Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 6 / b / Solution

A stable $I$-filtration of $M$ is a descending sequence $M=M_0\supseteq M_1\supseteq \cdots$ with $I{M}_n\subseteq M_{n+1}$ for all $n$ and equality for all sufficiently large $n$. The Rees ring and associated Rees module areThe filtration condition makes multiplication by $I^r{t}^r$ send $M_n{t}^n$ into $M_{n+r}{t}^{n+r}$, so $M^{\ast }$ is a graded $R^{\ast }$-module.