The Stacks project

Lemma 35.34.2. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i\in I}$ be an fpqc covering, see Topologies, Definition 34.9.1. 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 a closed immersion, then the descent datum is effective.

Proof. This is true because a closed immersion is an affine morphism (Morphisms, Lemma 29.11.9), and hence Lemma 35.34.1 applies. $\square$


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