DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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The Rees algebra of an ideal and the Rees module of a filtered module

Definition

Let $R$ be a commutative ring and $I \subset R$ an ideal. The Rees algebra of $I$ is the graded subring

R (I) : = ⨁_{n \geq 0} I^{n} t^{n} \subset R [t] .

Equivalently, it is the graded ring whose degree- $n$ piece is $I^{n}$ .

Let $M$ be an $R$ -module equipped with a descending filtration

M = M_{0} \supseteq M_{1} \supseteq M_{2} \supseteq \dots

satisfying $I M_{n} \subseteq M_{n + 1}$ for every $n \geq 0$ . The Rees module of this filtration is

R (M_{∙}) : = ⨁_{n \geq 0} M_{n} t^{n} \subset M [t],

viewed as a graded $R (I)$ -module.

For the $I$ -adic filtration $M_{n} = I^{n} M$ , the quotient

R (M_{∙}) / I R (M_{∙})

is naturally ${gr}_{I} (M)$ .

Depends on

Used by

Over a Noetherian ring, an ideal filtration is stable exactly when its Rees module is finite, and the Rees algebra is Noetherian Lemma

Dependency tree · one level

2 results within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stacks Project, Definition 10.70.1 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, Exercise (20.16) (standard reference, not scraped)