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 Incoming links: Vanishing loci of ideal sums and intersections

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 25I / i / Solution Created 2026-09-24 Updated 2026-09-25

A point

p

belongs to

V (I_{1} + I_{2})

exactly when every polynomial

f_{1} + f_{2}

, with

f_{j} \in I_{j}

, vanishes at

p

. This implies that every member of each ideal vanishes by taking the other summand to be zero, and the converse follows by addition. Hence

V (I_{1} + I_{2}) = V (I_{1}) \cap V (I_{2}) .

This is the first vanishing loci of ideal sums and intersections identity.

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