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.

Separated and complete filtered modules

Definition

Let $R$ be a commutative ring and let $M = F^{0} M \supseteq F^{1} M \supseteq F^{2} M \supseteq \dots$ be a decreasing filtration on an $R$ -module $M$ .

The module is separated for this filtration if $⋂_{n \geq 0} F^{n} M = 0.$

The module is complete for this filtration if it is separated and the canonical map $κ_{M} ⁣ : M ⟶ {\underset{\leftarrow}{l i m}}_{n \geq 1} M / F^{n} M, m ⟼ (m m o d F^{n} M)_{n \geq 1},$ is an isomorphism.

For the $I$ -adic filtration $F^{n} M : = I^{n} M$ , this says that $M$ is $I$ -adically separated when $⋂_{n \geq 0} I^{n} M = 0$ , and $I$ -adically complete when it is separated and $M \tilde{\to} {\underset{\leftarrow}{l i m}}_{n \geq 1} M / I^{n} M .$

Separated and complete filtered modules

Definition

Depends on

Dependency tree · 2 levels

Sources