The Stacks project

Lemma 34.10.12. Let $F$ be a contravariant functor on the category of schemes with values in sets. Then $F$ satisfies the sheaf property for the V topology if and only if it satisfies

  1. the sheaf property for every Zariski covering, and

  2. the sheaf property for any standard V covering.

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

  1. the sheaf property for a standard V covering of the form $\{ V \to U\} $, i.e., consisting of a single arrow.

Proof. Assume (1) and (2) hold. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a V 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 $i$.

We can find a standard V 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 V covering (Lemma 34.10.4) 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 $\{ T_ i \to T\} _{i = 1, \ldots , n}$ is a standard V covering, then $\coprod _{i = 1, \ldots , n} T_ i \to T$ is a morphism of affine schemes which is clearly also a standard V covering. 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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