The Stacks project

77.14 Crystals in quasi-coherent sheaves

Let $(I, \Phi , j)$ be a pair consisting of a set $I$ and a pre-relation $j : \Phi \to I \times I$. Assume given for every $i \in I$ a scheme $X_ i$ and for every $\phi \in \Phi $ a morphism of schemes $f_\phi : X_{i'} \to X_ i$ where $j(\phi ) = (i, i')$. Set $X = (\{ X_ i\} _{i \in I}, \{ f_\phi \} _{\phi \in \Phi })$. Define a crystal in quasi-coherent modules on $X$ as a rule which associates to every $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ a quasi-coherent sheaf $\mathcal{F}_ i$ on $X_ i$ and for every $\phi \in \Phi $ with $j(\phi ) = (i, i')$ an isomorphism

\[ \alpha _\phi : f_\phi ^*\mathcal{F}_ i \longrightarrow \mathcal{F}_{i'} \]

of quasi-coherent sheaves on $X_{i'}$. These crystals in quasi-coherent modules form an additive category $\textit{CQC}(X)$1. This category has colimits (proof is the same as the proof of Lemma 77.12.5). If all the morphisms $f_\phi $ are flat, then $\textit{CQC}(X)$ is abelian (proof is the same as the proof of Lemma 77.12.6). Let $\kappa $ be a cardinal. We say that a crystal in quasi-coherent modules $\mathcal{F}$ on $X$ is $\kappa $-generated if each $\mathcal{F}_ i$ is $\kappa $-generated (see Properties, Definition 28.23.1).

Lemma 77.14.1. In the situation above, if all the morphisms $f_\phi $ are flat, then there exists a cardinal $\kappa $ such that every object $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$ of $\textit{CQC}(X)$ is the directed colimit of its $\kappa $-generated submodules.

Proof. In the lemma and in this proof a submodule of $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$ means the data of a quasi-coherent submodule $\mathcal{G}_ i \subset \mathcal{F}_ i$ for all $i$ such that $\alpha _\phi (f_\phi ^*\mathcal{G}_ i) = \mathcal{G}_{i'}$ as subsheaves of $\mathcal{F}_{i'}$ for all $\phi \in \Phi $. This makes sense because since $f_\phi $ is flat the pullback $f^*_\phi $ is exact, i.e., preserves subsheaves. The proof will be a variant to the proof of Properties, Lemma 28.23.3. We urge the reader to read that proof first.

We claim that it suffices to prove the lemma in case all the schemes $X_ i$ are affine. To see this let

\[ J = \coprod \nolimits _{i \in I} \{ U \subset X_ i\text{ affine open}\} \]

and let

\begin{align*} \Psi = & \coprod \nolimits _{\phi \in \Phi } \{ (U, V) \mid U \subset X_ i, V \subset X_{i'}\text{ affine open with } f_\phi (U) \subset V \} \\ & \amalg \coprod \nolimits _{i \in I} \{ (U, U') \mid U, U' \subset X_ i\text{ affine open with } U \subset U' \} \end{align*}

endowed with the obvious map $\Psi \to J \times J$. Then our $(\mathcal{F}, \alpha )$ induces a crystal in quasi-coherent sheaves $(\{ \mathcal{H}_ j\} _{j \in J}, \{ \beta _\psi \} _{\psi \in \Psi })$ on $Y = (J, \Psi )$ by setting $\mathcal{H}_{(i, U)} = \mathcal{F}_ i|_ U$ for $(i, U) \in J$ and setting $\beta _\psi $ for $\psi \in \Psi $ equal to the restriction of $\alpha _\phi $ to $U$ if $\psi = (\phi , U, V)$ and equal to $\text{id} : (\mathcal{F}_ i|_{U'})|_ U \to \mathcal{F}_ i|_ U$ when $\psi = (i, U, U')$. Moreover, submodules of $(\{ \mathcal{H}_ j\} _{j \in J}, \{ \beta _\psi \} _{\psi \in \Psi })$ correspond $1$-to-$1$ with submodules of $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$. We omit the proof (hint: use Sheaves, Section 6.30). Moreover, it is clear that if $\kappa $ works for $Y$, then the same $\kappa $ works for $X$ (by the definition of $\kappa $-generated modules). Hence it suffices to proof the lemma for crystals in quasi-coherent sheaves on $Y$.

