Lemma 80.11.1. Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Let $\{ S_ i \to S\} _{i \in I}$ be a covering of $(\mathit{Sch}/S)_{fppf}$. Assume that

$F$ is a sheaf,

each $F_ i = h_{S_ i} \times F$ is an algebraic space, and

$\coprod _{i \in I} F_ i$ is an algebraic space (see Spaces, Lemma 65.8.4).

Then $F$ is an algebraic space.

## Comments (1)

Comment #1225 by David Corwin on