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

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Mittag-Leffler inverse systems

Definition

Let $M_{1} \overset{φ_{2}}{\leftarrow} M_{2} \overset{φ_{3}}{\leftarrow} M_{3} \overset{}{\leftarrow} \dots$ be an inverse system of $R$ -modules.

The system is Mittag-Leffler if for every $m \geq 1$ the descending family of images $im (φ_{n, m}) \subseteq M_{m} (n \geq m)$ eventually stabilizes. Explicitly, for each $m$ there exists $c (m) \geq m$ such that $im (φ_{n, m}) = im (φ_{c (m), m}) for every n \geq c (m) .$

In particular, any inverse system with surjective transition maps is Mittag-Leffler.

Depends on

Inverse systems and inverse limits of modules

Used by

Countable Mittag-Leffler systems preserve short exactness on inverse limits Theorem

Dependency tree · one level

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Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Exercise 22.6(3) (standard reference, not scraped)
The Stacks Project, Definition 10.86.1 (standard reference, not scraped)