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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