 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-101/4/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

(M_{n})

and

(M_{n}^{'})

are equivalent when each contains a fixed shift of the other: there are

a, b \geq 0

such that

M_{n + a} \subseteq M_{n}^{'}, M_{n + b}^{'} \subseteq M_{n}

for every

n \geq 0

For an

I

-filtration,

I M_{n} \subseteq M_{n + 1}

, so

I^{n} M_{0} \subseteq M_{n}

. If it is stable from

s

onward, then

M_{s + n} = I^{n} M_{s} \subseteq I^{n} M_{0} .

It is therefore equivalent to the I-adic filtration. Any two stable

I

-filtrations are consequently equivalent to each other.

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