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

{c}
{title2=$\mathcal R(I)$}
{wiki=Rees_algebra}

For an ideal $I\subseteq R$, the Rees ring is the graded subring
$$
\mathcal R(I)=\bigoplus_{n\geq0}I^nt^n\subseteq R[t].
$$
It packages all powers of $I$ into one algebra and turns stable filtrations into finitely generated graded modules.


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 Incoming links: Rees ring