $\square$

Comment #474 by Kestutis Cesnavicius on

The notation is almost the opposite of http://stacks.math.columbia.edu/tag/02W2 that is referenced just before for comparison. I wonder if it is worth changing it for consistency?

Comment #487 by on

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