The Stacks project

Lemma 99.13.4. Let $T$ be an algebraic space over $\mathbf{Z}$. Let $\mathcal{S}_ T$ denote the corresponding algebraic stack (Algebraic Stacks, Sections 94.7, 94.8, and 94.13). We have an equivalence of categories

\[ \left\{ \begin{matrix} \text{morphisms of algebraic spaces } \\ X \to T\text{ of finite type} \end{matrix} \right\} \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathit{Sch}_{fppf}}(\mathcal{S}_ T, \mathcal{S}\! \mathit{paces}'_{ft}) \]

and an equivalence of categories

\[ \left\{ \begin{matrix} \text{morphisms of algebraic spaces }X \to T \\ \text{of finite presentation, flat, and proper} \end{matrix} \right\} \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathit{Sch}_{fppf}}(\mathcal{S}_ T, \mathcal{S}\! \mathit{paces}'_{fp, flat, proper}) \]

Proof. We are going to deduce this lemma from the fact that it holds for schemes (essentially by construction of the stacks) and the fact that fppf descent data for algebraic spaces over algerbaic spaces are effective. We strongly encourage the reader to skip the proof.

The construction from left to right in either arrow is straightforward: given $X \to T$ of finite type the functor $\mathcal{S}_ T \to \mathcal{S}\! \mathit{paces}'_{ft}$ assigns to $U/T$ the base change $X_ U \to U$. We will explain how to construct a quasi-inverse.

If $T$ is a scheme, then there is a quasi-inverse by the $2$-Yoneda lemma, see Categories, Lemma 4.41.2. Let $p : U \to T$ be a surjective ├ętale morphism where $U$ is a scheme. Let $R = U \times _ T U$ with projections $s, t : R \to U$. Observe that we obtain morphisms

\[ \xymatrix{ \mathcal{S}_{U \times _ T U \times _ T U} \ar@<2ex>[r] \ar[r] \ar@<-2ex>[r] & \mathcal{S}_ R \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{S}_ U \ar[r] & \mathcal{S}_ T } \]

satisfying various compatibilities (on the nose).

Let $G : \mathcal{S}_ T \to \mathcal{S}\! \mathit{paces}'_{ft}$ be a functor over $\mathit{Sch}_{fppf}$. The restriction of $G$ to $\mathcal{S}_ U$ via the map displayed above corresponds to a finite type morphism $X_ U \to U$ of algebraic spaces via the $2$-Yoneda lemma. Since $p \circ s = p \circ t$ we see that $R \times _{s, U} X_ U$ and $R \times _{t, U} X_ U$ both correspond to the restriction of $G$ to $\mathcal{S}_ R$. Thus we obtain a canonical isomorphism $\varphi : X_ U \times _{U, t} R \to R \times _{s, U} X_ U$ over $R$. This isomorphism satisfies the cocycle condition by the various compatibilities of the diagram given above. Thus a descent datum which is effective by Bootstrap, Lemma 80.11.3 part (2). In other words, we obtain an object $X \to T$ of the right hand side category. We omit checking the construction $G \leadsto X$ is functorial and that it is quasi-inverse to the other construction. 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 to see that $X \to T$ is of finite presentation, flat, and proper. $\square$

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