The Stacks project

Lemma 34.9.13. Let $F$ be a contravariant functor on the category of schemes with values in sets. Then $F$ satisfies the sheaf property for the fpqc topology if and only if it satisfies

  1. the sheaf property for every Zariski covering, and

  2. the sheaf property for any standard fpqc covering.

Moreover, in the presence of (1) property (2) is equivalent to property

  1. the sheaf property for $\{ V \to U\} $ with $V$, $U$ affine and $V \to U$ faithfully flat.

Proof. Assume (1) and (2) hold. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be an fpqc covering. Let $s_ i \in F(T_ i)$ be a family of elements such that $s_ i$ and $s_ j$ map to the same element of $F(T_ i \times _ T T_ j)$. Let $W \subset T$ be the maximal open subset such that there exists a unique $s \in F(W)$ with $s|_{f_ i^{-1}(W)} = s_ i|_{f_ i^{-1}(W)}$ for all $i$. Such a maximal open exists because $F$ satisfies the sheaf property for Zariski coverings; in fact $W$ is the union of all opens with this property. Let $t \in T$. We will show $t \in W$. To do this we pick an affine open $t \in U \subset T$ and we will show there is a unique $s \in F(U)$ with $s|_{f_ i^{-1}(U)} = s_ i|_{f_ i^{-1}(U)}$ for all $i$.

By Lemma 34.9.8 we can find a standard fpqc covering $\{ U_ j \to U\} _{j = 1, \ldots , n}$ refining $\{ U \times _ T T_ i \to U\} $, say by morphisms $h_ j : U_ j \to T_{i_ j}$. By (2) we obtain a unique element $s \in F(U)$ such that $s|_{U_ j} = F(h_ j)(s_{i_ j})$. Note that for any scheme $V \to U$ over $U$ there is a unique section $s_ V \in F(V)$ which restricts to $F(h_ j \circ \text{pr}_2)(s_{i_ j})$ on $V \times _ U U_ j$ for $j = 1, \ldots , n$. Namely, this is true if $V$ is affine by (2) as $\{ V \times _ U U_ j \to V\} $ is a standard fpqc covering and in general this follows from (1) and the affine case by choosing an affine open covering of $V$. In particular, $s_ V = s|_ V$. Now, taking $V = U \times _ T T_ i$ and using that $s_{i_ j}|_{T_{i_ j} \times _ T T_ i} = s_ i|_{T_{i_ j} \times _ T T_ i}$ we conclude that $s|_{U \times _ T T_ i} = s_ V = s_ i|_{U \times _ T T_ i}$ which is what we had to show.

Proof of the equivalence of (2) and (2') in the presence of (1). Suppose $\{ T_ i \to T\} $ is a standard fpqc covering, then $\coprod T_ i \to T$ is a faithfully flat morphism of affine schemes. In the presence of (1) we have $F(\coprod T_ i) = \prod F(T_ i)$ and similarly $F((\coprod T_ i) \times _ T (\coprod T_ i)) = \prod F(T_ i \times _ T T_{i'})$. Thus the sheaf condition for $\{ T_ i \to T\} $ and $\{ \coprod T_ i \to T\} $ is the same. $\square$

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