Lemma 65.6.2 (025Z)—The Stacks project

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Part 4: Algebraic Spaces
Chapter 65: Algebraic Spaces
Section 65.6: Algebraic spaces
Lemma 65.6.2 (cite)

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Lemma 65.6.2. A scheme is an algebraic space. More precisely, given a scheme $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ the representable functor $h_ T$ is an algebraic space.

Proof. The functor $h_ T$ is a sheaf by our remarks in Section 65.2. The diagonal $h_ T \to h_ T \times h_ T = h_{T \times T}$ is representable because $(\mathit{Sch}/S)_{fppf}$ has fibre products. The identity map $h_ T \to h_ T$ is surjective étale. $\square$

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