Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 101 / 5 / b

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- Solution b

Solution

Consider

q_{3} = (x^{3}, y^{2} - z w) .

It is

(x, y^{2} - z w)

-primary. A direct ideal-intersection calculation gives

q_{1} \cap q_{2} \cap q_{3} = x^{2} [(y, z)^{2} \cap (x, g)] = x^{2} [x (y, z)^{2} + g (y, z)] = (x^{3} y^{2}, x^{3} yz, x^{3} z^{2}, x^{2} y (y^{2} - z w), x^{2} z (y^{2} - z w)) = I .

This decomposition is irredundant and its radicals are

(x), (y, z), (x, y^{2} - z w) .

Therefore

Ass (k [x, y, z, w] / I) = {(x), (y, z), (x, y^{2} - z w)} .

In particular,

p_{3} = (x, y^{2} - z w)

is the embedded associated prime and the decomposition also confirms

I = (x) \cap (y, z)

.

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