The Stacks project

Lemma 41.20.4. Let $f : X \to S$ be a morphism of schemes. Let $(V, \varphi )$ be a descent datum relative to $X/S$ with $V \to X$ étale. Let $S = \bigcup S_ i$ be an open covering. Assume that

  1. the pullback of the descent datum $(V, \varphi )$ to $X \times _ S S_ i/S_ i$ is effective,

  2. the functor ( for $X \times _ S (S_ i \cap S_ j) \to (S_ i \cap S_ j)$ is fully faithful, and

  3. the functor ( for $X \times _ S (S_ i \cap S_ j \cap S_ k) \to (S_ i \cap S_ j \cap S_ k)$ is faithful.

Then $(V, \varphi )$ is effective.

Proof. (Recall that pullbacks of descent data are defined in Descent, Definition 35.34.7.) Set $X_ i = X \times _ S S_ i$. Denote $(V_ i, \varphi _ i)$ the pullback of $(V, \varphi )$ to $X_ i/S_ i$. By assumption (1) we can find an étale morphism $U_ i \to S_ i$ which comes with an isomorphism $X_ i \times _{S_ i} U_ i \to V_ i$ compatible with $can$ and $\varphi _ i$. By assumption (2) we obtain isomorphisms $\psi _{ij} : U_ i \times _{S_ i} (S_ i \cap S_ j) \to U_ j \times _{S_ j} (S_ i \cap S_ j)$. By assumption (3) these isomorphisms satisfy the cocycle condition so that $(U_ i, \psi _{ij})$ is a descend datum for the Zariski covering $\{ S_ i \to S\} $. Then Descent, Lemma 35.35.10 (which is essentially just a reformulation of Schemes, Section 26.14) tells us that there exists a morphism of schemes $U \to S$ and isomorphisms $U \times _ S S_ i \to U_ i$ compatible with $\psi _{ij}$. The isomorphisms $U \times _ S S_ i \to U_ i$ determine corresponding isomorphisms $X_ i \times _ S U \to V_ i$ which glue to a morphism $X \times _ S U \to V$ compatible with the canonical descent datum and $\varphi $. $\square$

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