The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 077Z. Beware of the difference between the letter 'O' and the digit '0'.