Assume that all the schemes $X_ i$ are affine. Let $\kappa $ be an infinite cardinal larger than the cardinality of $I$ or $\Phi $. Let $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$ be an object of $\textit{CQC}(X)$. For each $i$ write $X_ i = \mathop{\mathrm{Spec}}(A_ i)$ and $M_ i = \Gamma (X_ i, \mathcal{F}_ i)$. For every $\phi \in \Phi $ with $j(\phi ) = (i, i')$ the map $\alpha _\phi $ translates into an $A_{i'}$-module isomorphism

\[ \alpha _\phi : M_ i \otimes _{A_ i} A_{i'} \longrightarrow M_{i'} \]

Using the axiom of choice choose a rule

\[ (\phi , m) \longmapsto S(\phi , m') \]

where the source is the collection of pairs $(\phi , m')$ such that $\phi \in \Phi $ with $j(\phi ) = (i, i')$ and $m' \in M_{i'}$ and where the output is a finite subset $S(\phi , m') \subset M_ i$ so that

\[ m' = \alpha _\phi \left(\sum \nolimits _{m \in S(\phi , m')} m \otimes a'_ m\right) \]

for some $a'_ m \in A_{i'}$.

Having made these choices we claim that any section of any $\mathcal{F}_ i$ over any $X_ i$ is in a $\kappa $-generated submodule. To see this suppose that we are given a collection $\mathcal{S} = \{ S_ i\} _{i \in I}$ of subsets $S_ i \subset M_ i$ each with cardinality at most $\kappa $. Then we define a new collection $\mathcal{S}' = \{ S'_ i\} _{i \in I}$ with

\[ S'_ i = S_ i \cup \bigcup \nolimits _{(\phi , m'),\ j(\phi ) = (i, i'),\ m' \in S_{i'}} S(\phi , m') \]

Note that each $S'_ i$ still has cardinality at most $\kappa $. Set $\mathcal{S}^{(0)} = \mathcal{S}$, $\mathcal{S}^{(1)} = \mathcal{S}'$ and by induction $\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'$. Then set $S_ i^{(\infty )} = \bigcup _{n \geq 0} S_ i^{(n)}$ and $\mathcal{S}^{(\infty )} = \{ S_ i^{(\infty )}\} _{i \in I}$. By construction, for every $\phi \in \Phi $ with $j(\phi ) = (i, i')$ and every $m' \in S^{(\infty )}_{i'}$ we can write $m'$ as a finite linear combination of images $\alpha _\phi (m \otimes 1)$ with $m \in S_ i^{(\infty )}$. Thus we see that setting $N_ i$ equal to the $A_ i$-submodule of $M_ i$ generated by $S_ i^{(\infty )}$ the corresponding quasi-coherent submodules $\widetilde{N_ i} \subset \mathcal{F}_ i$ form a $\kappa $-generated submodule. This finishes the proof. $\square$

Lemma 77.14.2. Let $B \to S$ as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $s$, $t$ are flat, then there exists a set $T$ and a family of objects $(\mathcal{F}_ t, \alpha _ t)_{t \in T}$ of $\mathit{QCoh}(U, R, s, t, c)$ such that every object $(\mathcal{F}, \alpha )$ is the directed colimit of its submodules isomorphic to one of the objects $(\mathcal{F}_ t, \alpha _ t)$.

Proof. This lemma is a generalization of Groupoids, Lemma 39.15.7 which deals with the case of a groupoid in schemes. We can't quite use the same argument, so we use the material on “crystals of quasi-coherent sheaves” we developed above.

