The Stacks project

Definition 74.22.1. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.

1. Let $V \to Y$ be a morphism of algebraic spaces. A descent datum for $V/Y/X$ is an isomorphism $\varphi : V \times _ X Y \to Y \times _ X V$ of algebraic spaces over $Y \times _ X Y$ satisfying the cocycle condition that the diagram

   \[ \xymatrix{ V \times _ X Y \times _ X Y \ar[rd]^{\varphi _{01}} \ar[rr]_{\varphi _{02}} & & Y \times _ X Y \times _ X V\\ & Y \times _ X V \times _ X Y \ar[ru]^{\varphi _{12}} } \]

   commutes (with obvious notation).

2. We also say that the pair $(V/Y, \varphi )$ is a descent datum relative to $Y \to X$.

3. A morphism $f : (V/Y, \varphi ) \to (V'/Y, \varphi ')$ of descent data relative to $Y \to X$ is a morphism $f : V \to V'$ of algebraic spaces over $Y$ such that the diagram

   \[ \xymatrix{ V \times _ X Y \ar[r]_{\varphi } \ar[d]_{f \times \text{id}_ Y} & Y \times _ X V \ar[d]^{\text{id}_ Y \times f} \\ V' \times _ X Y \ar[r]^{\varphi '} & Y \times _ X V' } \]

   commutes.