Definition 35.31.1. Let $f : X \to S$ be a morphism of schemes.

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

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

commutes (with obvious notation).

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

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

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

commutes.

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