35.34 Descent data for schemes over schemes
Most of the arguments in this section are formal relying only on the definition of a descent datum. In Simplicial Spaces, Section 85.27 we will examine the relationship with simplicial schemes which will somewhat clarify the situation.
Definition 35.34.1. Let $f : X \to S$ be a morphism of schemes.
Let $V \to X$ be a scheme over $X$. A descent datum for $V/X/S$ is an isomorphism $\varphi : V \times _ S X \to X \times _ S V$ of schemes over $X \times _ S X$ satisfying the cocycle condition that the diagram
\[ \xymatrix{ V \times _ S X \times _ S X \ar[rd]^{\varphi _{01}} \ar[rr]_{\varphi _{02}} & & X \times _ S X \times _ S V\\ & X \times _ S V \times _ S X \ar[ru]^{\varphi _{12}} } \]
commutes (with obvious notation).
We also say that the pair $(V/X, \varphi )$ is a descent datum relative to $X \to S$.
A morphism $f : (V/X, \varphi ) \to (V'/X, \varphi ')$ of descent data relative to $X \to S$ is a morphism $f : V \to V'$ of schemes over $X$ such that the diagram
\[ \xymatrix{ V \times _ S X \ar[r]_{\varphi } \ar[d]_{f \times \text{id}_ X} & X \times _ S V \ar[d]^{\text{id}_ X \times f} \\ V' \times _ S X \ar[r]^{\varphi '} & X \times _ S V' } \]
commutes.
There are all kinds of “miraculous” identities which arise out of the definition above. For example the pullback of $\varphi $ via the diagonal morphism $\Delta : X \to X \times _ S X$ can be seen as a morphism $\Delta ^*\varphi : V \to V$. This because $X \times _{\Delta , X \times _ S X} (V \times _ S X) = V$ and also $X \times _{\Delta , X \times _ S X} (X \times _ S V) = V$. In fact, $\Delta ^*\varphi $ is equal to the identity. This is a good exercise if you are unfamiliar with this material.
Here is the definition in case you have a family of morphisms with fixed target.
Definition 35.34.3. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i \in I}$ be a family of morphisms with target $S$.
A descent datum $(V_ i, \varphi _{ij})$ relative to the family $\{ X_ i \to S\} $ is given by a scheme $V_ i$ over $X_ i$ for each $i \in I$, an isomorphism $\varphi _{ij} : V_ i \times _ S X_ j \to X_ i \times _ S V_ j$ of schemes over $X_ i \times _ S X_ j$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram
\[ \xymatrix{ V_ i \times _ S X_ j \times _ S X_ k \ar[rd]^{\text{pr}_{01}^*\varphi _{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi _{ik}} & & X_ i \times _ S X_ j \times _ S V_ k\\ & X_ i \times _ S V_ j \times _ S X_ k \ar[ru]^{\text{pr}_{12}^*\varphi _{jk}} } \]
of schemes over $X_ i \times _ S X_ j \times _ S X_ k$ commutes (with obvious notation).
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 of $X_ i$-schemes $\psi _ i : V_ i \to V'_ i$ such that all the diagrams
\[ \xymatrix{ V_ i \times _ S X_ j \ar[r]_{\varphi _{ij}} \ar[d]_{\psi _ i \times \text{id}} & X_ i \times _ S V_ j \ar[d]^{\text{id} \times \psi _ j} \\ V'_ i \times _ S X_ j \ar[r]^{\varphi '_{ij}} & X_ i \times _ S V'_ j } \]
commute.
This is the notion that comes up naturally for example when the question arises whether the fibred category of relative curves is a stack in the fpqc topology (it isn't – at least not if you stick to schemes).
The reason we will usually work with the version of a family consisting of a single morphism is the following lemma.
Lemma 35.34.5. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i \in I}$ be a family of morphisms with target $S$. Set $X = \coprod _{i \in I} X_ i$, and consider it as an $S$-scheme. There is a canonical equivalence of categories
\[ \begin{matrix} \text{category of descent data }
\\ \text{relative to the family } \{ X_ i \to S\} _{i \in I}
\end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data}
\\ \text{ relative to } X/S
\end{matrix} \]
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 $X \times _ S X = \coprod _{ij} X_ i \times _ S X_ j$ and similarly for higher fibre products. Giving a morphism $V \to X$ 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 35.34.6. Pullback of descent data for schemes over schemes.
Let
\[ \xymatrix{ X' \ar[r]_ f \ar[d]_{a'} & X \ar[d]^ a \\ S' \ar[r]^ h & S } \]
be a commutative diagram of morphisms of schemes. The construction
\[ (V \to X, \varphi ) \longmapsto f^*(V \to X, \varphi ) = (V' \to X', \varphi ') \]
where $V' = X' \times _ X V$ and where $\varphi '$ is defined as the composition
\[ \xymatrix{ V' \times _{S'} X' \ar@{=}[r] & (X' \times _ X V) \times _{S'} X' \ar@{=}[r] & (X' \times _{S'} X') \times _{X \times _ S X} (V \times _ S X) \ar[d]^{\text{id} \times \varphi } \\ X' \times _{S'} V' \ar@{=}[r] & X' \times _{S'} (X' \times _ X V) & (X' \times _{S'} X') \times _{X \times _ S X} (X \times _ S V) \ar@{=}[l] } \]
defines a functor from the category of descent data relative to $X \to S$ to the category of descent data relative to $X' \to S'$.
