The Stacks project

Lemma 80.11.1. In Situation 80.10.6 the functor (80.11.0.1) restricts to an equivalence

  1. from the category of algebraic spaces affine over $X$ to the full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ affine,

  2. from the category of closed immersions $X' \to X$ to the full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ closed immersions, and

  3. same statement as in (2) for finite morphisms.

Proof. The category of algebraic spaces affine over $X$ is equivalent to the category of quasi-coherent sheaves $\mathcal{A}$ of $\mathcal{O}_ X$-algebras. The full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ affine is equivalent to the category of algebra objects of $\mathit{QCoh}(Y \to X, Z)$. In both cases this follows from Morphisms of Spaces, Lemma 66.20.7 with quasi-inverse given by the relative spectrum construction (Morphisms of Spaces, Definition 66.20.8) which commutes with arbitrary base change. Thus part (1) of the lemma follows from Proposition 80.10.9.

Fully faithfulness in part (2) follows from part (1). For essential surjectivity, we reduce by part (1) to proving that $X' \to X$ is a closed immersion if and only if both $U \times _ X X' \to U$ and $Y \times _ X X' \to Y$ are closed immersions. By Lemma 80.10.11 $\{ U \to X, Y \to X\} $ can be refined by an fpqc covering. Hence the result follows from Descent on Spaces, Lemma 73.11.17.

For (3) use the argument proving (2) and Descent on Spaces, Lemma 73.11.23. $\square$


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