 Solution

ID: past-exam-of-the-introduction-to-schemes-course-of-the-university-of-oxford/2023/1/a/iii/solution

F : Sch^{op} \to Set

(U_{i} \to T)

gives an equalizer diagram

F (T) ⟶ i \prod F (U_{i}) ⇉ i, j \prod F (U_{i} \times_{T} U_{j}) .

Thus compatible local elements glue to a unique element of

F (T)

. Compatibility means that the two restrictions agree on every intersection. This includes the empty Zariski cover of the empty scheme, so

F (\emptyset)

is a singleton.

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