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

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).

We also say that the pair $(V/X, \varphi )$ is a

*descent datum relative to $X \to S$*.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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