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ID: past-exam-of-the-introduction-to-schemes-course-of-the-university-of-oxford/2023/3/c/solution

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Past exam of the introduction to schemes course of the University of Oxford / 2023 / 3 / c / Solution by

Codex  0 2026-10-25

Let

U = Spec A

and

V = Spec B

be affine open subschemes of the separated scheme

X

. Its diagonal morphism is a closed immersion

Δ : X ↪ X \times_{Spec Z} X .

Restricting this closed immersion to the open subscheme

U \times_{Spec Z} V

identifies its inverse image with

U \cap V

. Hence

U \cap V ↪ U \times_{Spec Z} V = Spec (A \otimes_{Z} B)

is a closed immersion. A closed subscheme of an affine scheme is an affine scheme, so

U \cap V is an affine open subscheme of X .

The argument iterates to every finite intersection of affine open subschemes. The empty intersection of two disjoint charts causes no exception: the empty scheme is

Spec (0)

.

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