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35.2 Descent data for quasi-coherent sheaves

In this chapter we will use the convention where the projection maps $\text{pr}_ i : X \times \ldots \times X \to X$ are labeled starting with $i = 0$. Hence we have $\text{pr}_0, \text{pr}_1 : X \times X \to X$, $\text{pr}_0, \text{pr}_1, \text{pr}_2 : X \times X \times X \to X$, etc.

Definition 35.2.1. Let $S$ be a scheme. Let $\{ f_ i : S_ i \to S\} _{i \in I}$ be a family of morphisms with target $S$.

  1. A descent datum $(\mathcal{F}_ i, \varphi _{ij})$ for quasi-coherent sheaves with respect to the given family is given by a quasi-coherent sheaf $\mathcal{F}_ i$ on $S_ i$ for each $i \in I$, an isomorphism of quasi-coherent $\mathcal{O}_{S_ i \times _ S S_ j}$-modules $\varphi _{ij} : \text{pr}_0^*\mathcal{F}_ i \to \text{pr}_1^*\mathcal{F}_ j$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram

    \[ \xymatrix{ \text{pr}_0^*\mathcal{F}_ i \ar[rd]_{\text{pr}_{01}^*\varphi _{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi _{ik}} & & \text{pr}_2^*\mathcal{F}_ k \\ & \text{pr}_1^*\mathcal{F}_ j \ar[ru]_{\text{pr}_{12}^*\varphi _{jk}} & } \]

    of $\mathcal{O}_{S_ i \times _ S S_ j \times _ S S_ k}$-modules commutes. This is called the cocycle condition.

  2. A morphism $\psi : (\mathcal{F}_ i, \varphi _{ij}) \to (\mathcal{F}'_ i, \varphi '_{ij})$ of descent data is given by a family $\psi = (\psi _ i)_{i\in I}$ of morphisms of $\mathcal{O}_{S_ i}$-modules $\psi _ i : \mathcal{F}_ i \to \mathcal{F}'_ i$ such that all the diagrams

    \[ \xymatrix{ \text{pr}_0^*\mathcal{F}_ i \ar[r]_{\varphi _{ij}} \ar[d]_{\text{pr}_0^*\psi _ i} & \text{pr}_1^*\mathcal{F}_ j \ar[d]^{\text{pr}_1^*\psi _ j} \\ \text{pr}_0^*\mathcal{F}'_ i \ar[r]^{\varphi '_{ij}} & \text{pr}_1^*\mathcal{F}'_ j \\ } \]


A good example to keep in mind is the following. Suppose that $S = \bigcup S_ i$ is an open covering. In that case we have seen descent data for sheaves of sets in Sheaves, Section 6.33 where we called them “glueing data for sheaves of sets with respect to the given covering”. Moreover, we proved that the category of glueing data is equivalent to the category of sheaves on $S$. We will show the analogue in the setting above when $\{ S_ i \to S\} _{i\in I}$ is an fpqc covering.

In the extreme case where the covering $\{ S \to S\} $ is given by $\text{id}_ S$ a descent datum is necessarily of the form $(\mathcal{F}, \text{id}_\mathcal {F})$. The cocycle condition guarantees that the identity on $\mathcal{F}$ is the only permitted map in this case. The following lemma shows in particular that to every quasi-coherent sheaf of $\mathcal{O}_ S$-modules there is associated a unique descent datum with respect to any given family.

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 $(g_ i, g'_ i) : U_ i \to V_{\alpha (i)} \times _ V V_{\alpha '(i)}$. $\square$

Any family $\mathcal{U} = \{ S_ i \to S\} _{i \in I}$ is a refinement of the trivial covering $\{ S \to S\} $ in a unique way. For a quasi-coherent sheaf $\mathcal{F}$ on $S$ we denote simply $(\mathcal{F}|_{S_ i}, can)$ the descent datum with respect to $\mathcal{U}$ obtained by the procedure above.

Definition 35.2.3. Let $S$ be a scheme. Let $\{ S_ i \to S\} _{i \in I}$ be a family of morphisms with target $S$.

  1. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module. We call the unique descent on $\mathcal{F}$ datum with respect to the covering $\{ S \to S\} $ the trivial descent datum.

  2. The pullback of the trivial descent datum to $\{ S_ i \to S\} $ is called the canonical descent datum. Notation: $(\mathcal{F}|_{S_ i}, can)$.

  3. A descent datum $(\mathcal{F}_ i, \varphi _{ij})$ for quasi-coherent sheaves with respect to the given covering is said to be effective if there exists a quasi-coherent sheaf $\mathcal{F}$ on $S$ such that $(\mathcal{F}_ i, \varphi _{ij})$ is isomorphic to $(\mathcal{F}|_{S_ i}, can)$.

Lemma 35.2.4. Let $S$ be a scheme. Let $S = \bigcup U_ i$ be an open covering. Any descent datum on quasi-coherent sheaves for the family $\mathcal{U} = \{ U_ i \to S\} $ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_ S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.

Proof. This follows immediately from Sheaves, Section 6.33 and the fact that being quasi-coherent is a local property, see Modules, Definition 17.10.1. $\square$

To prove more we first need to study the case of modules over rings.

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