The Stacks project

Lemma 35.2.2. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ and $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ be families of morphisms of schemes with fixed target. Let $(g, \alpha : I \to J, (g_ i)) : \mathcal{U} \to \mathcal{V}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1. Let $(\mathcal{F}_ j, \varphi _{jj'})$ be a descent datum for quasi-coherent sheaves with respect to the family $\{ V_ j \to V\} _{j \in J}$. Then

  1. The system

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

    is a descent datum with respect to the family $\{ U_ i \to U\} _{i \in I}$.

  2. This construction is functorial in the descent datum $(\mathcal{F}_ j, \varphi _{jj'})$.

  3. Given a second morphism $(g', \alpha ' : I \to J, (g'_ i))$ of families of maps with fixed target with $g = g'$ there exists a functorial isomorphism of descent data

    \[ (g_ i^*\mathcal{F}_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')}) \cong ((g'_ i)^*\mathcal{F}_{\alpha '(i)}, (g'_ i \times g'_{i'})^*\varphi _{\alpha '(i)\alpha '(i')}). \]

Proof. Omitted. Hint: The maps $g_ i^*\mathcal{F}_{\alpha (i)} \to (g'_ i)^*\mathcal{F}_{\alpha '(i)}$ which give the isomorphism of descent data in part (3) are the pullbacks of the maps $\varphi _{\alpha (i)\alpha '(i)}$ by the morphisms $(g_ i, g'_ i) : U_ i \to V_{\alpha (i)} \times _ V V_{\alpha '(i)}$. $\square$

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