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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The associated graded ring and associated graded module of an ideal-adic filtration

Definition

Let $R$ be a commutative ring, let $I \subset R$ be an ideal, and let $M$ be an $R$ -module. The associated graded ring of the $I$ -adic filtration is

{gr}_{I} (R) : = ⨁_{n \geq 0} I^{n} / I^{n + 1} .

Multiplication is induced by multiplication in $R$ :

(a + I^{m + 1}) (b + I^{n + 1}) = a b + I^{m + n + 1} .

The associated graded module is

{gr}_{I} (M) : = ⨁_{n \geq 0} I^{n} M / I^{n + 1} M,

viewed as a graded ${gr}_{I} (R)$ -module by

(a + I^{m + 1}) (x + I^{n + 1} M) = a x + I^{m + n + 1} M .

Depends on

Nonnegatively graded rings and modules, homogeneous elements, and twists

Used by

Dependency tree · one level

1 result 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, Section 10.57: Graded modules (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, §20 (standard reference, not scraped)