Choose a scheme $W$ and a surjective étale morphism $W \to U$. Choose a scheme $V$ and a surjective étale morphism $V \to W \times _{U, s} R$. Choose a scheme $V'$ and a surjective étale morphism $V' \to R \times _{t, U} W$. Consider the collection of schemes

\[ I = \{ W, W \times _ U W, V, V', V \times _ R V'\} \]

and the set of morphisms of schemes

\[ \Phi = \{ \text{pr}_ i : W \times _ U W \to W, V \to W, V' \to W, V \times _ R V' \to V, V \times _ R V' \to V'\} \]

Set $X = (I, \Phi )$. Recall that we have defined a category $\textit{CQC}(X)$ of crystals of quasi-coherent sheaves on $X$. There is a functor

\[ \mathit{QCoh}(U, R, s, t, c) \longrightarrow \textit{CQC}(X) \]

which assigns to $(\mathcal{F}, \alpha )$ the sheaf $\mathcal{F}|_ W$ on $W$, the sheaf $\mathcal{F}|_{W \times _ U W}$ on $W \times _ U W$, the pullback of $\mathcal{F}$ via $V \to W \times _{U, s} R \to W \to U$ on $V$, the pullback of $\mathcal{F}$ via $V' \to R \times _{t, U} W \to W \to U$ on $V'$, and finally the pullback of $\mathcal{F}$ via $V \times _ R V' \to V \to W \times _{U, s} R \to W \to U$ on $V \times _ R V'$. As comparison maps $\{ \alpha _\phi \} _{\phi \in \Phi }$ we use the obvious ones (coming from associativity of pullbacks) except for the map $\phi = \text{pr}_{V'} : V \times _ R V' \to V'$ we use the pullback of $\alpha : t^*\mathcal{F} \to s^*\mathcal{F}$ to $V \times _ R V'$. This makes sense because of the following commutative diagram

\[ \xymatrix{ & V \times _ R V' \ar[ld] \ar[rd] \\ V \ar[rd] \ar[dd] & & V' \ar[ld] \ar[dd] \\ & R \ar@<-1ex>[dd]_ s \ar@<1ex>[dd]^ t \\ W \ar[rd] & & W \ar[ld] \\ & U } \]

The functor displayed above isn't an equivalence of categories. However, since $W \to U$ is surjective étale it is faithful2. Since all the morphisms in the diagram above are flat we see that it is an exact functor of abelian categories. Moreover, we claim that given $(\mathcal{F}, \alpha )$ with image $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$ there is a $1$-to-$1$ correspondence between quasi-coherent submodules of $(\mathcal{F}, \alpha )$ and $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$. Namely, given a submodule of $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$ compatibility of the submodule over $W$ with the projection maps $W \times _ U W \to W$ will guarantee the submodule comes from a quasi-coherent submodule of $\mathcal{F}$ (by Properties of Spaces, Proposition 65.32.1) and compatibility with $\alpha _{\text{pr}_{V'}}$ will insure this subsheaf is compatible with $\alpha $ (details omitted).

Choose a cardinal $\kappa $ as in Lemma 77.14.1 for the system $X = (I, \Phi )$. It is clear from Properties, Lemma 28.23.2 that there is a set of isomorphism classes of $\kappa $-generated crystals in quasi-coherent sheaves on $X$. Hence the result is clear. $\square$

[1] We could single out a set of triples $\phi , \phi ', \phi '' \in \Phi $ with $j(\phi ) = (i, i')$, $j(\phi ') = (i', i'')$, and $j(\phi '') = (i, i'')$ such that $f_{\phi ''} = f_\phi \circ f_{\phi '}$ and require that $\alpha _{\phi '} \circ f_{\phi '}^*\alpha _\phi = \alpha _{\phi ''}$ for these triples. This would define an additive subcategory. For example the data $(I, \Phi )$ could be the set of objects and arrows of an index category and $X$ could be a diagram of schemes over this index category. The result of Lemma 77.14.1 immediately gives the corresponding result in the subcategory.
[2] In fact the functor is fully faithful, but we won't need this.

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