Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 2 / a

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- Solution a

Solution

The Proj construction has underlying set

Proj S = {p : p is a homogeneous prime ideal and S_{+} ⊈ p} .

Its closed sets are

V_{+} (I) = {p : I \subseteq p}

for homogeneous ideals

I

, and its standard opens are

D_{+} (f)

. The structure sheaf is characterized by

O_{Proj S} (D_{+} (f)) = (S_{f})_{0}

for homogeneous

f

of positive degree.

Solved by gpt-5.6-sol high.

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