## 74.22 Descent data for spaces over spaces

This section is the analogue of Descent, Section 35.34 for algebraic spaces. Most of the arguments in this section are formal relying only on the definition of a descent datum.

Definition 74.22.1. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.

1. Let $V \to Y$ be a morphism of algebraic spaces. A descent datum for $V/Y/X$ is an isomorphism $\varphi : V \times _ X Y \to Y \times _ X V$ of algebraic spaces over $Y \times _ X Y$ satisfying the cocycle condition that the diagram

$\xymatrix{ V \times _ X Y \times _ X Y \ar[rd]^{\varphi _{01}} \ar[rr]_{\varphi _{02}} & & Y \times _ X Y \times _ X V\\ & Y \times _ X V \times _ X Y \ar[ru]^{\varphi _{12}} }$

commutes (with obvious notation).

2. We also say that the pair $(V/Y, \varphi )$ is a descent datum relative to $Y \to X$.

3. A morphism $f : (V/Y, \varphi ) \to (V'/Y, \varphi ')$ of descent data relative to $Y \to X$ is a morphism $f : V \to V'$ of algebraic spaces over $Y$ such that the diagram

$\xymatrix{ V \times _ X Y \ar[r]_{\varphi } \ar[d]_{f \times \text{id}_ Y} & Y \times _ X V \ar[d]^{\text{id}_ Y \times f} \\ V' \times _ X Y \ar[r]^{\varphi '} & Y \times _ X V' }$

commutes.

Remark 74.22.2. Let $S$ be a scheme. Let $Y \to X$ be a morphism of algebraic spaces over $S$. Let $(V/Y, \varphi )$ be a descent datum relative to $Y \to X$. We may think of the isomorphism $\varphi$ as an isomorphism

$(Y \times _ X Y) \times _{\text{pr}_0, Y} V \longrightarrow (Y \times _ X Y) \times _{\text{pr}_1, Y} V$

of algebraic spaces over $Y \times _ X Y$. So loosely speaking one may think of $\varphi$ as a map $\varphi : \text{pr}_0^*V \to \text{pr}_1^*V$1. The cocycle condition then says that $\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.

Here is the definition in case you have a family of morphisms with fixed target.

Definition 74.22.3. Let $S$ be a scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$.

1. A descent datum $(V_ i, \varphi _{ij})$ relative to the family $\{ X_ i \to X\}$ is given by an algebraic space $V_ i$ over $X_ i$ for each $i \in I$, an isomorphism $\varphi _{ij} : V_ i \times _ X X_ j \to X_ i \times _ X V_ j$ of algebraic spaces over $X_ i \times _ X X_ j$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram

$\xymatrix{ V_ i \times _ X X_ j \times _ X X_ k \ar[rd]^{\text{pr}_{01}^*\varphi _{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi _{ik}} & & X_ i \times _ X X_ j \times _ X V_ k\\ & X_ i \times _ X V_ j \times _ X X_ k \ar[ru]^{\text{pr}_{12}^*\varphi _{jk}} }$

of algebraic spaces over $X_ i \times _ X X_ j \times _ X X_ k$ commutes (with obvious notation).

2. A morphism $\psi : (V_ i, \varphi _{ij}) \to (V'_ i, \varphi '_{ij})$ of descent data is given by a family $\psi = (\psi _ i)_{i \in I}$ of morphisms $\psi _ i : V_ i \to V'_ i$ of algebraic spaces over $X_ i$ such that all the diagrams

$\xymatrix{ V_ i \times _ X X_ j \ar[r]_{\varphi _{ij}} \ar[d]_{\psi _ i \times \text{id}} & X_ i \times _ X V_ j \ar[d]^{\text{id} \times \psi _ j} \\ V'_ i \times _ X X_ j \ar[r]^{\varphi '_{ij}} & X_ i \times _ X V'_ j }$

commute.

