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37.55 Descending separated locally quasi-finite morphisms

In this section we show that “separated locally quasi-finite morphisms satisfy descent for fppf-coverings”. See Descent, Definition 35.36.1 for terminology. This is in the marvellous (for many reasons) paper by Raynaud and Gruson hidden in the proof of [Lemma 5.7.1, GruRay]. It can also be found in [Murre-representation], and [Exposé X, Lemma 5.4, SGA3] under the additional hypothesis that the morphism is locally of finite presentation. Here is the formal statement.

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. $\square$

Comments (2)

Comment #6249 by ToumaKazusa on

Why we have ? Though this fact may be dispensible in the proof.

Comment #6381 by on

The inclusion holds because is a descent datum relative to . Namely, this assumption implies that the restriction of to is the identity of , see discussion following Definition 35.34.1. This implies for a point with image and we have . To be pedantic: here is the point given by , and the map coming from the identiy on and the inclusion which we have as maps to .

If you are familiar with groupoids, then another way to see this is to recall that the descent datum produces a groupoid scheme, see Lemma 39.21.3, and recall the corresponding fact for groupoid schemes: if is a groupoid scheme and is a subset, then we have .

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