Lemma 35.34.6. Pullback of descent data for schemes over schemes.

Let

\[ \xymatrix{ X' \ar[r]_ f \ar[d]_{a'} & X \ar[d]^ a \\ S' \ar[r]^ h & S } \]be a commutative diagram of morphisms of schemes. The construction

\[ (V \to X, \varphi ) \longmapsto f^*(V \to X, \varphi ) = (V' \to X', \varphi ') \]where $V' = X' \times _ X V$ and where $\varphi '$ is defined as the composition

\[ \xymatrix{ V' \times _{S'} X' \ar@{=}[r] & (X' \times _ X V) \times _{S'} X' \ar@{=}[r] & (X' \times _{S'} X') \times _{X \times _ S X} (V \times _ S X) \ar[d]^{\text{id} \times \varphi } \\ X' \times _{S'} V' \ar@{=}[r] & X' \times _{S'} (X' \times _ X V) & (X' \times _{S'} X') \times _{X \times _ S X} (X \times _ S V) \ar@{=}[l] } \]defines a functor from the category of descent data relative to $X \to S$ to the category of descent data relative to $X' \to S'$.

Given two morphisms $f_ i : X' \to X$, $i = 0, 1$ making the diagram commute the functors $f_0^*$ and $f_1^*$ are canonically isomorphic.

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