The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

8.3 Descent data in fibred categories

In this section we define the notion of a descent datum in the abstract setting of a fibred category. Before we do so we point out that this is completely analogous to descent data for quasi-coherent sheaves (Descent, Section 34.2) and descent data for schemes over schemes (Descent, Section 34.31).

We will use the convention where the projection maps $\text{pr}_ i : X \times \ldots \times X \to X$ are labeled starting with $i = 0$. Hence we have $\text{pr}_0, \text{pr}_1 : X \times X \to X$, $\text{pr}_0, \text{pr}_1, \text{pr}_2 : X \times X \times X \to X$, etc.

Definition 8.3.1. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice of pullbacks as in Categories, Definition 4.32.6. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ be a family of morphisms of $\mathcal{C}$. Assume all the fibre products $U_ i \times _ U U_ j$, and $U_ i \times _ U U_ j \times _ U U_ k$ exist.

  1. A descent datum $(X_ i, \varphi _{ij})$ in $\mathcal{S}$ relative to the family $\{ f_ i : U_ i \to U\} $ is given by an object $X_ i$ of $\mathcal{S}_{U_ i}$ for each $i \in I$, an isomorphism $\varphi _{ij} : \text{pr}_0^*X_ i \to \text{pr}_1^*X_ j$ in $\mathcal{S}_{U_ i \times _ U U_ j}$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram

    \[ \xymatrix{ \text{pr}_0^*X_ i \ar[rd]_{\text{pr}_{01}^*\varphi _{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi _{ik}} & & \text{pr}_2^*X_ k \\ & \text{pr}_1^*X_ j \ar[ru]_{\text{pr}_{12}^*\varphi _{jk}} & } \]

    in the category $\mathcal{S}_{U_ i \times _ U U_ j \times _ U U_ k}$ commutes. This is called the cocycle condition.

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

    \[ \xymatrix{ \text{pr}_0^*X_ i \ar[r]_{\varphi _{ij}} \ar[d]_{\text{pr}_0^*\psi _ i} & \text{pr}_1^*X_ j \ar[d]^{\text{pr}_1^*\psi _ j} \\ \text{pr}_0^*X'_ i \ar[r]^{\varphi '_{ij}} & \text{pr}_1^*X'_ j \\ } \]

    in the categories $\mathcal{S}_{U_ i \times _ U U_ j}$ commute.

  3. The category of descent data relative to $\mathcal{U}$ is denoted $DD(\mathcal{U})$.

The fibre products $U_ i \times _ U U_ j$ and $U_ i \times _ U U_ j \times _ U U_ k$ will exist if each of the morphisms $f_ i : U_ i \to U$ is representable, see Categories, Definition 4.6.4. Recall that in a site one of the conditions for a covering $\{ U_ i \to U\} $ is that each of the morphisms is representable, see Sites, Definition 7.6.2 part (3). In fact the main interest in the definition above is where $\mathcal{C}$ is a site and $\{ U_ i \to U\} $ is a covering of $\mathcal{C}$. However, a descent datum is just an abstract gadget that can be defined as above. This is useful: for example, given a fibred category over $\mathcal{C}$ one can look at the collection of families with respect to which descent data are effective, and try to use these as the family of coverings for a site.

Remarks 8.3.2. Two remarks on Definition 8.3.1 are in order. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Let $\{ f_ i : U_ i \to U\} _{i \in I}$, and $(X_ i, \varphi _{ij})$ be as in Definition 8.3.1.

  1. There is a diagonal morphism $\Delta : U_ i \to U_ i \times _ U U_ i$. We can pull back $\varphi _{ii}$ via this morphism to get an automorphism $\Delta ^\ast \varphi _{ii} \in \text{Aut}_{U_ i}(x_ i)$. On pulling back the cocycle condition for the triple $(i, i, i)$ by $\Delta _{123} : U_ i \to U_ i \times _ U U_ i \times _ U U_ i$ we deduce that $\Delta ^\ast \varphi _{ii} \circ \Delta ^\ast \varphi _{ii} = \Delta ^\ast \varphi _{ii}$; thus $\Delta ^\ast \varphi _{ii} = \text{id}_{x_ i}$.

