DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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.

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Exact sequences and short exact sequences of modules

Definition

A sequence of left $R$ -modules and homomorphisms $\dots ⟶ M_{i - 1} \overset{d_{i - 1}}{\to} M_{i} \overset{d_{i}}{\to} M_{i + 1} ⟶ \dots$ is exact at $M_{i}$ if $im d_{i - 1} = \ker d_{i}$ (Module homomorphism and isomorphism, kernel, image and cokernel). It is exact if it is exact at every displayed module at which two arrows meet.

A short exact sequence is an exact sequence $0 ⟶ A \overset{i}{\to} B \overset{p}{\to} C ⟶ 0.$ Thus $i$ is injective, $p$ is surjective, and $im i = \ker p$ (Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel).

Depends on

Used by

Split short exact sequences, sections, and retractions Definition
The endpoints of a short exact sequence encode injectivity and surjectivity Lemma
The injective and surjective Four Lemmas Lemma
Covariant and contravariant Hom are left exact Theorem
The Snake Lemma for modules Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)