Remark 74.22.4. Let $S$ be a scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. Let $(V_ i, \varphi _{ij})$ be a descent datum relative to $\{ X_ i \to X\}$. We may think of the isomorphisms $\varphi _{ij}$ as isomorphisms

$(X_ i \times _ X X_ j) \times _{\text{pr}_0, X_ i} V_ i \longrightarrow (X_ i \times _ X X_ j) \times _{\text{pr}_1, X_ j} V_ j$

of algebraic spaces over $X_ i \times _ X X_ j$. So loosely speaking one may think of $\varphi _{ij}$ as an isomorphism $\text{pr}_0^*V_ i \to \text{pr}_1^*V_ j$ over $X_ i \times _ X X_ j$. The cocycle condition then says that $\text{pr}_{02}^*\varphi _{ik} = \text{pr}_{12}^*\varphi _{jk} \circ \text{pr}_{01}^*\varphi _{ij}$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.

The reason we will usually work with the version of a family consisting of a single morphism is the following lemma.

Lemma 74.22.5. Let $S$ be a scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. Set $Y = \coprod _{i \in I} X_ i$. There is a canonical equivalence of categories

$\begin{matrix} \text{category of descent data } \\ \text{relative to the family } \{ X_ i \to X\} _{i \in I} \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } Y/X \end{matrix}$

which maps $(V_ i, \varphi _{ij})$ to $(V, \varphi )$ with $V = \coprod _{i\in I} V_ i$ and $\varphi = \coprod \varphi _{ij}$.

Proof. Observe that $Y \times _ X Y = \coprod _{ij} X_ i \times _ X X_ j$ and similarly for higher fibre products. Giving a morphism $V \to Y$ is exactly the same as giving a family $V_ i \to X_ i$. And giving a descent datum $\varphi$ is exactly the same as giving a family $\varphi _{ij}$. $\square$

Lemma 74.22.6. Pullback of descent data. Let $S$ be a scheme.

1. Let

$\xymatrix{ Y' \ar[r]_ f \ar[d]_{a'} & Y \ar[d]^ a \\ X' \ar[r]^ h & X }$

be a commutative diagram of algebraic spaces over $S$. The construction

$(V \to Y, \varphi ) \longmapsto f^*(V \to Y, \varphi ) = (V' \to Y', \varphi ')$

where $V' = Y' \times _ Y V$ and where $\varphi '$ is defined as the composition

$\xymatrix{ V' \times _{X'} Y' \ar@{=}[r] & (Y' \times _ Y V) \times _{X'} Y' \ar@{=}[r] & (Y' \times _{X'} Y') \times _{Y \times _ X Y} (V \times _ X Y) \ar[d]^{\text{id} \times \varphi } \\ Y' \times _{X'} V' \ar@{=}[r] & Y' \times _{X'} (Y' \times _ Y V) & (Y' \times _ X Y') \times _{Y \times _ X Y} (Y \times _ X V) \ar@{=}[l] }$

defines a functor from the category of descent data relative to $Y \to X$ to the category of descent data relative to $Y' \to X'$.

2. Given two morphisms $f_ i : Y' \to Y$, $i = 0, 1$ making the diagram commute the functors $f_0^*$ and $f_1^*$ are canonically isomorphic.

Proof. We omit the proof of (1), but we remark that the morphism $\varphi '$ is the morphism $(f \times f)^*\varphi$ in the notation introduced in Remark 74.22.2. For (2) we indicate which morphism $f_0^*V \to f_1^*V$ gives the functorial isomorphism. Namely, since $f_0$ and $f_1$ both fit into the commutative diagram we see there is a unique morphism $r : Y' \to Y \times _ X Y$ with $f_ i = \text{pr}_ i \circ r$. Then we take

\begin{eqnarray*} f_0^*V & = & Y' \times _{f_0, Y} V \\ & = & Y' \times _{\text{pr}_0 \circ r, Y} V \\ & = & Y' \times _{r, Y \times _ X Y} (Y \times _ X Y) \times _{\text{pr}_0, Y} V \\ & \xrightarrow {\varphi } & Y' \times _{r, Y \times _ X Y} (Y \times _ X Y) \times _{\text{pr}_1, Y} V \\ & = & Y' \times _{\text{pr}_1 \circ r, Y} V \\ & = & Y' \times _{f_1, Y} V \\ & = & f_1^*V \end{eqnarray*}

We omit the verification that this works. $\square$

Definition 74.22.7. With $S, X, X', Y, Y', f, a, a', h$ as in Lemma 74.22.6 the functor

$(V, \varphi ) \longmapsto f^*(V, \varphi )$

constructed in that lemma is called the pullback functor on descent data.

