The Stacks project

Remark 73.22.4. Let $S$ be a scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. Let $(V_ i, \varphi _{ij})$ be a descent datum relative to $\{ X_ i \to X\} $. We may think of the isomorphisms $\varphi _{ij}$ as isomorphisms

\[ (X_ i \times _ X X_ j) \times _{\text{pr}_0, X_ i} V_ i \longrightarrow (X_ i \times _ X X_ j) \times _{\text{pr}_1, X_ j} V_ j \]

of algebraic spaces over $X_ i \times _ X X_ j$. So loosely speaking one may think of $\varphi _{ij}$ as an isomorphism $\text{pr}_0^*V_ i \to \text{pr}_1^*V_ j$ over $X_ i \times _ X X_ j$. The cocycle condition then says that $\text{pr}_{02}^*\varphi _{ik} = \text{pr}_{12}^*\varphi _{jk} \circ \text{pr}_{01}^*\varphi _{ij}$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.


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