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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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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

Dependency tree · one level

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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)