The Stacks project

Lemma 63.20.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There is an anti-equivalence of categories

\[ \begin{matrix} \text{algebraic spaces} \\ \text{affine over }X \end{matrix} \longleftrightarrow \begin{matrix} \text{quasi-coherent sheaves} \\ \text{of }\mathcal{O}_ X\text{-algebras} \end{matrix} \]

which associates to $f : Y \to X$ the sheaf $f_*\mathcal{O}_ Y$. Moreover, this equivalence is compatible with arbitrary base change.

Proof. This lemma is the analogue of Morphisms, Lemma 28.11.5. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. We will construct an affine morphism of algebraic spaces $\pi : Y = \underline{\mathop{\mathrm{Spec}}}_ X(\mathcal{A}) \to X$ with $\pi _*\mathcal{O}_ Y \cong \mathcal{A}$. To do this, choose a scheme $U$ and a surjective ├ętale morphism $\varphi : U \to X$. As usual denote $R = U \times _ X U$ with projections $s, t : R \to U$. Denote $\psi : R \to X$ the composition $\psi = \varphi \circ s = \varphi \circ t$. By the aforementioned lemma there exists an affine morphisms of schemes $\pi _0 : V \to U$ and $\pi _1 : W \to R$ with $\pi _{0, *}\mathcal{O}_ V \cong \varphi ^*\mathcal{A}$ and $\pi _{1, *}\mathcal{O}_ W \cong \psi ^*\mathcal{A}$. Since the construction is compatible with base change there exist morphisms $s', t' : W \to V$ such that the diagrams

\[ \vcenter { \xymatrix{ W \ar[r]_{s'} \ar[d] & V \ar[d] \\ R \ar[r]^ s & U } } \quad \text{and}\quad \vcenter { \xymatrix{ W \ar[r]_{t'} \ar[d] & V \ar[d] \\ R \ar[r]^ t & U } } \]

are cartesian. It follows that $s', t'$ are ├ętale. It is a formal consequence of the above that $(t', s') : W \to V \times _ S V$ is a monomorphism. We omit the verification that $W \to V \times _ S V$ is an equivalence relation (hint: think about the pullback of $\mathcal{A}$ to $U \times _ X U \times _ X U = R \times _{s, U, t} R$). The quotient sheaf $Y = V/W$ is an algebraic space, see Spaces, Theorem 61.10.5. By Groupoids, Lemma 38.20.7 we see that $Y \times _ X U \cong V$. Hence $Y \to X$ is affine by Lemma 63.20.3. Finally, the isomorphism of

\[ (Y \times _ X U \to U)_*\mathcal{O}_{Y \times _ X U} = \pi _{0, *}\mathcal{O}_ V \cong \varphi ^*\mathcal{A} \]

is compatible with glueing isomorphisms, whence $(Y \to X)_*\mathcal{O}_ Y \cong \mathcal{A}$ by Properties of Spaces, Proposition 62.32.1. We omit the verification that this construction is compatible with base change. $\square$


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