Presentations of algebraic spaces

A presentation of an algebraic space $F$ over $S$ (Algebraic spaces over a scheme, defined as fppf sheaves) is a pair consisting of an $S$-scheme $U$, an etale equivalence relation $j=(t,s)\text{\hspace{0.17em}⁣}:R\to U{\times }_{S}U$ on $U$ over $S$ (Groupoids in schemes, relations and etale equivalence relations) and a surjective etale morphism $U\to F$ such that $R=U{\times }_{F}U$, that is, such that $j$ identifies $R$ with the kernel pair of $U\to F$ (Fibre product of schemes). By Surjective etale maps from schemes give presentations every surjective etale morphism from a scheme to $F$ yields a presentation: the kernel pair $U{\times }_{F}U$ is an etale equivalence relation and $F$ is its quotient sheaf. Conversely a presentation determines $F$ as $U\mathrm{/}R$.

A presentation is quasi-compact when $U$ is quasi-compact. This depends on the chosen cover: under the inherited Axiom of Choice (The Axiom of Choice), a nonempty affine scheme $S$ has both the presentation $S\to S$ and the non-quasi-compact presentation ${\coprod }_{n\ge 0}S\to S$. The open components of the latter source have no finite subcover.

A presentation is separated, locally separated or locally quasi-finite when $j$ is respectively a closed immersion (Closed immersions of schemes), an immersion, or separated and locally quasi-finite. These diagonal conditions are independent of the presentation: $j$ is the base change of ${\mathrm{\Delta }}_F$ along the surjective etale cover $U{\times }_{S}U\to F{\times }_{S}F$, and closed immersions and immersions are fppf local on the target (Stacks, Descent Lemmas 35.23.21 and 35.24.1). Moreover the separated locally quasi-finite condition on $j$ holds for every presentation: $j$ is a monomorphism, hence separated, and is locally of finite type because $s$ is etale; its fibres have at most one point, so it is locally quasi-finite (Stacks Lemma 65.13.1). This condition concerns the diagonal and does not say that $F\to S$ is locally quasi-finite.