The Stacks project

Lemma 76.11.5. In Situation 76.11.4.

  1. The functor $F_ n$ satisfies the sheaf property for the fpqc topology.

  2. If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_ n$ is limit preserving.

Proof. Proof of (1). Suppose that $\{ T_ i \to T\} $ is an fpqc covering of a scheme $T$ over $Y$. We have to show that if $F_ n(T_ i)$ is nonempty for all $i$, then $F_ n(T)$ is nonempty. Choose a diagram as in part (1) of Lemma 76.11.2. Denote $F'_ n$ the corresponding functor for $\varphi ^*\mathcal{F}$ and the morphism $U \to V$. By More on Flatness, Lemma 38.20.12 we have the sheaf property for $F'_ n$. Thus we get the sheaf property for $F_ n$ because for $T \to Y$ we have $F_ n(T) = F'_ n(V \times _ Y T)$ by Lemma 76.11.2 and because $\{ V \times _ Y T_ i \to V \times _ Y T\} $ is an fpqc covering.

Proof of (2). Suppose that $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ is a filtered limit of affine schemes $T_ i$ over $Y$ and assume that $F_ n(T)$ is nonempty. We have to show that $F_ n(T_ i)$ is nonempty for some $i$. Choose a diagram as in part (1) of Lemma 76.11.2. Fix $i \in I$ and choose an affine open $W_ i \subset V \times _ Y T_ i$ mapping surjectively onto $T_ i$. For $i' \geq i$ let $W_{i'}$ be the inverse image of $W_ i$ in $V \times _ Y T_{i'}$ and let $W \subset V \times _ Y T$ be the inverse image of $W_ i$. Then $W = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} W_ i$ is a filtered limit of affine schemes over $V$. By Lemma 76.11.2 again it suffices to show that $F'_ n(W_{i'})$ is nonempty for some $i' \geq i$. But we know that $F'_ n(W)$ is nonempty because of our assumption that $F_ n(T) = F'_ n(V \times _ Y T)$ is nonempty. Thus we can apply More on Flatness, Lemma 38.20.12 to conclude. $\square$


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