The Stacks project

Lemma 38.20.8. In Situation 38.20.7.

  1. The functor $H_ p$ satisfies the sheaf property for the fpqc topology.

  2. If $\mathcal{F}$ is of finite presentation, then functor $H_ p$ is limit preserving.

Proof. Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc1 covering of schemes over $S$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and denote $\mathcal{F}_ i$ the pullback of $\mathcal{F}$ to $X_ i$. Assume that $\mathcal{F}_ i$ satisfies ( for all $i$. Pick $t \in T$ and let $\xi _ t \in X_ T$ denote the generic point of $X_ t$. We have to show that $\mathcal{F}$ is free in a neighbourhood of $\xi _ t$. For some $i \in I$ we can find a $t_ i \in T_ i$ mapping to $t$. Let $\xi _ i \in X_ i$ denote the generic point of $X_{t_ i}$, so that $\xi _ i$ maps to $\xi _ t$. The fact that $\mathcal{F}_ i$ is free of rank $p$ in a neighbourhood of $\xi _ i$ implies that $(\mathcal{F}_ i)_{x_ i} \cong \mathcal{O}_{X_ i, x_ i}^{\oplus p}$ which implies that $\mathcal{F}_{T, \xi _ t} \cong \mathcal{O}_{X_ T, \xi _ t}^{\oplus p}$ as $\mathcal{O}_{X_ T, \xi _ t} \to \mathcal{O}_{X_ i, x_ i}$ is flat, see for example Algebra, Lemma 10.78.6. Thus there exists an affine neighbourhood $U$ of $\xi _ t$ in $X_ T$ and a surjection $\mathcal{O}_ U^{\oplus p} \to \mathcal{F}_ U = \mathcal{F}_ T|_ U$, see Modules, Lemma 17.9.4. After shrinking $T$ we may assume that $U \to T$ is surjective. Hence $U \to T$ is a smooth morphism of affines with geometrically irreducible fibres. Moreover, for every $t' \in T$ we see that the induced map

\[ \alpha : \mathcal{O}_{U, \xi _{t'}}^{\oplus p} \longrightarrow \mathcal{F}_{U, \xi _{t'}} \]

is an isomorphism (since by the same argument as before the module on the right is free of rank $p$). It follows from Lemma 38.10.1 that

\[ \Gamma (U, \mathcal{O}_ U^{\oplus p}) \otimes _{\Gamma (T, \mathcal{O}_ T)} \mathcal{O}_{T, t'} \longrightarrow \Gamma (U, \mathcal{F}_ U) \otimes _{\Gamma (T, \mathcal{O}_ T)} \mathcal{O}_{T, t'} \]

is injective for every $t' \in T$. Hence we see the surjection $\alpha $ is an isomorphism. This finishes the proof of (1).

Assume that $\mathcal{F}$ is of finite presentation. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $S$-schemes and assume that $\mathcal{F}_ T$ satisfies ( Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and denote $\mathcal{F}_ i$ the pullback of $\mathcal{F}$ to $X_ i$. Let $U \subset X_ T$ denote the open subscheme of points where $\mathcal{F}_ T$ is flat over $T$, see More on Morphisms, Theorem 37.15.1. By assumption every generic point of every fibre is a point of $U$, i.e., $U \to T$ is a smooth surjective morphism with geometrically irreducible fibres. We may shrink $U$ a bit and assume that $U$ is quasi-compact. Using Limits, Lemma 32.4.11 we can find an $i \in I$ and a quasi-compact open $U_ i \subset X_ i$ whose inverse image in $X_ T$ is $U$. After increasing $i$ we may assume that $\mathcal{F}_ i|_{U_ i}$ is flat over $T_ i$, see Limits, Lemma 32.10.4. In particular, $\mathcal{F}_ i|_{U_ i}$ is finite locally free hence defines a locally constant rank function $\rho : U_ i \to \{ 0, 1, 2, \ldots \} $. Let $(U_ i)_ p \subset U_ i$ denote the open and closed subset where $\rho $ has value $p$. Let $V_ i \subset T_ i$ be the image of $(U_ i)_ p$; note that $V_ i$ is open and quasi-compact. By assumption the image of $T \to T_ i$ is contained in $V_ i$. Hence there exists an $i' \geq i$ such that $T_{i'} \to T_ i$ factors through $V_ i$ by Limits, Lemma 32.4.11. Then $\mathcal{F}_{i'}$ satisfies ( as desired. Some details omitted. $\square$

[1] It is quite easy to show that $H_ p$ is a sheaf for the fppf topology using that flat morphisms of finite presentation are open. This is all we really need later on. But it is kind of fun to prove directly that it also satisfies the sheaf condition for the fpqc topology.

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