The Stacks project

Definition 26.15.3. Let $F$ be a contravariant functor on the category of schemes with values in sets.

1. We say that $F$ satisfies the sheaf property for the Zariski topology if for every scheme $T$ and every open covering $T = \bigcup _{i \in I} U_ i$, and for any collection of elements $\xi _ i \in F(U_ i)$ such that $\xi _ i|_{U_ i \cap U_ j} = \xi _ j|_{U_ i \cap U_ j}$ there exists a unique element $\xi \in F(T)$ such that $\xi _ i = \xi |_{U_ i}$ in $F(U_ i)$.

2. A subfunctor $H \subset F$ is a rule that associates to every scheme $T$ a subset $H(T) \subset F(T)$ such that the maps $F(f) : F(T) \to F(T')$ maps $H(T)$ into $H(T')$ for all morphisms of schemes $f : T' \to T$.

3. Let $H \subset F$ be a subfunctor. We say that $H \subset F$ is representable by open immersions if for all pairs $(T, \xi )$, where $T$ is a scheme and $\xi \in F(T)$ there exists an open subscheme $U_\xi \subset T$ with the following property:

   • A morphism $f : T' \to T$ factors through $U_\xi$ if and only if $f^*\xi \in H(T')$.

4. Let $I$ be a set. For each $i \in I$ let $H_ i \subset F$ be a subfunctor. We say that the collection $(H_ i)_{i \in I}$ covers $F$ if and only if for every $\xi \in F(T)$ there exists an open covering $T = \bigcup U_ i$ such that $\xi |_{U_ i} \in H_ i(U_ i)$.