The Stacks project

Remark 26.15.5. Suppose the functor $F$ is defined on all locally ringed spaces, and if conditions of Lemma 26.15.4 are replaced by the following:

1. $F$ satisfies the sheaf property on the category of locally ringed spaces,

2. there exists a set $I$ and a collection of subfunctors $F_ i \subset F$ such that

   1. each $F_ i$ is representable by a scheme,

   2. each $F_ i \subset F$ is representable by open immersions on the category of locally ringed spaces, and

   3. the collection $(F_ i)_{i \in I}$ covers $F$ as a functor on the category of locally ringed spaces.

We leave it to the reader to spell this out further. Then the end result is that the functor $F$ is representable in the category of locally ringed spaces and that the representing object is a scheme.