Lemma of the day

Let S be a scheme. Let {Xi → X}i ∈ I be an fppf covering of algebraic spaces over S. Assume I is countable (we can allow larger index sets if we bound the size of the algebraic spaces or if we don’t worry about set theoretic issues). Then any descent datum for algebraic spaces relative to {Xi → X}i ∈ I is effective. See Lemma Tag 0ADV.

Slogan: fppf descent data for algebraic spaces are effective.