  2. There is a morphism $\Delta _{13}: U_ i \times _ U U_ j \to U_ i \times _ U U_ j \times _ U U_ i$ and we can pull back the cocycle condition for the triple $(i, j, i)$ to get the identity $(\sigma ^\ast \varphi _{ji}) \circ \varphi _{ij} = \text{id}_{\text{pr}_0^\ast x_ i}$, where $\sigma : U_ i \times _ U U_ j \to U_ j \times _ U U_ i$ is the switching morphism.

Lemma 8.3.3. (Pullback of descent data.) Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice pullbacks as in Categories, Definition 4.32.6. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$, and $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ be a families of morphisms of $\mathcal{C}$ with fixed target. Assume all the fibre products $U_ i \times _ U U_{i'}$, $U_ i \times _ U U_{i'} \times _ U U_{i''}$, $V_ j \times _ V V_{j'}$, and $V_ j \times _ V V_{j'} \times _ V V_{j''}$ exist. Let $\alpha : I \to J$, $h : U \to V$ 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.

  1. Let $(Y_ j, \varphi _{jj'})$ be a descent datum relative to the family $\{ V_ j \to V\} $. The system

    \[ \left( g_ i^*Y_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')} \right) \]

    is a descent datum relative to $\mathcal{U}$.

  2. This construction defines a functor between descent data relative to $\mathcal{V}$ and descent data relative to $\mathcal{U}$.

  3. Given a second $\alpha ' : I \to J$, $h' : U \to V$ 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.

Proof. Omitted. $\square$

Definition 8.3.4. With $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$, $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$, $\alpha : I \to J$, $h : U \to V$, and $g_ i : U_ i \to V_{\alpha (i)}$ as in Lemma 8.3.3 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.

Given $h : U \to V$, if there exists a morphism $\tilde h : \mathcal{U} \to \mathcal{V}$ covering $h$ then $\tilde h^*$ is independent of the choice of $\tilde h$ as we saw in Lemma 8.3.3. Hence we will sometimes simply write $h^*$ to indicate the pullback functor.

Definition 8.3.5. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice of pullbacks as in Categories, Definition 4.32.6. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ be a family of morphisms with target $U$. Assume all the fibre products $U_ i \times _ U U_ j$ and $U_ i \times _ U U_ j \times _ U U_ k$ exist.

  1. Given an object $X$ of $\mathcal{S}_ U$ the trivial descent datum is the descent datum $(X, \text{id}_ X)$ with respect to the family $\{ \text{id}_ U : U \to U\} $.

  2. Given an object $X$ of $\mathcal{S}_ U$ we have a canonical descent datum on the family of objects $f_ i^*X$ by pulling back the trivial descent datum $(X, \text{id}_ X)$ via the obvious map $\{ f_ i : U_ i \to U\} \to \{ \text{id}_ U : U \to U\} $. We denote this descent datum $(f_ i^*X, can)$.

  3. A descent datum $(X_ i, \varphi _{ij})$ relative to $\{ f_ i : U_ i \to U\} $ is called effective if there exists an object $X$ of $\mathcal{S}_ U$ such that $(X_ i, \varphi _{ij})$ is isomorphic to $(f_ i^*X, can)$.

Note that the rule that associates to $X \in \mathcal{S}_ U$ its canonical descent datum relative to $\mathcal{U}$ defines a functor

\[ \mathcal{S}_ U \longrightarrow DD(\mathcal{U}). \]

A descent datum is effective if and only if it is in the essential image of this functor. Let us make explicit the canonical descent datum as follows.

Lemma 8.3.6. In the situation of Definition 8.3.5 part (2) the maps $can_{ij} : \text{pr}_0^*f_ i^*X \to \text{pr}_1^*f_ j^*X$ are equal to $(\alpha _{\text{pr}_1, f_ j})_ X \circ (\alpha _{\text{pr}_0, f_ i})_ X^{-1}$ where $\alpha _{\cdot , \cdot }$ is as in Categories, Lemma 4.32.7 and where we use the equality $f_ i \circ \text{pr}_0 = f_ j \circ \text{pr}_1$ as maps $U_ i \times _ U U_ j \to U$.

Proof. Omitted. $\square$


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