## 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 84.27 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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