The Stacks project

Remark 94.7.3. Ignoring set theoretical difficulties1 $\mathcal{S}\! \mathit{paces}$ also satisfies descent for objects and hence is a stack. Namely, we have to show that given

  1. an fppf covering $\{ U_ i \to U\} _{i \in I}$,

  2. for each $i \in I$ an algebraic space $X_ i/U_ i$, and

  3. for each $i, j \in I$ an isomorphism $\varphi _{ij} : X_ i \times _ U U_ j \to U_ i \times _ U X_ j$ of algebraic spaces over $U_ i \times _ U U_ j$ satisfying the cocycle condition over $U_ i \times _ U U_ j \times _ U U_ k$,

there exists an algebraic space $X/U$ and isomorphisms $X_{U_ i} \cong X_ i$ over $U_ i$ recovering the isomorphisms $\varphi _{ij}$. First, note that by Sites, Lemma 7.26.4 there exists a sheaf $X$ on $(\mathit{Sch}/U)_{fppf}$ recovering the $X_ i$ and the $\varphi _{ij}$. Then by Bootstrap, Lemma 79.11.1 we see that $X$ is an algebraic space (if we ignore the set theoretic condition of that lemma). We will use this argument in the next section to show that if we consider only algebraic spaces of finite type, then we obtain a stack.

[1] The difficulty is not that $\mathcal{S}\! \mathit{paces}$ is a proper class, since by our definition of an algebraic space over $S$ there is only a set worth of isomorphism classes of algebraic spaces over $S$. It is rather that arbitrary disjoint unions of algebraic spaces may end up being too large, hence lie outside of our chosen “partial universe” of sets.

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