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.

Inverse systems and inverse limits of modules

Definition

Let $R$ be a commutative ring.

An inverse system of $R$ -modules indexed by $N_{\geq 1}$ is a sequence of $R$ -modules $M_{1} \overset{φ_{2}}{\leftarrow} M_{2} \overset{φ_{3}}{\leftarrow} M_{3} \overset{φ_{4}}{\leftarrow} \dots$ together with $R$ -linear transition maps and their composites $φ_{n, m} : = φ_{m + 1} \circ \dots \circ φ_{n} ⁣ : M_{n} \to M_{m} (n \geq m),$ and $φ_{m, m} : = {id}_{M_{m}}$ .

Its inverse limit is the submodule $\underset{\leftarrow}{l i m} M_{n} : = {(x_{n})_{n \geq 1} \in \prod_{n \geq 1} M_{n} : φ_{n} (x_{n}) = x_{n - 1} for every n \geq 2} .$

The coordinate maps $π_{n} ⁣ : \underset{\leftarrow}{l i m} M_{n} \to M_{n}, (x_{m})_{m} \mapsto x_{n},$ are called the limit projections.

Used by

Mittag-Leffler inverse systems Definition
Prime-to-residue-characteristic Tate modules and inertia Definition
Separated and complete filtered modules Definition
The I-adic completion of a module Definition
The p-adic integers as an inverse limit and as a completion Example
Inverse limits preserve kernels Theorem
Universal property of an inverse limit of modules Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §22.5 (standard reference, not scraped)
The Stacks Project, Section 10.87 (standard reference, not scraped)