74.3 Descent data for quasi-coherent sheaves
This section is the analogue of Descent, Section 35.2 for algebraic spaces. It makes sense to read that section first.
Definition 74.3.1. Let S be a scheme. Let \{ f_ i : X_ i \to X\} _{i \in I} be a family of morphisms of algebraic spaces over S with fixed target X.
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 X_ i for each i \in I, an isomorphism of quasi-coherent \mathcal{O}_{X_ i \times _ X X_ 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}_{X_ i \times _ X X_ j \times _ X X_ k}-modules commutes. This is called the cocycle condition.
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}_{X_ 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 \\ }
commute.
Lemma 74.3.2. Let S be a scheme. 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 algebraic spaces over S with fixed targets. 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
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}.
This construction is functorial in the descent datum (\mathcal{F}_ j, \varphi _{jj'}).
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
Let g : U \to V be a morphism of algebraic spaces. The lemma above tells us that there is a well defined pullback functor between the categories of descent data relative to families of maps with target V and U provided there is a morphism between those families of maps which “lives over g”.
Definition 74.3.3. Let S be a scheme. Let \{ U_ i \to U\} _{i \in I} be a family of morphisms of algebraic spaces over S with fixed target.
Let \mathcal{F} be a quasi-coherent \mathcal{O}_ U-module. We call the unique descent on \mathcal{F} datum with respect to the covering \{ U \to U\} the trivial descent datum.
The pullback of the trivial descent datum to \{ U_ i \to U\} is called the canonical descent datum. Notation: (\mathcal{F}|_{U_ i}, can).
A descent datum (\mathcal{F}_ i, \varphi _{ij}) for quasi-coherent sheaves with respect to the given family is said to be effective if there exists a quasi-coherent sheaf \mathcal{F} on U such that (\mathcal{F}_ i, \varphi _{ij}) is isomorphic to (\mathcal{F}|_{U_ i}, can).
Lemma 74.3.4. Let S be a scheme. Let U be an algebraic space over S. Let \{ U_ i \to U\} be a Zariski covering of U, see Topologies on Spaces, Definition 73.3.1. Any descent datum on quasi-coherent sheaves for the family \mathcal{U} = \{ U_ i \to U\} is effective. Moreover, the functor from the category of quasi-coherent \mathcal{O}_ U-modules to the category of descent data with respect to \{ U_ i \to U\} is fully faithful.
Proof.
Omitted.
\square
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