The Stacks project

Lemma 99.13.3. The category $\mathcal{S}\! \mathit{paces}'_{ft}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$. The same is true for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$.

Proof. The reason this lemma holds is the slogan: any fppf descent datum for algebraic spaces is effective, see Bootstrap, Section 80.11. More precisely, the lemma for $\mathcal{S}\! \mathit{paces}'_{ft}$ follows from Examples of Stacks, Lemma 95.8.1 as we saw in Examples of Stacks, Section 95.12. However, let us review the proof. We need to check conditions (1), (2), and (3) of Stacks, Definition 8.5.1.

Property (1) we have seen in Lemma 99.13.1.

Property (2) follows from Lemma 99.13.2 in the case of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$. In the case of $\mathcal{S}\! \mathit{paces}'_{ft}$ it follows from Examples of Stacks, Lemma 95.7.2 (and this is really the “correct” reference).

Condition (3) for $\mathcal{S}\! \mathit{paces}'_{ft}$ is checked as follows. Suppose given

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

  2. for each $i \in I$ an algebraic space $X_ i$ of finite type over $U_ i$, 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 $U_ i \times _ U U_ j \times _ U U_ k$.

We have to show there exists an algebraic space $X$ of finite type over $U$ and isomorphisms $X_{U_ i} \cong X_ i$ over $U_ i$ recovering the isomorphisms $\varphi _{ij}$. This follows from Bootstrap, Lemma 80.11.3 part (2). By Descent on Spaces, Lemma 74.11.11 we see that $X \to U$ is of finite type. In the case of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ one additionally uses Descent on Spaces, Lemma 74.11.12, 74.11.13, and 74.11.19 in the last step. $\square$

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