Given two morphisms $f_ i : X' \to X$, $i = 0, 1$ 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 35.34.2. 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 $r : X' \to X \times _ S X$ with $f_ i = \text{pr}_ i \circ r$. Then we take
\begin{eqnarray*} f_0^*V & = & X' \times _{f_0, X} V \\ & = & X' \times _{\text{pr}_0 \circ r, X} V \\ & = & X' \times _{r, X \times _ S X} (X \times _ S X) \times _{\text{pr}_0, X} V \\ & \xrightarrow {\varphi } & X' \times _{r, X \times _ S X} (X \times _ S X) \times _{\text{pr}_1, X} V \\ & = & X' \times _{\text{pr}_1 \circ r, X} V \\ & = & X' \times _{f_1, X} V \\ & = & f_1^*V \end{eqnarray*}
We omit the verification that this works.
$\square$
Definition 35.34.7. With $S, S', X, X', f, a, a', h$ as in Lemma 35.34.6 the functor
\[ (V, \varphi ) \longmapsto f^*(V, \varphi ) \]
constructed in that lemma is called the pullback functor on descent data.
Lemma 35.34.8 (Pullback of descent data for schemes over families). Let $\mathcal{U} = \{ U_ i \to S'\} _{i \in I}$ and $\mathcal{V} = \{ V_ j \to S\} _{j \in J}$ be families of morphisms with fixed target. Let $\alpha : I \to J$, $h : S' \to S$ and $g_ i : U_ i \to V_{\alpha (i)}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1.
Let $(Y_ j, \varphi _{jj'})$ be a descent datum relative to the family $\{ V_ j \to S'\} $. The system
\[ \left( g_ i^*Y_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')} \right) \]
(with notation as in Remark 35.34.4) is a descent datum relative to $\mathcal{V}$.
This construction defines a functor between descent data relative to $\mathcal{U}$ and descent data relative to $\mathcal{V}$.
Given a second $\alpha ' : I \to J$, $h' : S' \to S$ and $g'_ i : U_ i \to V_{\alpha '(i)}$ morphism of families of maps with fixed target, then if $h = h'$ the two resulting functors between descent data are canonically isomorphic.
These functors agree, via Lemma 35.34.5, with the pullback functors constructed in Lemma 35.34.6.
Proof.
This follows from Lemma 35.34.6 via the correspondence of Lemma 35.34.5.
$\square$
Definition 35.34.9. With $\mathcal{U} = \{ U_ i \to S'\} _{i \in I}$, $\mathcal{V} = \{ V_ j \to S\} _{j \in J}$, $\alpha : I \to J$, $h : S' \to S$, and $g_ i : U_ i \to V_{\alpha (i)}$ as in Lemma 35.34.8 the functor
\[ (Y_ j, \varphi _{jj'}) \longmapsto (g_ i^*Y_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')}) \]
constructed in that lemma is called the pullback functor on descent data.
If $\mathcal{U}$ and $\mathcal{V}$ have the same target $S$, and if $\mathcal{U}$ refines $\mathcal{V}$ (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 35.34.8 that the choice of the pair does not matter (up to a canonical isomorphism).
Definition 35.34.10. Let $S$ be a scheme. Let $f : X \to S$ be a morphism of schemes.
Given a scheme $U$ over $S$ we have the trivial descent datum of $U$ relative to $\text{id} : S \to S$, namely the identity morphism on $U$.
By Lemma 35.34.6 we get a canonical descent datum on $X \times _ S U$ relative to $X \to S$ by pulling back the trivial descent datum via $f$. We often denote $(X \times _ S U, can)$ this descent datum.
A descent datum $(V, \varphi )$ relative to $X/S$ is called effective if $(V, \varphi )$ is isomorphic to the canonical descent datum $(X \times _ S U, can)$ for some scheme $U$ over $S$.
Thus being effective means there exists a scheme $U$ over $S$ and an isomorphism $\psi : V \to X \times _ S U$ of $X$-schemes such that $\varphi $ is equal to the composition
\[ V \times _ S X \xrightarrow {\psi \times \text{id}_ X} X \times _ S U \times _ S X = X \times _ S X \times _ S U \xrightarrow {\text{id}_ X \times \psi ^{-1}} X \times _ S V \]
Definition 35.34.11. Let $S$ be a scheme. Let $\{ X_ i \to S\} $ be a family of morphisms with target $S$.
Given a scheme $U$ over $S$ we have a canonical descent datum on the family of schemes $X_ i \times _ S U$ by pulling back the trivial descent datum for $U$ relative to $\{ \text{id} : S \to S\} $. We denote this descent datum $(X_ i \times _ S U, can)$.
A descent datum $(V_ i, \varphi _{ij})$ relative to $\{ X_ i \to S\} $ is called effective if there exists a scheme $U$ over $S$ such that $(V_ i, \varphi _{ij})$ is isomorphic to $(X_ i \times _ S U, can)$.
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