The Stacks project

Lemma 35.34.8 (Pullback of descent data for schemes over families). Let $\mathcal{U} = \{ U_ i \to S'\} _{i \in I}$ and $\mathcal{V} = \{ V_ j \to S\} _{j \in J}$ be families of morphisms with fixed target. Let $\alpha : I \to J$, $h : S' \to S$ and $g_ i : U_ i \to V_{\alpha (i)}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1.

  1. Let $(Y_ j, \varphi _{jj'})$ be a descent datum relative to the family $\{ V_ j \to S'\} $. The system

    \[ \left( g_ i^*Y_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')} \right) \]

    (with notation as in Remark 35.34.4) is a descent datum relative to $\mathcal{V}$.

  2. This construction defines a functor between descent data relative to $\mathcal{U}$ and descent data relative to $\mathcal{V}$.

  3. Given a second $\alpha ' : I \to J$, $h' : S' \to S$ and $g'_ i : U_ i \to V_{\alpha '(i)}$ morphism of families of maps with fixed target, then if $h = h'$ the two resulting functors between descent data are canonically isomorphic.

  4. These functors agree, via Lemma 35.34.5, with the pullback functors constructed in Lemma 35.34.6.

Proof. This follows from Lemma 35.34.6 via the correspondence of Lemma 35.34.5. $\square$

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