 Solution

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

e \in R

D (e) \subseteq Spec R

. Since

e (1 - e) = 0

, a prime ideal contains at least one of

e, 1 - e

, and it cannot contain both. Therefore

D (e) = V (1 - e), Spec R = D (e) ⊔ D (1 - e) .

Thus

D (e)

Conversely, let

C

T = Spec R

. The structure sheaf glues the section

1

C

and the section

0

on its complement to a unique global section

e \in Γ (T, O_{T}) = R

. Checking on these two open subsets gives

e^{2} = e

, and

D (e) = C

. Conversely, the restriction of an idempotent

e

D (e)

1

because

e

is a unit there, and its restriction to

D (1 - e)

0

. Thus the two constructions are inverse, including for the zero ring:

{e \in R : e^{2} = e} ⟷ {C \subseteq Spec R : C clopen}, e ⟼ D (e) .

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