Lemma 74.22.8. Let $S$ be a scheme. Let $\mathcal{U}' = \{ X'_ i \to X'\} _{i \in I'}$ and $\mathcal{U} = \{ X_ j \to X\} _{i \in I}$ be families of morphisms with fixed target. Let $\alpha : I' \to I$, $g : X' \to X$ and $g_ i : X'_ i \to X_{\alpha (i)}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1.

1. Let $(V_ i, \varphi _{ij})$ be a descent datum relative to the family $\mathcal{U}$. The system

$\left( g_ i^*V_{\alpha (i)}, (g_ i \times g_ j)^*\varphi _{\alpha (i) \alpha (j)} \right)$

(with notation as in Remark 74.22.4) is a descent datum relative to $\mathcal{U}'$.

2. This construction defines a functor between the category of descent data relative to $\mathcal{U}$ and the category of descent data relative to $\mathcal{U}'$.

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

4. These functors agree, via Lemma 74.22.5, with the pullback functors constructed in Lemma 74.22.6.

Proof. This follows from Lemma 74.22.6 via the correspondence of Lemma 74.22.5. $\square$

Definition 74.22.9. With $\mathcal{U}' = \{ X'_ i \to X'\} _{i \in I'}$, $\mathcal{U} = \{ X_ i \to X\} _{i \in I}$, $\alpha : I' \to I$, $g : X' \to X$, and $g_ i : X'_ i \to X_{\alpha (i)}$ as in Lemma 74.22.8 the functor

$(V_ i, \varphi _{ij}) \longmapsto (g_ i^*V_{\alpha (i)}, (g_ i \times g_ j)^*\varphi _{\alpha (i) \alpha (j)})$

constructed in that lemma is called the pullback functor on descent data.

If $\mathcal{U}$ and $\mathcal{U}'$ have the same target $X$, and if $\mathcal{U}'$ refines $\mathcal{U}$ (see Sites, Definition 7.8.1) but no explicit pair $(\alpha , g_ i)$ is given, then we can still talk about the pullback functor since we have seen in Lemma 74.22.8 that the choice of the pair does not matter (up to a canonical isomorphism).

Definition 74.22.10. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.

1. Given an algebraic space $U$ over $X$ we have the trivial descent datum of $U$ relative to $\text{id} : X \to X$, namely the identity morphism on $U$.

2. By Lemma 74.22.6 we get a canonical descent datum on $Y \times _ X U$ relative to $Y \to X$ by pulling back the trivial descent datum via $f$. We often denote $(Y \times _ X U, can)$ this descent datum.

3. A descent datum $(V, \varphi )$ relative to $Y/X$ is called effective if $(V, \varphi )$ is isomorphic to the canonical descent datum $(Y \times _ X U, can)$ for some algebraic space $U$ over $X$.

Thus being effective means there exists an algebraic space $U$ over $X$ and an isomorphism $\psi : V \to Y \times _ X U$ over $Y$ such that $\varphi$ is equal to the composition

$V \times _ X Y \xrightarrow {\psi \times \text{id}_ Y} Y \times _ X U \times _ S Y = Y \times _ X Y \times _ X U \xrightarrow {\text{id}_ Y \times \psi ^{-1}} Y \times _ X V$

There is a slight problem here which is that this definition (in spirit) conflicts with the definition given in Descent, Definition 35.34.10 in case $Y$ and $X$ are schemes. However, it will always be clear from context which version we mean.

Definition 74.22.11. Let $S$ be a scheme. Let $\{ X_ i \to X\}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$.

1. Given an algebraic space $U$ over $X$ we have a canonical descent datum on the family of algebraic spaces $X_ i \times _ X U$ by pulling back the trivial descent datum for $U$ relative to $\{ \text{id} : S \to S\}$. We denote this descent datum $(X_ i \times _ X U, can)$.

2. A descent datum $(V_ i, \varphi _{ij})$ relative to $\{ X_ i \to S\}$ is called effective if there exists an algebraic space $U$ over $X$ such that $(V_ i, \varphi _{ij})$ is isomorphic to $(X_ i \times _ X U, can)$.

[1] Unfortunately, we have chosen the “wrong” direction for our arrow here. In Definitions 74.22.1 and 74.22.3 we should have the opposite direction to what was done in Definition 74.3.1 by the general principle that “functions” and “spaces” are dual.

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