The Stacks project

95.12 The stack in groupoids of finite type algebraic spaces

Let $p : \mathcal{S}\! \mathit{paces}_{ft} \to (\mathit{Sch}/S)_{fppf}$ be the stack introduced in Section 95.8 (using the abuse of notation introduced there). We can turn this into a stack in groupoids $p' : \mathcal{S}\! \mathit{paces}_{ft}' \to (\mathit{Sch}/S)_{fppf}$ by the procedure of Categories, Lemma 4.35.3, see Stacks, Lemma 8.5.3. In this particular case this simply means $\mathcal{S}\! \mathit{paces}_{ft}'$ has the same objects as $\mathcal{S}\! \mathit{paces}_{ft}$, i.e., finite type morphisms $X \to U$ where $X$ is an algebraic space over $S$ and $U$ is a scheme over $S$. But the morphisms $(f, g) : X/U \to Y/V$ are now commutative diagrams

\[ \xymatrix{ X \ar[d] \ar[r]_ f & Y \ar[d] \\ U \ar[r]^ g & V } \]

which are cartesian.

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