A stable $I$-filtration of $M$ is a descending sequence $M=M_0\supseteq M_1\supseteq\cdots$ with $IM_n\subseteq M_{n+1}$ for all $n$ and equality for all sufficiently large $n$. The <Rees ring> and associated Rees module are
$$
R^*=\bigoplus_{n\geq0}I^nt^n,
\qquad
M^*=\bigoplus_{n\geq0}M_nt^n.
$$
The filtration condition makes multiplication by $I^rt^r$ send $M_nt^n$ into $M_{n+r}t^{n+r}$, so $M^*$ is a graded $R^*$-module.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-101/6/b/solution