The Stacks project

Remark 85.7.5. Let $S$ be a scheme and let $(\mathit{Sch}/S)_{fppf}$ be a big fppf site as in Topologies, Definition 34.7.8. As our set theoretic condition on $X$ in Definitions 85.5.1 and 85.7.1 we take: there exist objects $U, R$ of $(\mathit{Sch}/S)_{fppf}$, a morphism $U \to X$ which is a surjection of fppf sheaves, and a morphism $R \to U \times _ X U$ which is a surjection of fppf sheaves. In other words, we require our sheaf to be a coequalizer of two maps between representable sheaves. Here are some observations which imply this notion behaves reasonably well:

  1. Suppose $X = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } X_\lambda $ and the system satisfies conditions (1) and (2) of Definition 85.5.1. Then $U = \coprod _{\lambda \in \Lambda } X_\lambda \to X$ is a surjection of fppf sheaves. Moreover, $U \times _ X U$ is a closed subscheme of $U \times _ S U$ by Lemma 85.5.2. Hence if $U$ is representable by an object of $(\mathit{Sch}/S)_{fppf}$ then $U \times _ S U$ is too (see Sets, Lemma 3.9.9) and the set theoretic condition is satisfied. This is always the case if $\Lambda $ is countable, see Sets, Lemma 3.9.9.

  2. Sanity check. Let $\{ X_ i \to X\} _{i \in I}$ be as in Definition 85.7.1 (with the set theoretic condition as formulated above) and assume that each $X_ i$ is actually an affine scheme. Then $X$ is an algebraic space. Namely, if we choose a larger big fppf site $(\mathit{Sch}'/S)_{fppf}$ such that $U' = \coprod X_ i$ and $R' = \coprod X_ i \times _ X X_ j$ are representable by objects in it, then $X' = U'/R'$ will be an object of the category of algebraic spaces for this choice. Then an application of Spaces, Lemma 63.15.2 shows that $X$ is an algebraic space for $(\mathit{Sch}/S)_{fppf}$.

  3. Let $\{ X_ i \to X\} _{i \in I}$ be a family of maps of sheaves satisfying conditions (1), (2), (3) of Definition 85.7.1. For each $i$ we can pick $U_ i \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and $U_ i \to X_ i$ which is a surjection of sheaves. Thus if $I$ is not too large (for example countable) then $U = \coprod U_ i \to X$ is a surjection of sheaves and $U$ is representable by an object of $(\mathit{Sch}/S)_{fppf}$. To get $R \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ surjecting onto $U \times _ X U$ it suffices to assume the diagonal $\Delta : X \to X \times _ S X$ is not too wild, for example this always works if the diagonal of $X$ is quasi-compact, i.e., $X$ is quasi-separated.


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